TPTP Problem File: SLH0710^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Finite_Fields/0003_Finite_Fields_Preliminary_Results/prob_00389_013874__17981608_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1537 ( 113 unt; 252 typ; 0 def)
% Number of atoms : 5401 (1120 equ; 0 cnn)
% Maximal formula atoms : 17 ( 4 avg)
% Number of connectives : 21867 ( 235 ~; 60 |; 112 &;17845 @)
% ( 0 <=>;3615 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Number of types : 24 ( 23 usr)
% Number of type conns : 467 ( 467 >; 0 *; 0 +; 0 <<)
% Number of symbols : 232 ( 229 usr; 11 con; 0-4 aty)
% Number of variables : 3526 ( 78 ^;3400 !; 48 ?;3526 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:19:44.939
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
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% Explicit typings (229)
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abelia2778853791629620336t_unit: partia2956882679547061052t_unit > $o ).
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abelia3891852623213500406t_unit: partia2670972154091845814t_unit > $o ).
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abelia5304159692179083286t_unit: partia7496981018696276118t_unit > $o ).
thf(sy_c_Ring_Oabelian__group_001tf__a_001tf__b,type,
abelian_group_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oabelian__monoid_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
abelia3641329199688042803t_unit: partia2956882679547061052t_unit > $o ).
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abelia226231641709521465t_unit: partia2670972154091845814t_unit > $o ).
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thf(sy_c_Ring_Oabelian__monoid_001tf__a_001tf__b,type,
abelian_monoid_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Ocring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
cring_5991999922451032090t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring_Ocring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Ring_Ocring_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
cring_3470013030684506304t_unit: partia7496981018696276118t_unit > $o ).
thf(sy_c_Ring_Ocring_001tf__a_001tf__b,type,
cring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Odomain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
domain7810152921033798211t_unit: partia2956882679547061052t_unit > $o ).
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domain6553523120543210313t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
domain1617769409708967785t_unit: partia7496981018696276118t_unit > $o ).
thf(sy_c_Ring_Odomain_001tf__a_001tf__b,type,
domain_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Ofield_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
field_1861437471013600865t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
field_6388047844668329575t_unit: partia2670972154091845814t_unit > $o ).
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field_1540243473349940225t_unit: partia4960592913263135132t_unit > $o ).
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field_26233345952514695t_unit: partia7496981018696276118t_unit > $o ).
thf(sy_c_Ring_Ofield_001tf__a_001tf__b,type,
field_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_l1939023646219158831t_unit: partia2956882679547061052t_unit > $o ).
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ring_l6212528067271185461t_unit: partia2670972154091845814t_unit > $o ).
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ring_s8247141995668492373t_unit: partia7496981018696276118t_unit > $o ).
thf(sy_c_Ring_Oring_001tf__a_001tf__b,type,
ring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
add_li174743652000525320t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a > list_list_a ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
add_li7652885771158616974t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Ring_Oring_Oadd_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
add_se2486902527185523630t_unit: partia7496981018696276118t_unit > set_list_a > set_list_a > set_list_a ).
thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
add_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
zero_l4142658623432671053t_unit: partia2670972154091845814t_unit > list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
zero_s2910681146719230829t_unit: partia7496981018696276118t_unit > set_list_a ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring_Osemiring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
semiri2871908745932252451t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Osemiring_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
semiri4000464634269493571t_unit: partia7496981018696276118t_unit > $o ).
thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
semiring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Omult__of_001tf__a_001tf__b,type,
ring_mult_of_a_b: partia2175431115845679010xt_a_b > partia8223610829204095565t_unit ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_p715737262848045090t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_p8098905331641078952t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001tf__a_001tf__b,type,
ring_p8803135361686045600in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r360171070648044744t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r932985474545269838t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r5115406448772830318t_unit: partia7496981018696276118t_unit > set_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001tf__a_001tf__b,type,
ring_r999134135267193926le_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r5437400583859147359t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r6430282645014804837t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r1091214237498979717t_unit: partia7496981018696276118t_unit > set_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001tf__a_001tf__b,type,
ring_ring_prime_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Subrings_Osubcring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subcri7763218559781929323t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subcri7783154434480317835t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001tf__a_001tf__b,type,
subcring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubdomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subdom7821232466298058046t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subdom3220114454046903646t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001tf__a_001tf__b,type,
subdomain_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subfie1779122896746047282t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subfie4339374749748326226t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subrin6918843898125473962t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubring_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subrin5643252653130547402t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).
thf(sy_c_Subrings_Osubring_001tf__a_001tf__b,type,
subring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member5524387281408368019list_a: list_set_list_a > set_list_set_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_K,type,
k: set_a ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_p,type,
p: list_a ).
thf(sy_v_q,type,
q: list_a ).
% Relevant facts (1277)
thf(fact_0_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_1_assms_I1_J,axiom,
subfield_a_b @ k @ r ).
% assms(1)
thf(fact_2_assms_I3_J,axiom,
member_list_a @ q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).
% assms(3)
thf(fact_3_assms_I2_J,axiom,
member_list_a @ p @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).
% assms(2)
thf(fact_4_zero__pdivides,axiom,
! [P: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P )
= ( P = nil_a ) ) ).
% zero_pdivides
thf(fact_5_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_6_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_7_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_8_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_9_is__abelian__group,axiom,
abelian_group_a_b @ r ).
% is_abelian_group
thf(fact_10_comm__monoid__axioms,axiom,
comm_m952295370001973751xt_a_b @ r ).
% comm_monoid_axioms
thf(fact_11_associated__sym,axiom,
! [A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( associ5860276527279195403xt_a_b @ r @ B @ A ) ) ).
% associated_sym
thf(fact_12_monoid__axioms,axiom,
monoid8385113658579753027xt_a_b @ r ).
% monoid_axioms
thf(fact_13_is__cring,axiom,
cring_a_b @ r ).
% is_cring
thf(fact_14_rupture__def,axiom,
( polyno5459750281392823787re_a_b
= ( ^ [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] : ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ R @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) ) ) ) ).
% rupture_def
thf(fact_15_univ__poly__is__principal,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% univ_poly_is_principal
thf(fact_16_domain_Ozero__pdivides,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( polyno8016796738000020810t_unit @ R2 @ nil_list_a @ P )
= ( P = nil_list_a ) ) ) ).
% domain.zero_pdivides
thf(fact_17_domain_Ozero__pdivides,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a] :
( ( domain_a_b @ R2 )
=> ( ( polyno5814909790663948098es_a_b @ R2 @ nil_a @ P )
= ( P = nil_a ) ) ) ).
% domain.zero_pdivides
thf(fact_18_domain_Ozero__pdivides__zero,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( polyno8016796738000020810t_unit @ R2 @ nil_list_a @ nil_list_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_19_domain_Ozero__pdivides__zero,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R2 )
=> ( polyno5814909790663948098es_a_b @ R2 @ nil_a @ nil_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_20_exists__unique__long__division,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( Q != nil_a )
=> ? [X: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ X )
& ! [Y: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ Y )
=> ( Y = X ) ) ) ) ) ) ) ).
% exists_unique_long_division
thf(fact_21_pprimeE_I1_J,axiom,
! [K2: set_a,P: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
=> ( P != nil_a ) ) ) ) ).
% pprimeE(1)
thf(fact_22_long__division__zero_I1_J,axiom,
! [K2: set_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% long_division_zero(1)
thf(fact_23_univ__poly__zero__closed,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ).
% univ_poly_zero_closed
thf(fact_24_long__division__closed_I1_J,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_25_rupture__surj__composed__with__pmod,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P ) @ Q )
= ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P ) @ ( polynomial_pmod_a_b @ r @ Q @ P ) ) ) ) ) ) ).
% rupture_surj_composed_with_pmod
thf(fact_26_pmod__zero__iff__pdivides,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ P @ Q )
= nil_a )
= ( polyno5814909790663948098es_a_b @ r @ Q @ P ) ) ) ) ) ).
% pmod_zero_iff_pdivides
thf(fact_27_domain_Ouniv__poly__is__principal,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ring_p715737262848045090t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_28_domain_Ouniv__poly__is__principal,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_29_long__division__zero_I2_J,axiom,
! [K2: set_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polynomial_pmod_a_b @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% long_division_zero(2)
thf(fact_30_cring_Oassociated__iff__same__ideal,axiom,
! [R2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( associ8407585678920448409t_unit @ R2 @ A @ B )
= ( ( cgenid9131348535277946915t_unit @ R2 @ A )
= ( cgenid9131348535277946915t_unit @ R2 @ B ) ) ) ) ) ) ).
% cring.associated_iff_same_ideal
thf(fact_31_cring_Oassociated__iff__same__ideal,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( associ5860276527279195403xt_a_b @ R2 @ A @ B )
= ( ( cgenid547466209912283029xt_a_b @ R2 @ A )
= ( cgenid547466209912283029xt_a_b @ R2 @ B ) ) ) ) ) ) ).
% cring.associated_iff_same_ideal
thf(fact_32_long__division__closed_I2_J,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ) ) ) ).
% long_division_closed(2)
thf(fact_33_pdivides__zero,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P @ nil_a ) ) ) ).
% pdivides_zero
thf(fact_34_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_35_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_36_associated__iff__same__ideal,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
= ( ( cgenid547466209912283029xt_a_b @ r @ A )
= ( cgenid547466209912283029xt_a_b @ r @ B ) ) ) ) ) ).
% associated_iff_same_ideal
thf(fact_37_assoc__subst,axiom,
! [A: a,B: a,F: a > a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ! [A2: a,B2: a] :
( ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ A2 @ B2 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ ( F @ B2 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% assoc_subst
thf(fact_38_associated__trans,axiom,
! [A: a,B: a,C: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ( associ5860276527279195403xt_a_b @ r @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ C ) ) ) ) ) ).
% associated_trans
thf(fact_39_univ__poly__is__cring,axiom,
! [K2: set_a] :
( ( subring_a_b @ K2 @ r )
=> ( cring_3148771470849435808t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% univ_poly_is_cring
thf(fact_40_univ__poly__is__domain,axiom,
! [K2: set_a] :
( ( subring_a_b @ K2 @ r )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% univ_poly_is_domain
thf(fact_41_associated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ A ) ) ).
% associated_refl
thf(fact_42_carrier__polynomial__shell,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% carrier_polynomial_shell
thf(fact_43_ring_Opdiv_Ocong,axiom,
polynomial_pdiv_a_b = polynomial_pdiv_a_b ).
% ring.pdiv.cong
thf(fact_44_ring_Opmod_Ocong,axiom,
polynomial_pmod_a_b = polynomial_pmod_a_b ).
% ring.pmod.cong
thf(fact_45_ring_Olong__divides_Ocong,axiom,
polyno2806191415236617128es_a_b = polyno2806191415236617128es_a_b ).
% ring.long_divides.cong
thf(fact_46_domain_Ouniv__poly__is__cring,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( cring_5991999922451032090t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) ) ) ).
% domain.univ_poly_is_cring
thf(fact_47_domain_Ouniv__poly__is__cring,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( cring_3148771470849435808t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ) ).
% domain.univ_poly_is_cring
thf(fact_48_domain_Ouniv__poly__is__domain,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( domain7810152921033798211t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) ) ) ).
% domain.univ_poly_is_domain
thf(fact_49_domain_Ouniv__poly__is__domain,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ) ).
% domain.univ_poly_is_domain
thf(fact_50_domain_Oring__primeE_I1_J,axiom,
! [R2: partia7496981018696276118t_unit,P: set_list_a] :
( ( domain1617769409708967785t_unit @ R2 )
=> ( ( member_set_list_a @ P @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( ring_r1091214237498979717t_unit @ R2 @ P )
=> ( P
!= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_51_domain_Oring__primeE_I1_J,axiom,
! [R2: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ring_r6430282645014804837t_unit @ R2 @ P )
=> ( P
!= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_52_domain_Oring__primeE_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_ring_prime_a_b @ R2 @ P )
=> ( P
!= ( zero_a_b @ R2 ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_53_mem__Collect__eq,axiom,
! [A: list_a,P3: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A: a,P3: a > $o] :
( ( member_a @ A @ ( collect_a @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
! [A: set_a,P3: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A3: set_list_a] :
( ( collect_list_a
@ ^ [X2: list_a] : ( member_list_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A3: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_59_ring_Ocarrier__polynomial__shell,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ) ) ).
% ring.carrier_polynomial_shell
thf(fact_60_ring_Ocarrier__polynomial__shell,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( ring_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ) ) ).
% ring.carrier_polynomial_shell
thf(fact_61_domain_Opdivides__zero,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( polyno8016796738000020810t_unit @ R2 @ P @ nil_list_a ) ) ) ) ).
% domain.pdivides_zero
thf(fact_62_domain_Opdivides__zero,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( polyno5814909790663948098es_a_b @ R2 @ P @ nil_a ) ) ) ) ).
% domain.pdivides_zero
thf(fact_63_domain_Olong__division__closed_I2_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( member_list_list_a @ ( polyno1727750685288865234t_unit @ R2 @ P @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) ) ) ) ) ) ).
% domain.long_division_closed(2)
thf(fact_64_domain_Olong__division__closed_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ R2 @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ) ) ) ) ).
% domain.long_division_closed(2)
thf(fact_65_domain_Olong__division__zero_I2_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( polyno1727750685288865234t_unit @ R2 @ nil_list_a @ Q )
= nil_list_a ) ) ) ) ).
% domain.long_division_zero(2)
thf(fact_66_domain_Olong__division__zero_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( polynomial_pmod_a_b @ R2 @ nil_a @ Q )
= nil_a ) ) ) ) ).
% domain.long_division_zero(2)
thf(fact_67_domain_Olong__division__closed_I1_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( member_list_list_a @ ( polyno5893782122288709345t_unit @ R2 @ P @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_68_domain_Olong__division__closed_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ R2 @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_69_domain_OpprimeE_I1_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P )
=> ( P != nil_list_a ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_70_domain_OpprimeE_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P )
=> ( P != nil_a ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_71_domain_Opmod__zero__iff__pdivides,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R2 @ P @ Q )
= nil_list_a )
= ( polyno8016796738000020810t_unit @ R2 @ Q @ P ) ) ) ) ) ) ).
% domain.pmod_zero_iff_pdivides
thf(fact_72_domain_Opmod__zero__iff__pdivides,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( ( polynomial_pmod_a_b @ R2 @ P @ Q )
= nil_a )
= ( polyno5814909790663948098es_a_b @ R2 @ Q @ P ) ) ) ) ) ) ).
% domain.pmod_zero_iff_pdivides
thf(fact_73_domain_Olong__division__zero_I1_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( polyno5893782122288709345t_unit @ R2 @ nil_list_a @ Q )
= nil_list_a ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_74_domain_Olong__division__zero_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( polynomial_pdiv_a_b @ R2 @ nil_a @ Q )
= nil_a ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_75_principal__domain_Oaxioms_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_p8803135361686045600in_a_b @ R2 )
=> ( domain_a_b @ R2 ) ) ).
% principal_domain.axioms(1)
thf(fact_76_principal__domain_Oaxioms_I1_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( ring_p8098905331641078952t_unit @ R2 )
=> ( domain6553523120543210313t_unit @ R2 ) ) ).
% principal_domain.axioms(1)
thf(fact_77_domain_Orupture__surj__composed__with__pmod,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P ) @ Q )
= ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P ) @ ( polyno1727750685288865234t_unit @ R2 @ Q @ P ) ) ) ) ) ) ) ).
% domain.rupture_surj_composed_with_pmod
thf(fact_78_domain_Orupture__surj__composed__with__pmod,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P ) @ Q )
= ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P ) @ ( polynomial_pmod_a_b @ R2 @ Q @ P ) ) ) ) ) ) ) ).
% domain.rupture_surj_composed_with_pmod
thf(fact_79_domain_Oexists__unique__long__division,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( Q != nil_list_a )
=> ? [X: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ R2 @ P @ Q @ X )
& ! [Y: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ R2 @ P @ Q @ Y )
=> ( Y = X ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_80_domain_Oexists__unique__long__division,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( Q != nil_a )
=> ? [X: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ R2 @ P @ Q @ X )
& ! [Y: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ R2 @ P @ Q @ Y )
=> ( Y = X ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_81_same__pmod__iff__pdivides,axiom,
! [K2: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q )
= ( polynomial_pmod_a_b @ r @ B @ Q ) )
= ( polyno5814909790663948098es_a_b @ r @ Q @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K2 ) @ A @ B ) ) ) ) ) ) ) ).
% same_pmod_iff_pdivides
thf(fact_82_univ__poly__a__minus__consistent,axiom,
! [K2: set_a,Q: list_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ Q )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) ) ) ) ).
% univ_poly_a_minus_consistent
thf(fact_83_cgenideal__is__principalideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I ) @ r ) ) ).
% cgenideal_is_principalideal
thf(fact_84_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_85_long__divisionI,axiom,
! [K2: set_a,P: list_a,Q: list_a,B: list_a,R3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B @ R3 ) )
=> ( ( produc6837034575241423639list_a @ B @ R3 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ) ).
% long_divisionI
thf(fact_86_long__divisionE,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( Q != nil_a )
=> ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ).
% long_divisionE
thf(fact_87_pprime__iff__pirreducible,axiom,
! [K2: set_a,P: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_88_cgenideal__pirreducible,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ Q )
=> ( ( member_list_a @ Q @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ Q ) ) ) ) ) ) ).
% cgenideal_pirreducible
thf(fact_89_exists__long__division,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( Q != nil_a )
=> ~ ! [B2: list_a] :
( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ! [R4: list_a] :
( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B2 @ R4 ) ) ) ) ) ) ) ) ).
% exists_long_division
thf(fact_90_pirreducibleE_I1_J,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
=> ( P != nil_a ) ) ) ) ).
% pirreducibleE(1)
thf(fact_91_pprimeE_I3_J,axiom,
! [K2: set_a,P: list_a,Q: list_a,R3: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ Q @ R3 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
| ( polyno5814909790663948098es_a_b @ r @ P @ R3 ) ) ) ) ) ) ) ) ).
% pprimeE(3)
thf(fact_92_poly__add_Ocases,axiom,
! [X3: produc9164743771328383783list_a] :
~ ! [P1: list_a,P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).
% poly_add.cases
thf(fact_93_ring__primeE_I1_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( P
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_94_subring__props_I2_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K2 ) ) ).
% subring_props(2)
thf(fact_95_univ__poly__is__ring,axiom,
! [K2: set_a] :
( ( subring_a_b @ K2 @ r )
=> ( ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% univ_poly_is_ring
thf(fact_96_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_97_ring_Opoly__add_Ocases,axiom,
! [R2: partia2670972154091845814t_unit,X3: produc7709606177366032167list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ~ ! [P1: list_list_a,P22: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ P1 @ P22 ) ) ) ).
% ring.poly_add.cases
thf(fact_98_ring_Opoly__add_Ocases,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: produc9164743771328383783list_a] :
( ( ring_a_b @ R2 )
=> ~ ! [P1: list_a,P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ P1 @ P22 ) ) ) ).
% ring.poly_add.cases
thf(fact_99_univ__poly__zero,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a] :
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R2 @ K2 ) )
= nil_a ) ).
% univ_poly_zero
thf(fact_100_domain_Ouniv__poly__is__ring,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ring_l1939023646219158831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) ) ) ).
% domain.univ_poly_is_ring
thf(fact_101_domain_Ouniv__poly__is__ring,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ) ).
% domain.univ_poly_is_ring
thf(fact_102_domain_Oring__irreducibleE_I1_J,axiom,
! [R2: partia7496981018696276118t_unit,R3: set_list_a] :
( ( domain1617769409708967785t_unit @ R2 )
=> ( ( member_set_list_a @ R3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( ring_r5115406448772830318t_unit @ R2 @ R3 )
=> ( R3
!= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_103_domain_Oring__irreducibleE_I1_J,axiom,
! [R2: partia2670972154091845814t_unit,R3: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ring_r932985474545269838t_unit @ R2 @ R3 )
=> ( R3
!= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_104_domain_Oring__irreducibleE_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,R3: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_r999134135267193926le_a_b @ R2 @ R3 )
=> ( R3
!= ( zero_a_b @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_105_principal__domain_Oprimeness__condition,axiom,
! [R2: partia2670972154091845814t_unit,P: list_a] :
( ( ring_p8098905331641078952t_unit @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ring_r932985474545269838t_unit @ R2 @ P )
= ( ring_r6430282645014804837t_unit @ R2 @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_106_principal__domain_Oprimeness__condition,axiom,
! [R2: partia2175431115845679010xt_a_b,P: a] :
( ( ring_p8803135361686045600in_a_b @ R2 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_r999134135267193926le_a_b @ R2 @ P )
= ( ring_ring_prime_a_b @ R2 @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_107_domain_OpirreducibleE_I1_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P )
=> ( P != nil_list_a ) ) ) ) ) ).
% domain.pirreducibleE(1)
thf(fact_108_domain_OpirreducibleE_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P )
=> ( P != nil_a ) ) ) ) ) ).
% domain.pirreducibleE(1)
thf(fact_109_domain_Oexists__long__division,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( Q != nil_list_a )
=> ~ ! [B2: list_list_a] :
( ( member_list_list_a @ B2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ! [R4: list_list_a] :
( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ~ ( polyno6947042923167803568t_unit @ R2 @ P @ Q @ ( produc8696003437204565271list_a @ B2 @ R4 ) ) ) ) ) ) ) ) ) ).
% domain.exists_long_division
thf(fact_110_domain_Oexists__long__division,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( Q != nil_a )
=> ~ ! [B2: list_a] :
( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ! [R4: list_a] :
( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ R2 @ P @ Q @ ( produc6837034575241423639list_a @ B2 @ R4 ) ) ) ) ) ) ) ) ) ).
% domain.exists_long_division
thf(fact_111_domain_OpprimeE_I3_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a,R3: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( polyno8016796738000020810t_unit @ R2 @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ Q @ R3 ) )
=> ( ( polyno8016796738000020810t_unit @ R2 @ P @ Q )
| ( polyno8016796738000020810t_unit @ R2 @ P @ R3 ) ) ) ) ) ) ) ) ) ).
% domain.pprimeE(3)
thf(fact_112_domain_OpprimeE_I3_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a,R3: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R2 @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ Q @ R3 ) )
=> ( ( polyno5814909790663948098es_a_b @ R2 @ P @ Q )
| ( polyno5814909790663948098es_a_b @ R2 @ P @ R3 ) ) ) ) ) ) ) ) ) ).
% domain.pprimeE(3)
thf(fact_113_domain_Ocgenideal__pirreducible,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ Q )
=> ( ( member_list_list_a @ Q @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P ) )
=> ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P @ Q ) ) ) ) ) ) ) ).
% domain.cgenideal_pirreducible
thf(fact_114_domain_Ocgenideal__pirreducible,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ Q )
=> ( ( member_list_a @ Q @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P @ Q ) ) ) ) ) ) ) ).
% domain.cgenideal_pirreducible
thf(fact_115_domain_Opprime__iff__pirreducible,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_116_domain_Opprime__iff__pirreducible,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_117_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,Q: list_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P @ Q )
= ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ P @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_118_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,Q: list_a,P: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P @ Q )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ P @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_119_domain_Olong__divisionE,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( Q != nil_list_a )
=> ( polyno6947042923167803568t_unit @ R2 @ P @ Q @ ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ R2 @ P @ Q ) @ ( polyno1727750685288865234t_unit @ R2 @ P @ Q ) ) ) ) ) ) ) ) ).
% domain.long_divisionE
thf(fact_120_domain_Olong__divisionE,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( Q != nil_a )
=> ( polyno2806191415236617128es_a_b @ R2 @ P @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ R2 @ P @ Q ) @ ( polynomial_pmod_a_b @ R2 @ P @ Q ) ) ) ) ) ) ) ) ).
% domain.long_divisionE
thf(fact_121_domain_Olong__divisionI,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a,B: list_list_a,R3: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno6947042923167803568t_unit @ R2 @ P @ Q @ ( produc8696003437204565271list_a @ B @ R3 ) )
=> ( ( produc8696003437204565271list_a @ B @ R3 )
= ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ R2 @ P @ Q ) @ ( polyno1727750685288865234t_unit @ R2 @ P @ Q ) ) ) ) ) ) ) ) ) ).
% domain.long_divisionI
thf(fact_122_domain_Olong__divisionI,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a,B: list_a,R3: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ R2 @ P @ Q @ ( produc6837034575241423639list_a @ B @ R3 ) )
=> ( ( produc6837034575241423639list_a @ B @ R3 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ R2 @ P @ Q ) @ ( polynomial_pmod_a_b @ R2 @ P @ Q ) ) ) ) ) ) ) ) ) ).
% domain.long_divisionI
thf(fact_123_domain_Osame__pmod__iff__pdivides,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R2 @ A @ Q )
= ( polyno1727750685288865234t_unit @ R2 @ B @ Q ) )
= ( polyno8016796738000020810t_unit @ R2 @ Q @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ A @ B ) ) ) ) ) ) ) ) ).
% domain.same_pmod_iff_pdivides
thf(fact_124_domain_Osame__pmod__iff__pdivides,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,A: list_a,B: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( ( polynomial_pmod_a_b @ R2 @ A @ Q )
= ( polynomial_pmod_a_b @ R2 @ B @ Q ) )
= ( polyno5814909790663948098es_a_b @ R2 @ Q @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ A @ B ) ) ) ) ) ) ) ) ).
% domain.same_pmod_iff_pdivides
thf(fact_125_associated__polynomials__imp__same__is__root,axiom,
! [P: list_a,Q: list_a,X3: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X3 )
= ( polyno4133073214067823460ot_a_b @ r @ Q @ X3 ) ) ) ) ) ).
% associated_polynomials_imp_same_is_root
thf(fact_126_is__root__poly__mult__imp__is__root,axiom,
! [P: list_a,Q: list_a,X3: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ X3 )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X3 )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X3 ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_127_ring__irreducibleE_I1_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( R3
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_128_rupture__is__field__iff__pirreducible,axiom,
! [K2: set_a,P: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ r @ K2 @ P ) )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P ) ) ) ) ).
% rupture_is_field_iff_pirreducible
thf(fact_129_pprimeI,axiom,
! [K2: set_a,P: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ! [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ Q2 @ R4 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q2 )
| ( polyno5814909790663948098es_a_b @ r @ P @ R4 ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P ) ) ) ) ) ) ).
% pprimeI
thf(fact_130_pdiv__pmod,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ Q @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% pdiv_pmod
thf(fact_131_associated__polynomials__imp__same__roots,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q )
=> ( ( polynomial_roots_a_b @ r @ P )
= ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ).
% associated_polynomials_imp_same_roots
thf(fact_132_ring__primeE_I3_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( prime_a_ring_ext_a_b @ r @ P ) ) ) ).
% ring_primeE(3)
thf(fact_133_pirreducible__pow__pdivides__iff,axiom,
! [K2: set_a,P: list_a,Q: list_a,R3: list_a,N: nat] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
=> ( ~ ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K2 ) @ P @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ Q @ R3 ) )
= ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K2 ) @ P @ N ) @ R3 ) ) ) ) ) ) ) ) ).
% pirreducible_pow_pdivides_iff
thf(fact_134_pirreducibleI,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ! [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ Q2 @ R4 ) )
=> ( ( member_list_a @ Q2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
| ( member_list_a @ R4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P ) ) ) ) ) ) ).
% pirreducibleI
thf(fact_135_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_136_ring__primeI,axiom,
! [P: a] :
( ( P
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P )
=> ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% ring_primeI
thf(fact_137_polynomial__pow__not__zero,axiom,
! [P: list_a,N: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N )
!= nil_a ) ) ) ).
% polynomial_pow_not_zero
thf(fact_138_subring__polynomial__pow__not__zero,axiom,
! [K2: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K2 ) @ P @ N )
!= nil_a ) ) ) ) ).
% subring_polynomial_pow_not_zero
thf(fact_139_long__division__add_I2_J,axiom,
! [K2: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( polynomial_pmod_a_b @ r @ A @ Q ) @ ( polynomial_pmod_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(2)
thf(fact_140_long__division__add__iff,axiom,
! [K2: set_a,A: list_a,B: list_a,C: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q )
= ( polynomial_pmod_a_b @ r @ B @ Q ) )
= ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ A @ C ) @ Q )
= ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ).
% long_division_add_iff
thf(fact_141_pirreducibleE_I2_J,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ) ) ) ).
% pirreducibleE(2)
thf(fact_142_long__division__add_I1_J,axiom,
! [K2: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( polynomial_pdiv_a_b @ r @ A @ Q ) @ ( polynomial_pdiv_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(1)
thf(fact_143_pprimeE_I2_J,axiom,
! [K2: set_a,P: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ) ) ) ).
% pprimeE(2)
thf(fact_144_pirreducibleE_I3_J,axiom,
! [K2: set_a,P: list_a,Q: list_a,R3: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ Q @ R3 ) )
=> ( ( member_list_a @ Q @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
| ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ) ) ) ) ) ) ) ).
% pirreducibleE(3)
thf(fact_145_ring_Oroots_Ocong,axiom,
polynomial_roots_a_b = polynomial_roots_a_b ).
% ring.roots.cong
thf(fact_146_ring_Ois__root_Ocong,axiom,
polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).
% ring.is_root.cong
thf(fact_147_domain_Opow__non__zero,axiom,
! [R2: partia7496981018696276118t_unit,X3: set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R2 )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( X3
!= ( zero_s2910681146719230829t_unit @ R2 ) )
=> ( ( pow_se8252319793075206062it_nat @ R2 @ X3 @ N )
!= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ) ).
% domain.pow_non_zero
thf(fact_148_domain_Opow__non__zero,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( X3
!= ( zero_l4142658623432671053t_unit @ R2 ) )
=> ( ( pow_li1142815632869257134it_nat @ R2 @ X3 @ N )
!= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ) ).
% domain.pow_non_zero
thf(fact_149_domain_Opow__non__zero,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,N: nat] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( X3
!= ( zero_a_b @ R2 ) )
=> ( ( pow_a_1026414303147256608_b_nat @ R2 @ X3 @ N )
!= ( zero_a_b @ R2 ) ) ) ) ) ).
% domain.pow_non_zero
thf(fact_150_cring_Oideal__eq__carrier__iff,axiom,
! [R2: partia2670972154091845814t_unit,A: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ( partia5361259788508890537t_unit @ R2 )
= ( cgenid9131348535277946915t_unit @ R2 @ A ) )
= ( member_list_a @ A @ ( units_2932844235741507942t_unit @ R2 ) ) ) ) ) ).
% cring.ideal_eq_carrier_iff
thf(fact_151_cring_Oideal__eq__carrier__iff,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ( partia707051561876973205xt_a_b @ R2 )
= ( cgenid547466209912283029xt_a_b @ R2 @ A ) )
= ( member_a @ A @ ( units_a_ring_ext_a_b @ R2 ) ) ) ) ) ).
% cring.ideal_eq_carrier_iff
thf(fact_152_domain_Oring__irreducibleE_I4_J,axiom,
! [R2: partia2670972154091845814t_unit,R3: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ring_r932985474545269838t_unit @ R2 @ R3 )
=> ~ ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_153_domain_Oring__irreducibleE_I4_J,axiom,
! [R2: partia2175431115845679010xt_a_b,R3: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_r999134135267193926le_a_b @ R2 @ R3 )
=> ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_154_domain_Ozero__is__prime_I1_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( prime_2011924034616061926t_unit @ R2 @ ( zero_l4142658623432671053t_unit @ R2 ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_155_domain_Ozero__is__prime_I1_J,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R2 )
=> ( prime_5738381090551951334t_unit @ R2 @ ( zero_s2910681146719230829t_unit @ R2 ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_156_domain_Ozero__is__prime_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R2 )
=> ( prime_a_ring_ext_a_b @ R2 @ ( zero_a_b @ R2 ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_157_ring__prime__def,axiom,
( ring_r1091214237498979717t_unit
= ( ^ [R: partia7496981018696276118t_unit,A4: set_list_a] :
( ( A4
!= ( zero_s2910681146719230829t_unit @ R ) )
& ( prime_5738381090551951334t_unit @ R @ A4 ) ) ) ) ).
% ring_prime_def
thf(fact_158_ring__prime__def,axiom,
( ring_r6430282645014804837t_unit
= ( ^ [R: partia2670972154091845814t_unit,A4: list_a] :
( ( A4
!= ( zero_l4142658623432671053t_unit @ R ) )
& ( prime_2011924034616061926t_unit @ R @ A4 ) ) ) ) ).
% ring_prime_def
thf(fact_159_ring__prime__def,axiom,
( ring_ring_prime_a_b
= ( ^ [R: partia2175431115845679010xt_a_b,A4: a] :
( ( A4
!= ( zero_a_b @ R ) )
& ( prime_a_ring_ext_a_b @ R @ A4 ) ) ) ) ).
% ring_prime_def
thf(fact_160_domain_Oring__associated__iff,axiom,
! [R2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( associ8407585678920448409t_unit @ R2 @ A @ B )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ R2 ) )
& ( A
= ( mult_l7073676228092353617t_unit @ R2 @ X2 @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_161_domain_Oring__associated__iff,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( associ5860276527279195403xt_a_b @ R2 @ A @ B )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ R2 ) )
& ( A
= ( mult_a_ring_ext_a_b @ R2 @ X2 @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_162_domain_Oring__irreducibleE_I5_J,axiom,
! [R2: partia2670972154091845814t_unit,R3: list_a,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ring_r932985474545269838t_unit @ R2 @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( R3
= ( mult_l7073676228092353617t_unit @ R2 @ A @ B ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ R2 ) )
| ( member_list_a @ B @ ( units_2932844235741507942t_unit @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_163_domain_Oring__irreducibleE_I5_J,axiom,
! [R2: partia2175431115845679010xt_a_b,R3: a,A: a,B: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_r999134135267193926le_a_b @ R2 @ R3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( R3
= ( mult_a_ring_ext_a_b @ R2 @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ R2 ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_164_domain_Opolynomial__pow__not__zero,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ P @ N )
!= nil_list_a ) ) ) ) ).
% domain.polynomial_pow_not_zero
thf(fact_165_domain_Opolynomial__pow__not__zero,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a,N: nat] :
( ( domain_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ P @ N )
!= nil_a ) ) ) ) ).
% domain.polynomial_pow_not_zero
thf(fact_166_domain_Osubring__polynomial__pow__not__zero,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( P != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P @ N )
!= nil_list_a ) ) ) ) ) ).
% domain.subring_polynomial_pow_not_zero
thf(fact_167_domain_Osubring__polynomial__pow__not__zero,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,N: nat] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R2 @ K2 ) @ P @ N )
!= nil_a ) ) ) ) ) ).
% domain.subring_polynomial_pow_not_zero
thf(fact_168_ring_Oring__primeI,axiom,
! [R2: partia7496981018696276118t_unit,P: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( P
!= ( zero_s2910681146719230829t_unit @ R2 ) )
=> ( ( prime_5738381090551951334t_unit @ R2 @ P )
=> ( ring_r1091214237498979717t_unit @ R2 @ P ) ) ) ) ).
% ring.ring_primeI
thf(fact_169_ring_Oring__primeI,axiom,
! [R2: partia2670972154091845814t_unit,P: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( P
!= ( zero_l4142658623432671053t_unit @ R2 ) )
=> ( ( prime_2011924034616061926t_unit @ R2 @ P )
=> ( ring_r6430282645014804837t_unit @ R2 @ P ) ) ) ) ).
% ring.ring_primeI
thf(fact_170_ring_Oring__primeI,axiom,
! [R2: partia2175431115845679010xt_a_b,P: a] :
( ( ring_a_b @ R2 )
=> ( ( P
!= ( zero_a_b @ R2 ) )
=> ( ( prime_a_ring_ext_a_b @ R2 @ P )
=> ( ring_ring_prime_a_b @ R2 @ P ) ) ) ) ).
% ring.ring_primeI
thf(fact_171_domain_Oring__primeE_I3_J,axiom,
! [R2: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ring_r6430282645014804837t_unit @ R2 @ P )
=> ( prime_2011924034616061926t_unit @ R2 @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_172_domain_Oring__primeE_I3_J,axiom,
! [R2: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_ring_prime_a_b @ R2 @ P )
=> ( prime_a_ring_ext_a_b @ R2 @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_173_domain_OpirreducibleE_I2_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P )
=> ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) ) ) ) ) ) ).
% domain.pirreducibleE(2)
thf(fact_174_domain_OpirreducibleE_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ) ) ) ) ).
% domain.pirreducibleE(2)
thf(fact_175_domain_OpprimeE_I2_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P )
=> ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) ) ) ) ) ) ).
% domain.pprimeE(2)
thf(fact_176_domain_OpprimeE_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ) ) ) ) ).
% domain.pprimeE(2)
thf(fact_177_domain_Olong__division__add__iff,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,A: list_list_a,B: list_list_a,C: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R2 @ A @ Q )
= ( polyno1727750685288865234t_unit @ R2 @ B @ Q ) )
= ( ( polyno1727750685288865234t_unit @ R2 @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ A @ C ) @ Q )
= ( polyno1727750685288865234t_unit @ R2 @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ) ).
% domain.long_division_add_iff
thf(fact_178_domain_Olong__division__add__iff,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,A: list_a,B: list_a,C: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( ( polynomial_pmod_a_b @ R2 @ A @ Q )
= ( polynomial_pmod_a_b @ R2 @ B @ Q ) )
= ( ( polynomial_pmod_a_b @ R2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ A @ C ) @ Q )
= ( polynomial_pmod_a_b @ R2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ) ).
% domain.long_division_add_iff
thf(fact_179_domain_Olong__division__add_I2_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( polyno1727750685288865234t_unit @ R2 @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ A @ B ) @ Q )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ ( polyno1727750685288865234t_unit @ R2 @ A @ Q ) @ ( polyno1727750685288865234t_unit @ R2 @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(2)
thf(fact_180_domain_Olong__division__add_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,A: list_a,B: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( polynomial_pmod_a_b @ R2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ ( polynomial_pmod_a_b @ R2 @ A @ Q ) @ ( polynomial_pmod_a_b @ R2 @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(2)
thf(fact_181_domain_Olong__division__add_I1_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( polyno5893782122288709345t_unit @ R2 @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ A @ B ) @ Q )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ ( polyno5893782122288709345t_unit @ R2 @ A @ Q ) @ ( polyno5893782122288709345t_unit @ R2 @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_182_domain_Olong__division__add_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,A: list_a,B: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( polynomial_pdiv_a_b @ R2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ ( polynomial_pdiv_a_b @ R2 @ A @ Q ) @ ( polynomial_pdiv_a_b @ R2 @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_183_domain_OpirreducibleE_I3_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a,R3: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( P
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ Q @ R3 ) )
=> ( ( member_list_list_a @ Q @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
| ( member_list_list_a @ R3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) ) ) ) ) ) ) ) ) ) ).
% domain.pirreducibleE(3)
thf(fact_184_domain_OpirreducibleE_I3_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a,R3: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ Q @ R3 ) )
=> ( ( member_list_a @ Q @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
| ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ) ) ) ) ) ) ) ) ).
% domain.pirreducibleE(3)
thf(fact_185_domain_Oassociated__polynomials__imp__same__roots,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ P @ Q )
=> ( ( polyno7858422826990252003t_unit @ R2 @ P )
= ( polyno7858422826990252003t_unit @ R2 @ Q ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_roots
thf(fact_186_domain_Oassociated__polynomials__imp__same__roots,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ P @ Q )
=> ( ( polynomial_roots_a_b @ R2 @ P )
= ( polynomial_roots_a_b @ R2 @ Q ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_roots
thf(fact_187_domain_OpirreducibleI,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( P != nil_list_a )
=> ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ! [Q2: list_list_a,R4: list_list_a] :
( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( P
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ Q2 @ R4 ) )
=> ( ( member_list_list_a @ Q2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
| ( member_list_list_a @ R4 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) ) ) ) ) )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P ) ) ) ) ) ) ) ).
% domain.pirreducibleI
thf(fact_188_domain_OpirreducibleI,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ! [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ Q2 @ R4 ) )
=> ( ( member_list_a @ Q2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
| ( member_list_a @ R4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P ) ) ) ) ) ) ) ).
% domain.pirreducibleI
thf(fact_189_domain_Orupture__is__field__iff__pirreducible,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( field_1540243473349940225t_unit @ ( polyno859807163042199155t_unit @ R2 @ K2 @ P ) )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P ) ) ) ) ) ).
% domain.rupture_is_field_iff_pirreducible
thf(fact_190_domain_Orupture__is__field__iff__pirreducible,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ R2 @ K2 @ P ) )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P ) ) ) ) ) ).
% domain.rupture_is_field_iff_pirreducible
thf(fact_191_domain_Opirreducible__pow__pdivides__iff,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a,R3: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P )
=> ( ~ ( polyno8016796738000020810t_unit @ R2 @ P @ Q )
=> ( ( polyno8016796738000020810t_unit @ R2 @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P @ N ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ Q @ R3 ) )
= ( polyno8016796738000020810t_unit @ R2 @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P @ N ) @ R3 ) ) ) ) ) ) ) ) ) ).
% domain.pirreducible_pow_pdivides_iff
thf(fact_192_domain_Opirreducible__pow__pdivides__iff,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a,R3: list_a,N: nat] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P )
=> ( ~ ( polyno5814909790663948098es_a_b @ R2 @ P @ Q )
=> ( ( polyno5814909790663948098es_a_b @ R2 @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R2 @ K2 ) @ P @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ Q @ R3 ) )
= ( polyno5814909790663948098es_a_b @ R2 @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R2 @ K2 ) @ P @ N ) @ R3 ) ) ) ) ) ) ) ) ) ).
% domain.pirreducible_pow_pdivides_iff
thf(fact_193_domain_Opdiv__pmod,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( P
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ Q @ ( polyno5893782122288709345t_unit @ R2 @ P @ Q ) ) @ ( polyno1727750685288865234t_unit @ R2 @ P @ Q ) ) ) ) ) ) ) ).
% domain.pdiv_pmod
thf(fact_194_domain_Opdiv__pmod,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ Q @ ( polynomial_pdiv_a_b @ R2 @ P @ Q ) ) @ ( polynomial_pmod_a_b @ R2 @ P @ Q ) ) ) ) ) ) ) ).
% domain.pdiv_pmod
thf(fact_195_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,X3: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ R2 @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ P @ Q ) @ X3 )
=> ( ( polyno6951661231331188332t_unit @ R2 @ P @ X3 )
| ( polyno6951661231331188332t_unit @ R2 @ Q @ X3 ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_196_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,X3: a] :
( ( domain_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ R2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ P @ Q ) @ X3 )
=> ( ( polyno4133073214067823460ot_a_b @ R2 @ P @ X3 )
| ( polyno4133073214067823460ot_a_b @ R2 @ Q @ X3 ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_197_domain_Oassociated__polynomials__imp__same__is__root,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,X3: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ P @ Q )
=> ( ( polyno6951661231331188332t_unit @ R2 @ P @ X3 )
= ( polyno6951661231331188332t_unit @ R2 @ Q @ X3 ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_is_root
thf(fact_198_domain_Oassociated__polynomials__imp__same__is__root,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,X3: a] :
( ( domain_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ P @ Q )
=> ( ( polyno4133073214067823460ot_a_b @ R2 @ P @ X3 )
= ( polyno4133073214067823460ot_a_b @ R2 @ Q @ X3 ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_is_root
thf(fact_199_domain_OpprimeI,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( P != nil_list_a )
=> ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ! [Q2: list_list_a,R4: list_list_a] :
( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( polyno8016796738000020810t_unit @ R2 @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ Q2 @ R4 ) )
=> ( ( polyno8016796738000020810t_unit @ R2 @ P @ Q2 )
| ( polyno8016796738000020810t_unit @ R2 @ P @ R4 ) ) ) ) )
=> ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P ) ) ) ) ) ) ) ).
% domain.pprimeI
thf(fact_200_domain_OpprimeI,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ! [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R2 @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ Q2 @ R4 ) )
=> ( ( polyno5814909790663948098es_a_b @ R2 @ P @ Q2 )
| ( polyno5814909790663948098es_a_b @ R2 @ P @ R4 ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P ) ) ) ) ) ) ) ).
% domain.pprimeI
thf(fact_201_cringI,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( abelia3891852623213500406t_unit @ R2 )
=> ( ( comm_m1219397618491936389t_unit @ R2 )
=> ( ! [X: list_a,Y2: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ ( add_li7652885771158616974t_unit @ R2 @ X @ Y2 ) @ Z )
= ( add_li7652885771158616974t_unit @ R2 @ ( mult_l7073676228092353617t_unit @ R2 @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ R2 @ Y2 @ Z ) ) ) ) ) )
=> ( cring_3148771470849435808t_unit @ R2 ) ) ) ) ).
% cringI
thf(fact_202_cringI,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ R2 )
=> ( ( comm_m952295370001973751xt_a_b @ R2 )
=> ( ! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( add_a_b @ R2 @ X @ Y2 ) @ Z )
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R2 @ Y2 @ Z ) ) ) ) ) )
=> ( cring_a_b @ R2 ) ) ) ) ).
% cringI
thf(fact_203_ringI,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( abelia3891852623213500406t_unit @ R2 )
=> ( ( monoid5589397312508706001t_unit @ R2 )
=> ( ! [X: list_a,Y2: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ ( add_li7652885771158616974t_unit @ R2 @ X @ Y2 ) @ Z )
= ( add_li7652885771158616974t_unit @ R2 @ ( mult_l7073676228092353617t_unit @ R2 @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ R2 @ Y2 @ Z ) ) ) ) ) )
=> ( ! [X: list_a,Y2: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ Z @ ( add_li7652885771158616974t_unit @ R2 @ X @ Y2 ) )
= ( add_li7652885771158616974t_unit @ R2 @ ( mult_l7073676228092353617t_unit @ R2 @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ R2 @ Z @ Y2 ) ) ) ) ) )
=> ( ring_l6212528067271185461t_unit @ R2 ) ) ) ) ) ).
% ringI
thf(fact_204_ringI,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ R2 )
=> ( ( monoid8385113658579753027xt_a_b @ R2 )
=> ( ! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( add_a_b @ R2 @ X @ Y2 ) @ Z )
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R2 @ Y2 @ Z ) ) ) ) ) )
=> ( ! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ Z @ ( add_a_b @ R2 @ X @ Y2 ) )
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R2 @ Z @ Y2 ) ) ) ) ) )
=> ( ring_a_b @ R2 ) ) ) ) ) ).
% ringI
thf(fact_205_pdivides__imp__roots__incl,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).
% pdivides_imp_roots_incl
thf(fact_206_polynomial__pow__division,axiom,
! [P: list_a,N: nat,M: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ M ) ) ) ) ).
% polynomial_pow_division
thf(fact_207_var__pow__closed,axiom,
! [K2: set_a,N: nat] :
( ( subring_a_b @ K2 @ r )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K2 ) @ ( var_a_b @ r ) @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ) ).
% var_pow_closed
thf(fact_208_monoid_OassociatedI2_H,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,U: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( A
= ( mult_l7073676228092353617t_unit @ G @ B @ U ) )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( associ8407585678920448409t_unit @ G @ A @ B ) ) ) ) ) ).
% monoid.associatedI2'
thf(fact_209_monoid_OassociatedI2_H,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,U: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( A
= ( mult_a_ring_ext_a_b @ G @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A @ B ) ) ) ) ) ).
% monoid.associatedI2'
thf(fact_210_monoid_OassociatedI2_H,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,U: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( A
= ( mult_a_Product_unit @ G @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ A @ B ) ) ) ) ) ).
% monoid.associatedI2'
thf(fact_211_monoid_OassociatedI2,axiom,
! [G: partia2670972154091845814t_unit,U: list_a,A: list_a,B: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( A
= ( mult_l7073676228092353617t_unit @ G @ B @ U ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( associ8407585678920448409t_unit @ G @ A @ B ) ) ) ) ) ).
% monoid.associatedI2
thf(fact_212_monoid_OassociatedI2,axiom,
! [G: partia2175431115845679010xt_a_b,U: a,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ G @ B @ U ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A @ B ) ) ) ) ) ).
% monoid.associatedI2
thf(fact_213_monoid_OassociatedI2,axiom,
! [G: partia8223610829204095565t_unit,U: a,A: a,B: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ U @ ( units_a_Product_unit @ G ) )
=> ( ( A
= ( mult_a_Product_unit @ G @ B @ U ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ A @ B ) ) ) ) ) ).
% monoid.associatedI2
thf(fact_214_divides__pirreducible__condition,axiom,
! [K2: set_a,Q: list_a,P: list_a] :
( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ Q )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ Q )
=> ( ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
| ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ Q ) ) ) ) ) ).
% divides_pirreducible_condition
thf(fact_215_roots__mem__iff__is__root,axiom,
! [P: list_a,X3: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a @ X3 @ ( set_mset_a @ ( polynomial_roots_a_b @ r @ P ) ) )
= ( polyno4133073214067823460ot_a_b @ r @ P @ X3 ) ) ) ).
% roots_mem_iff_is_root
thf(fact_216_associated__polynomials__imp__same__length,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ Q )
=> ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q ) ) ) ) ) ) ).
% associated_polynomials_imp_same_length
thf(fact_217_Units__pow__closed,axiom,
! [X3: a,D: nat] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ D ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_pow_closed
thf(fact_218_pow__mult__distrib,axiom,
! [X3: a,Y3: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 )
= ( mult_a_ring_ext_a_b @ r @ Y3 @ X3 ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y3 @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_219_nat__pow__distrib,axiom,
! [X3: a,Y3: a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y3 @ N ) ) ) ) ) ).
% nat_pow_distrib
thf(fact_220_nat__pow__comm,axiom,
! [X3: a,N: nat,M: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ M ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ M ) @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_221_group__commutes__pow,axiom,
! [X3: a,Y3: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 )
= ( mult_a_ring_ext_a_b @ r @ Y3 @ X3 ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ Y3 )
= ( mult_a_ring_ext_a_b @ r @ Y3 @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_222_unit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_223_prod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_224_prod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_225_r__distr,axiom,
! [X3: a,Y3: a,Z2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z2 @ ( add_a_b @ r @ X3 @ Y3 ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z2 @ X3 ) @ ( mult_a_ring_ext_a_b @ r @ Z2 @ Y3 ) ) ) ) ) ) ).
% r_distr
thf(fact_226_l__distr,axiom,
! [X3: a,Y3: a,Z2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X3 @ Y3 ) @ Z2 )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Z2 ) @ ( mult_a_ring_ext_a_b @ r @ Y3 @ Z2 ) ) ) ) ) ) ).
% l_distr
thf(fact_227_Units__closed,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_228_m__lcomm,axiom,
! [X3: a,Y3: a,Z2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( mult_a_ring_ext_a_b @ r @ Y3 @ Z2 ) )
= ( mult_a_ring_ext_a_b @ r @ Y3 @ ( mult_a_ring_ext_a_b @ r @ X3 @ Z2 ) ) ) ) ) ) ).
% m_lcomm
thf(fact_229_m__comm,axiom,
! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 )
= ( mult_a_ring_ext_a_b @ r @ Y3 @ X3 ) ) ) ) ).
% m_comm
thf(fact_230_m__assoc,axiom,
! [X3: a,Y3: a,Z2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ r @ X3 @ ( mult_a_ring_ext_a_b @ r @ Y3 @ Z2 ) ) ) ) ) ) ).
% m_assoc
thf(fact_231_a__lcomm,axiom,
! [X3: a,Y3: a,Z2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( add_a_b @ r @ Y3 @ Z2 ) )
= ( add_a_b @ r @ Y3 @ ( add_a_b @ r @ X3 @ Z2 ) ) ) ) ) ) ).
% a_lcomm
thf(fact_232_a__comm,axiom,
! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ Y3 )
= ( add_a_b @ r @ Y3 @ X3 ) ) ) ) ).
% a_comm
thf(fact_233_a__assoc,axiom,
! [X3: a,Y3: a,Z2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X3 @ Y3 ) @ Z2 )
= ( add_a_b @ r @ X3 @ ( add_a_b @ r @ Y3 @ Z2 ) ) ) ) ) ) ).
% a_assoc
thf(fact_234_add_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( add_a_b @ r @ A @ C )
= ( add_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.r_cancel
thf(fact_235_add_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( add_a_b @ r @ C @ A )
= ( add_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.l_cancel
thf(fact_236_subring__props_I6_J,axiom,
! [K2: set_a,H1: a,H2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H2 @ K2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H2 ) @ K2 ) ) ) ) ).
% subring_props(6)
thf(fact_237_subring__props_I7_J,axiom,
! [K2: set_a,H1: a,H2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H2 @ K2 )
=> ( member_a @ ( add_a_b @ r @ H1 @ H2 ) @ K2 ) ) ) ) ).
% subring_props(7)
thf(fact_238_Units__assoc,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ).
% Units_assoc
thf(fact_239_pow__non__zero,axiom,
! [X3: a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X3
!= ( zero_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N )
!= ( zero_a_b @ r ) ) ) ) ).
% pow_non_zero
thf(fact_240_m__rcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B @ A )
= ( mult_a_ring_ext_a_b @ r @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% m_rcancel
thf(fact_241_m__lcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( mult_a_ring_ext_a_b @ r @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% m_lcancel
thf(fact_242_integral__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_243_local_Ointegral,axiom,
! [A: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_244_local_Ominus__unique,axiom,
! [Y3: a,X3: a,Y4: a] :
( ( ( add_a_b @ r @ Y3 @ X3 )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X3 @ Y4 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y3 = Y4 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_245_add_Or__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X3 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_246_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X )
= X ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_247_add_Ol__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X3 )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_248_add_Oinv__comm,axiom,
! [X3: a,Y3: a] :
( ( ( add_a_b @ r @ X3 @ Y3 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y3 @ X3 )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_249_associatedI2,axiom,
! [U: a,A: a,B: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2
thf(fact_250_associatedI2_H,axiom,
! [A: a,B: a,U: a] :
( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2'
thf(fact_251_ring__associated__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
& ( A
= ( mult_a_ring_ext_a_b @ r @ X2 @ B ) ) ) ) ) ) ) ).
% ring_associated_iff
thf(fact_252_Units__cong,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_cong
thf(fact_253_mult__cong__r,axiom,
! [B: a,B3: a,A: a] :
( ( associ5860276527279195403xt_a_b @ r @ B @ B3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B3 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_254_mult__cong__l,axiom,
! [A: a,A5: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ A5 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A5 @ B ) ) ) ) ) ) ).
% mult_cong_l
thf(fact_255_univ__poly__is__monoid,axiom,
! [K2: set_a] :
( ( subring_a_b @ K2 @ r )
=> ( monoid5589397312508706001t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% univ_poly_is_monoid
thf(fact_256_univ__poly__is__abelian__group,axiom,
! [K2: set_a] :
( ( subring_a_b @ K2 @ r )
=> ( abelia3891852623213500406t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% univ_poly_is_abelian_group
thf(fact_257_ideal__eq__carrier__iff,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( partia707051561876973205xt_a_b @ r )
= ( cgenid547466209912283029xt_a_b @ r @ A ) )
= ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ideal_eq_carrier_iff
thf(fact_258_ring__irreducibleE_I5_J,axiom,
! [R3: a,A: a,B: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R3
= ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_259_ring__irreducibleE_I4_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_260_monoid__comm__monoidI,axiom,
( ! [X: a,Y2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X ) ) ) )
=> ( comm_m952295370001973751xt_a_b @ r ) ) ).
% monoid_comm_monoidI
thf(fact_261_var__closed_I1_J,axiom,
! [K2: set_a] :
( ( subring_a_b @ K2 @ r )
=> ( member_list_a @ ( var_a_b @ r ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ) ).
% var_closed(1)
thf(fact_262_Units__m__closed,axiom,
! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_263_pdivides__iff__shell,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ Q ) ) ) ) ) ).
% pdivides_iff_shell
thf(fact_264_subring__polynomial__pow__division,axiom,
! [K2: set_a,P: list_a,N: nat,M: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K2 ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K2 ) @ P @ M ) ) ) ) ) ).
% subring_polynomial_pow_division
thf(fact_265_Units__l__cancel,axiom,
! [X3: a,Y3: a,Z2: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 )
= ( mult_a_ring_ext_a_b @ r @ X3 @ Z2 ) )
= ( Y3 = Z2 ) ) ) ) ) ).
% Units_l_cancel
thf(fact_266_nat__pow__closed,axiom,
! [X3: a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% nat_pow_closed
thf(fact_267_m__closed,axiom,
! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_268_a__closed,axiom,
! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X3 @ Y3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_269_local_Oadd_Oright__cancel,axiom,
! [X3: a,Y3: a,Z2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y3 @ X3 )
= ( add_a_b @ r @ Z2 @ X3 ) )
= ( Y3 = Z2 ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_270_dividesI_H,axiom,
! [B: list_a,G: partia2670972154091845814t_unit,A: list_a,C: list_a] :
( ( B
= ( mult_l7073676228092353617t_unit @ G @ A @ C ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( factor1757716651909850160t_unit @ G @ A @ B ) ) ) ).
% dividesI'
thf(fact_271_dividesI_H,axiom,
! [B: a,G: partia2175431115845679010xt_a_b,A: a,C: a] :
( ( B
= ( mult_a_ring_ext_a_b @ G @ A @ C ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( factor8216151070175719842xt_a_b @ G @ A @ B ) ) ) ).
% dividesI'
thf(fact_272_dividesI_H,axiom,
! [B: a,G: partia8223610829204095565t_unit,A: a,C: a] :
( ( B
= ( mult_a_Product_unit @ G @ A @ C ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ A @ B ) ) ) ).
% dividesI'
thf(fact_273_r__null,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_274_l__null,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X3 )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_275_r__zero,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( zero_a_b @ r ) )
= X3 ) ) ).
% r_zero
thf(fact_276_l__zero,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X3 )
= X3 ) ) ).
% l_zero
thf(fact_277_add_Or__cancel__one_H,axiom,
! [X3: a,A: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X3
= ( add_a_b @ r @ A @ X3 ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_278_add_Or__cancel__one,axiom,
! [X3: a,A: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X3 )
= X3 )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_279_add_Ol__cancel__one_H,axiom,
! [X3: a,A: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X3
= ( add_a_b @ r @ X3 @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_280_add_Ol__cancel__one,axiom,
! [X3: a,A: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X3 @ A )
= X3 )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_281_associatedD,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( factor8216151070175719842xt_a_b @ G @ A @ B ) ) ).
% associatedD
thf(fact_282_associatedD,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( factor1757716651909850160t_unit @ G @ A @ B ) ) ).
% associatedD
thf(fact_283_associatedD,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ( factor3040189038382604065t_unit @ G @ A @ B ) ) ).
% associatedD
thf(fact_284_associatedE,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ~ ( ( factor8216151070175719842xt_a_b @ G @ A @ B )
=> ~ ( factor8216151070175719842xt_a_b @ G @ B @ A ) ) ) ).
% associatedE
thf(fact_285_associatedE,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ~ ( ( factor1757716651909850160t_unit @ G @ A @ B )
=> ~ ( factor1757716651909850160t_unit @ G @ B @ A ) ) ) ).
% associatedE
thf(fact_286_associatedE,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ~ ( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ~ ( factor3040189038382604065t_unit @ G @ B @ A ) ) ) ).
% associatedE
thf(fact_287_associated__def,axiom,
( associ5860276527279195403xt_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b,A4: a,B4: a] :
( ( factor8216151070175719842xt_a_b @ G2 @ A4 @ B4 )
& ( factor8216151070175719842xt_a_b @ G2 @ B4 @ A4 ) ) ) ) ).
% associated_def
thf(fact_288_associated__def,axiom,
( associ8407585678920448409t_unit
= ( ^ [G2: partia2670972154091845814t_unit,A4: list_a,B4: list_a] :
( ( factor1757716651909850160t_unit @ G2 @ A4 @ B4 )
& ( factor1757716651909850160t_unit @ G2 @ B4 @ A4 ) ) ) ) ).
% associated_def
thf(fact_289_associated__def,axiom,
( associ6879500422977059064t_unit
= ( ^ [G2: partia8223610829204095565t_unit,A4: a,B4: a] :
( ( factor3040189038382604065t_unit @ G2 @ A4 @ B4 )
& ( factor3040189038382604065t_unit @ G2 @ B4 @ A4 ) ) ) ) ).
% associated_def
thf(fact_290_divides__antisym,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( factor8216151070175719842xt_a_b @ G @ A @ B )
=> ( ( factor8216151070175719842xt_a_b @ G @ B @ A )
=> ( associ5860276527279195403xt_a_b @ G @ A @ B ) ) ) ).
% divides_antisym
thf(fact_291_divides__antisym,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( factor1757716651909850160t_unit @ G @ A @ B )
=> ( ( factor1757716651909850160t_unit @ G @ B @ A )
=> ( associ8407585678920448409t_unit @ G @ A @ B ) ) ) ).
% divides_antisym
thf(fact_292_divides__antisym,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ( ( factor3040189038382604065t_unit @ G @ B @ A )
=> ( associ6879500422977059064t_unit @ G @ A @ B ) ) ) ).
% divides_antisym
thf(fact_293_dividesD,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( factor1757716651909850160t_unit @ G @ A @ B )
=> ? [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
& ( B
= ( mult_l7073676228092353617t_unit @ G @ A @ X ) ) ) ) ).
% dividesD
thf(fact_294_dividesD,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( factor8216151070175719842xt_a_b @ G @ A @ B )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
& ( B
= ( mult_a_ring_ext_a_b @ G @ A @ X ) ) ) ) ).
% dividesD
thf(fact_295_dividesD,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ? [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
& ( B
= ( mult_a_Product_unit @ G @ A @ X ) ) ) ) ).
% dividesD
thf(fact_296_dividesE,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( factor1757716651909850160t_unit @ G @ A @ B )
=> ~ ! [C2: list_a] :
( ( B
= ( mult_l7073676228092353617t_unit @ G @ A @ C2 ) )
=> ~ ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ G ) ) ) ) ).
% dividesE
thf(fact_297_dividesE,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( factor8216151070175719842xt_a_b @ G @ A @ B )
=> ~ ! [C2: a] :
( ( B
= ( mult_a_ring_ext_a_b @ G @ A @ C2 ) )
=> ~ ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% dividesE
thf(fact_298_dividesE,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ~ ! [C2: a] :
( ( B
= ( mult_a_Product_unit @ G @ A @ C2 ) )
=> ~ ( member_a @ C2 @ ( partia6735698275553448452t_unit @ G ) ) ) ) ).
% dividesE
thf(fact_299_dividesI,axiom,
! [C: list_a,G: partia2670972154091845814t_unit,B: list_a,A: list_a] :
( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( B
= ( mult_l7073676228092353617t_unit @ G @ A @ C ) )
=> ( factor1757716651909850160t_unit @ G @ A @ B ) ) ) ).
% dividesI
thf(fact_300_dividesI,axiom,
! [C: a,G: partia2175431115845679010xt_a_b,B: a,A: a] :
( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( B
= ( mult_a_ring_ext_a_b @ G @ A @ C ) )
=> ( factor8216151070175719842xt_a_b @ G @ A @ B ) ) ) ).
% dividesI
thf(fact_301_dividesI,axiom,
! [C: a,G: partia8223610829204095565t_unit,B: a,A: a] :
( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( B
= ( mult_a_Product_unit @ G @ A @ C ) )
=> ( factor3040189038382604065t_unit @ G @ A @ B ) ) ) ).
% dividesI
thf(fact_302_factor__def,axiom,
( factor1757716651909850160t_unit
= ( ^ [G2: partia2670972154091845814t_unit,A4: list_a,B4: list_a] :
? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G2 ) )
& ( B4
= ( mult_l7073676228092353617t_unit @ G2 @ A4 @ X2 ) ) ) ) ) ).
% factor_def
thf(fact_303_factor__def,axiom,
( factor8216151070175719842xt_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b,A4: a,B4: a] :
? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G2 ) )
& ( B4
= ( mult_a_ring_ext_a_b @ G2 @ A4 @ X2 ) ) ) ) ) ).
% factor_def
thf(fact_304_factor__def,axiom,
( factor3040189038382604065t_unit
= ( ^ [G2: partia8223610829204095565t_unit,A4: a,B4: a] :
? [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ G2 ) )
& ( B4
= ( mult_a_Product_unit @ G2 @ A4 @ X2 ) ) ) ) ) ).
% factor_def
thf(fact_305_monoid_Odivides__refl,axiom,
! [G: partia2670972154091845814t_unit,A: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( factor1757716651909850160t_unit @ G @ A @ A ) ) ) ).
% monoid.divides_refl
thf(fact_306_monoid_Odivides__refl,axiom,
! [G: partia2175431115845679010xt_a_b,A: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( factor8216151070175719842xt_a_b @ G @ A @ A ) ) ) ).
% monoid.divides_refl
thf(fact_307_monoid_Odivides__refl,axiom,
! [G: partia8223610829204095565t_unit,A: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ A @ A ) ) ) ).
% monoid.divides_refl
thf(fact_308_monoid_Odivides__trans,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( factor1757716651909850160t_unit @ G @ A @ B )
=> ( ( factor1757716651909850160t_unit @ G @ B @ C )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( factor1757716651909850160t_unit @ G @ A @ C ) ) ) ) ) ).
% monoid.divides_trans
thf(fact_309_monoid_Odivides__trans,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( factor8216151070175719842xt_a_b @ G @ A @ B )
=> ( ( factor8216151070175719842xt_a_b @ G @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( factor8216151070175719842xt_a_b @ G @ A @ C ) ) ) ) ) ).
% monoid.divides_trans
thf(fact_310_monoid_Odivides__trans,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ( ( factor3040189038382604065t_unit @ G @ B @ C )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ A @ C ) ) ) ) ) ).
% monoid.divides_trans
thf(fact_311_ring_Ozero__divides,axiom,
! [R2: partia7496981018696276118t_unit,A: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( factor2800830226752492592t_unit @ R2 @ ( zero_s2910681146719230829t_unit @ R2 ) @ A )
= ( A
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ).
% ring.zero_divides
thf(fact_312_ring_Ozero__divides,axiom,
! [R2: partia2670972154091845814t_unit,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( factor1757716651909850160t_unit @ R2 @ ( zero_l4142658623432671053t_unit @ R2 ) @ A )
= ( A
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ).
% ring.zero_divides
thf(fact_313_ring_Ozero__divides,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a] :
( ( ring_a_b @ R2 )
=> ( ( factor8216151070175719842xt_a_b @ R2 @ ( zero_a_b @ R2 ) @ A )
= ( A
= ( zero_a_b @ R2 ) ) ) ) ).
% ring.zero_divides
thf(fact_314_monoid_Odivides__prod__r,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( factor1757716651909850160t_unit @ G @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( factor1757716651909850160t_unit @ G @ A @ ( mult_l7073676228092353617t_unit @ G @ B @ C ) ) ) ) ) ) ).
% monoid.divides_prod_r
thf(fact_315_monoid_Odivides__prod__r,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( factor8216151070175719842xt_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( factor8216151070175719842xt_a_b @ G @ A @ ( mult_a_ring_ext_a_b @ G @ B @ C ) ) ) ) ) ) ).
% monoid.divides_prod_r
thf(fact_316_monoid_Odivides__prod__r,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ A @ ( mult_a_Product_unit @ G @ B @ C ) ) ) ) ) ) ).
% monoid.divides_prod_r
thf(fact_317_monoid_Odivides__mult__lI,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( factor1757716651909850160t_unit @ G @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( factor1757716651909850160t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ C @ A ) @ ( mult_l7073676228092353617t_unit @ G @ C @ B ) ) ) ) ) ) ).
% monoid.divides_mult_lI
thf(fact_318_monoid_Odivides__mult__lI,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( factor8216151070175719842xt_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( factor8216151070175719842xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ C @ A ) @ ( mult_a_ring_ext_a_b @ G @ C @ B ) ) ) ) ) ) ).
% monoid.divides_mult_lI
thf(fact_319_monoid_Odivides__mult__lI,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ ( mult_a_Product_unit @ G @ C @ A ) @ ( mult_a_Product_unit @ G @ C @ B ) ) ) ) ) ) ).
% monoid.divides_mult_lI
thf(fact_320_comm__monoid_Odivides__prod__l,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( comm_m1219397618491936389t_unit @ G )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( factor1757716651909850160t_unit @ G @ A @ B )
=> ( factor1757716651909850160t_unit @ G @ A @ ( mult_l7073676228092353617t_unit @ G @ C @ B ) ) ) ) ) ) ) ).
% comm_monoid.divides_prod_l
thf(fact_321_comm__monoid_Odivides__prod__l,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( comm_m952295370001973751xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ A @ B )
=> ( factor8216151070175719842xt_a_b @ G @ A @ ( mult_a_ring_ext_a_b @ G @ C @ B ) ) ) ) ) ) ) ).
% comm_monoid.divides_prod_l
thf(fact_322_comm__monoid_Odivides__prod__l,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ( factor3040189038382604065t_unit @ G @ A @ ( mult_a_Product_unit @ G @ C @ B ) ) ) ) ) ) ) ).
% comm_monoid.divides_prod_l
thf(fact_323_comm__monoid_Odivides__mult__rI,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( comm_m1219397618491936389t_unit @ G )
=> ( ( factor1757716651909850160t_unit @ G @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( factor1757716651909850160t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ A @ C ) @ ( mult_l7073676228092353617t_unit @ G @ B @ C ) ) ) ) ) ) ) ).
% comm_monoid.divides_mult_rI
thf(fact_324_comm__monoid_Odivides__mult__rI,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( comm_m952295370001973751xt_a_b @ G )
=> ( ( factor8216151070175719842xt_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( factor8216151070175719842xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A @ C ) @ ( mult_a_ring_ext_a_b @ G @ B @ C ) ) ) ) ) ) ) ).
% comm_monoid.divides_mult_rI
thf(fact_325_comm__monoid_Odivides__mult__rI,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ ( mult_a_Product_unit @ G @ A @ C ) @ ( mult_a_Product_unit @ G @ B @ C ) ) ) ) ) ) ) ).
% comm_monoid.divides_mult_rI
thf(fact_326_monoid_Ounit__divides,axiom,
! [G: partia2670972154091845814t_unit,U: list_a,A: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( factor1757716651909850160t_unit @ G @ U @ A ) ) ) ) ).
% monoid.unit_divides
thf(fact_327_monoid_Ounit__divides,axiom,
! [G: partia2175431115845679010xt_a_b,U: a,A: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( factor8216151070175719842xt_a_b @ G @ U @ A ) ) ) ) ).
% monoid.unit_divides
thf(fact_328_monoid_Ounit__divides,axiom,
! [G: partia8223610829204095565t_unit,U: a,A: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ U @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ U @ A ) ) ) ) ).
% monoid.unit_divides
thf(fact_329_monoid_Odivides__cong__l,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,X4: list_a,Y3: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ X3 @ X4 )
=> ( ( factor1757716651909850160t_unit @ G @ X4 @ Y3 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( factor1757716651909850160t_unit @ G @ X3 @ Y3 ) ) ) ) ) ).
% monoid.divides_cong_l
thf(fact_330_monoid_Odivides__cong__l,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,X4: a,Y3: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ X3 @ X4 )
=> ( ( factor8216151070175719842xt_a_b @ G @ X4 @ Y3 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( factor8216151070175719842xt_a_b @ G @ X3 @ Y3 ) ) ) ) ) ).
% monoid.divides_cong_l
thf(fact_331_monoid_Odivides__cong__l,axiom,
! [G: partia8223610829204095565t_unit,X3: a,X4: a,Y3: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ X3 @ X4 )
=> ( ( factor3040189038382604065t_unit @ G @ X4 @ Y3 )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ X3 @ Y3 ) ) ) ) ) ).
% monoid.divides_cong_l
thf(fact_332_monoid_Odivides__cong__r,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Y4: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( factor1757716651909850160t_unit @ G @ X3 @ Y3 )
=> ( ( associ8407585678920448409t_unit @ G @ Y3 @ Y4 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( factor1757716651909850160t_unit @ G @ X3 @ Y4 ) ) ) ) ) ).
% monoid.divides_cong_r
thf(fact_333_monoid_Odivides__cong__r,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a,Y4: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( factor8216151070175719842xt_a_b @ G @ X3 @ Y3 )
=> ( ( associ5860276527279195403xt_a_b @ G @ Y3 @ Y4 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( factor8216151070175719842xt_a_b @ G @ X3 @ Y4 ) ) ) ) ) ).
% monoid.divides_cong_r
thf(fact_334_monoid_Odivides__cong__r,axiom,
! [G: partia8223610829204095565t_unit,X3: a,Y3: a,Y4: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ X3 @ Y3 )
=> ( ( associ6879500422977059064t_unit @ G @ Y3 @ Y4 )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ X3 @ Y4 ) ) ) ) ) ).
% monoid.divides_cong_r
thf(fact_335_comm__monoid_Odivides__unit,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,U: list_a] :
( ( comm_m1219397618491936389t_unit @ G )
=> ( ( factor1757716651909850160t_unit @ G @ A @ U )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ G ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ) ).
% comm_monoid.divides_unit
thf(fact_336_comm__monoid_Odivides__unit,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,U: a] :
( ( comm_m952295370001973751xt_a_b @ G )
=> ( ( factor8216151070175719842xt_a_b @ G @ A @ U )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ G ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% comm_monoid.divides_unit
thf(fact_337_comm__monoid_Odivides__unit,axiom,
! [G: partia8223610829204095565t_unit,A: a,U: a] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ A @ U )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ G ) )
=> ( member_a @ A @ ( units_a_Product_unit @ G ) ) ) ) ) ) ).
% comm_monoid.divides_unit
thf(fact_338_domain_Osubring__polynomial__pow__division,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,N: nat,M: nat] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P @ N ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P @ M ) ) ) ) ) ) ).
% domain.subring_polynomial_pow_division
thf(fact_339_domain_Osubring__polynomial__pow__division,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,N: nat,M: nat] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R2 @ K2 ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R2 @ K2 ) @ P @ M ) ) ) ) ) ) ).
% domain.subring_polynomial_pow_division
thf(fact_340_prime__def,axiom,
( prime_2011924034616061926t_unit
= ( ^ [G2: partia2670972154091845814t_unit,P2: list_a] :
( ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ G2 ) )
& ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ! [Y5: list_a] :
( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( factor1757716651909850160t_unit @ G2 @ P2 @ ( mult_l7073676228092353617t_unit @ G2 @ X2 @ Y5 ) )
=> ( ( factor1757716651909850160t_unit @ G2 @ P2 @ X2 )
| ( factor1757716651909850160t_unit @ G2 @ P2 @ Y5 ) ) ) ) ) ) ) ) ).
% prime_def
thf(fact_341_prime__def,axiom,
( prime_a_ring_ext_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b,P2: a] :
( ~ ( member_a @ P2 @ ( units_a_ring_ext_a_b @ G2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ P2 @ ( mult_a_ring_ext_a_b @ G2 @ X2 @ Y5 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ P2 @ X2 )
| ( factor8216151070175719842xt_a_b @ G2 @ P2 @ Y5 ) ) ) ) ) ) ) ) ).
% prime_def
thf(fact_342_prime__def,axiom,
( prime_a_Product_unit
= ( ^ [G2: partia8223610829204095565t_unit,P2: a] :
( ~ ( member_a @ P2 @ ( units_a_Product_unit @ G2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( factor3040189038382604065t_unit @ G2 @ P2 @ ( mult_a_Product_unit @ G2 @ X2 @ Y5 ) )
=> ( ( factor3040189038382604065t_unit @ G2 @ P2 @ X2 )
| ( factor3040189038382604065t_unit @ G2 @ P2 @ Y5 ) ) ) ) ) ) ) ) ).
% prime_def
thf(fact_343_primeI,axiom,
! [P: list_a,G: partia2670972154091845814t_unit] :
( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ G ) )
=> ( ! [A2: list_a,B2: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( factor1757716651909850160t_unit @ G @ P @ ( mult_l7073676228092353617t_unit @ G @ A2 @ B2 ) )
=> ( ( factor1757716651909850160t_unit @ G @ P @ A2 )
| ( factor1757716651909850160t_unit @ G @ P @ B2 ) ) ) ) )
=> ( prime_2011924034616061926t_unit @ G @ P ) ) ) ).
% primeI
thf(fact_344_primeI,axiom,
! [P: a,G: partia2175431115845679010xt_a_b] :
( ~ ( member_a @ P @ ( units_a_ring_ext_a_b @ G ) )
=> ( ! [A2: a,B2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ P @ ( mult_a_ring_ext_a_b @ G @ A2 @ B2 ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ P @ A2 )
| ( factor8216151070175719842xt_a_b @ G @ P @ B2 ) ) ) ) )
=> ( prime_a_ring_ext_a_b @ G @ P ) ) ) ).
% primeI
thf(fact_345_primeI,axiom,
! [P: a,G: partia8223610829204095565t_unit] :
( ~ ( member_a @ P @ ( units_a_Product_unit @ G ) )
=> ( ! [A2: a,B2: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ P @ ( mult_a_Product_unit @ G @ A2 @ B2 ) )
=> ( ( factor3040189038382604065t_unit @ G @ P @ A2 )
| ( factor3040189038382604065t_unit @ G @ P @ B2 ) ) ) ) )
=> ( prime_a_Product_unit @ G @ P ) ) ) ).
% primeI
thf(fact_346_primeE,axiom,
! [G: partia2670972154091845814t_unit,P: list_a] :
( ( prime_2011924034616061926t_unit @ G @ P )
=> ~ ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ G ) )
=> ~ ! [X5: list_a] :
( ( member_list_a @ X5 @ ( partia5361259788508890537t_unit @ G ) )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( factor1757716651909850160t_unit @ G @ P @ ( mult_l7073676228092353617t_unit @ G @ X5 @ Xa ) )
=> ( ( factor1757716651909850160t_unit @ G @ P @ X5 )
| ( factor1757716651909850160t_unit @ G @ P @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_347_primeE,axiom,
! [G: partia2175431115845679010xt_a_b,P: a] :
( ( prime_a_ring_ext_a_b @ G @ P )
=> ~ ( ~ ( member_a @ P @ ( units_a_ring_ext_a_b @ G ) )
=> ~ ! [X5: a] :
( ( member_a @ X5 @ ( partia707051561876973205xt_a_b @ G ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ P @ ( mult_a_ring_ext_a_b @ G @ X5 @ Xa ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ P @ X5 )
| ( factor8216151070175719842xt_a_b @ G @ P @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_348_primeE,axiom,
! [G: partia8223610829204095565t_unit,P: a] :
( ( prime_a_Product_unit @ G @ P )
=> ~ ( ~ ( member_a @ P @ ( units_a_Product_unit @ G ) )
=> ~ ! [X5: a] :
( ( member_a @ X5 @ ( partia6735698275553448452t_unit @ G ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ P @ ( mult_a_Product_unit @ G @ X5 @ Xa ) )
=> ( ( factor3040189038382604065t_unit @ G @ P @ X5 )
| ( factor3040189038382604065t_unit @ G @ P @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_349_ring_Odivides__zero,axiom,
! [R2: partia7496981018696276118t_unit,A: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R2 ) )
=> ( factor2800830226752492592t_unit @ R2 @ A @ ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ).
% ring.divides_zero
thf(fact_350_ring_Odivides__zero,axiom,
! [R2: partia2670972154091845814t_unit,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( factor1757716651909850160t_unit @ R2 @ A @ ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ).
% ring.divides_zero
thf(fact_351_ring_Odivides__zero,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( factor8216151070175719842xt_a_b @ R2 @ A @ ( zero_a_b @ R2 ) ) ) ) ).
% ring.divides_zero
thf(fact_352_ring_Odivides__mult,axiom,
! [R2: partia2670972154091845814t_unit,A: list_a,C: list_a,B: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( factor1757716651909850160t_unit @ R2 @ A @ B )
=> ( factor1757716651909850160t_unit @ R2 @ ( mult_l7073676228092353617t_unit @ R2 @ C @ A ) @ ( mult_l7073676228092353617t_unit @ R2 @ C @ B ) ) ) ) ) ) ).
% ring.divides_mult
thf(fact_353_ring_Odivides__mult,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,C: a,B: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( factor8216151070175719842xt_a_b @ R2 @ A @ B )
=> ( factor8216151070175719842xt_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ C @ A ) @ ( mult_a_ring_ext_a_b @ R2 @ C @ B ) ) ) ) ) ) ).
% ring.divides_mult
thf(fact_354_pdivides__def,axiom,
( polyno8016796738000020810t_unit
= ( ^ [R: partia2670972154091845814t_unit] : ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% pdivides_def
thf(fact_355_pdivides__def,axiom,
( polyno5814909790663948098es_a_b
= ( ^ [R: partia2175431115845679010xt_a_b] : ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% pdivides_def
thf(fact_356_domain_Ouniv__poly__is__monoid,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( monoid5729698748631984209t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) ) ) ).
% domain.univ_poly_is_monoid
thf(fact_357_domain_Ouniv__poly__is__monoid,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( monoid5589397312508706001t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ) ).
% domain.univ_poly_is_monoid
thf(fact_358_domain_Ouniv__poly__is__abelian__group,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( abelia2778853791629620336t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) ) ) ).
% domain.univ_poly_is_abelian_group
thf(fact_359_domain_Ouniv__poly__is__abelian__group,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( abelia3891852623213500406t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ) ).
% domain.univ_poly_is_abelian_group
thf(fact_360_cring_Ois__cring,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R2 )
=> ( cring_a_b @ R2 ) ) ).
% cring.is_cring
thf(fact_361_cring_Ois__cring,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( cring_3148771470849435808t_unit @ R2 ) ) ).
% cring.is_cring
thf(fact_362_domain_Opdivides__iff__shell,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( polyno8016796738000020810t_unit @ R2 @ P @ Q )
= ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P @ Q ) ) ) ) ) ) ).
% domain.pdivides_iff_shell
thf(fact_363_domain_Opdivides__iff__shell,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R2 @ P @ Q )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P @ Q ) ) ) ) ) ) ).
% domain.pdivides_iff_shell
thf(fact_364_domain_Oassociated__polynomials__imp__same__length,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P @ Q )
=> ( ( size_s349497388124573686list_a @ P )
= ( size_s349497388124573686list_a @ Q ) ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_length
thf(fact_365_domain_Oassociated__polynomials__imp__same__length,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P @ Q )
=> ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q ) ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_length
thf(fact_366_domain_Ovar__closed_I1_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( member_list_list_a @ ( var_li8453953174693405341t_unit @ R2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) ) ) ) ).
% domain.var_closed(1)
thf(fact_367_domain_Ovar__closed_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( member_list_a @ ( var_a_b @ R2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ) ) ).
% domain.var_closed(1)
thf(fact_368_domain_Oroots__mem__iff__is__root,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a,X3: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( member_list_a @ X3 @ ( set_mset_list_a @ ( polyno7858422826990252003t_unit @ R2 @ P ) ) )
= ( polyno6951661231331188332t_unit @ R2 @ P @ X3 ) ) ) ) ).
% domain.roots_mem_iff_is_root
thf(fact_369_domain_Oroots__mem__iff__is__root,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a,X3: a] :
( ( domain_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( member_a @ X3 @ ( set_mset_a @ ( polynomial_roots_a_b @ R2 @ P ) ) )
= ( polyno4133073214067823460ot_a_b @ R2 @ P @ X3 ) ) ) ) ).
% domain.roots_mem_iff_is_root
thf(fact_370_ring_Odivides__pirreducible__condition,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,Q: list_list_a,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ Q )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P @ Q )
=> ( ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
| ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P @ Q ) ) ) ) ) ) ).
% ring.divides_pirreducible_condition
thf(fact_371_ring_Odivides__pirreducible__condition,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,Q: list_a,P: list_a] :
( ( ring_a_b @ R2 )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ Q )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P @ Q )
=> ( ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
| ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P @ Q ) ) ) ) ) ) ).
% ring.divides_pirreducible_condition
thf(fact_372_domain_Opolynomial__pow__division,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a,N: nat,M: nat] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno8016796738000020810t_unit @ R2 @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ P @ N ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ P @ M ) ) ) ) ) ).
% domain.polynomial_pow_division
thf(fact_373_domain_Opolynomial__pow__division,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a,N: nat,M: nat] :
( ( domain_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno5814909790663948098es_a_b @ R2 @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ P @ M ) ) ) ) ) ).
% domain.polynomial_pow_division
thf(fact_374_domain_Ovar__pow__closed,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( member_list_list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ ( var_li8453953174693405341t_unit @ R2 ) @ N ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) ) ) ) ).
% domain.var_pow_closed
thf(fact_375_domain_Ovar__pow__closed,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,N: nat] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R2 @ K2 ) @ ( var_a_b @ R2 ) @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ) ) ).
% domain.var_pow_closed
thf(fact_376_monoid_Oassociated__sym,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( associ5860276527279195403xt_a_b @ G @ B @ A ) ) ) ).
% monoid.associated_sym
thf(fact_377_monoid_Oassociated__sym,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( associ8407585678920448409t_unit @ G @ B @ A ) ) ) ).
% monoid.associated_sym
thf(fact_378_monoid_Oassociated__sym,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ( associ6879500422977059064t_unit @ G @ B @ A ) ) ) ).
% monoid.associated_sym
thf(fact_379_cring_Oaxioms_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R2 )
=> ( ring_a_b @ R2 ) ) ).
% cring.axioms(1)
thf(fact_380_cring_Oaxioms_I1_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ring_l6212528067271185461t_unit @ R2 ) ) ).
% cring.axioms(1)
thf(fact_381_domain_Opdivides__imp__roots__incl,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ R2 @ P @ Q )
=> ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ R2 @ P ) @ ( polyno7858422826990252003t_unit @ R2 @ Q ) ) ) ) ) ) ) ).
% domain.pdivides_imp_roots_incl
thf(fact_382_domain_Opdivides__imp__roots__incl,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ R2 @ P @ Q )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ R2 @ P ) @ ( polynomial_roots_a_b @ R2 @ Q ) ) ) ) ) ) ) ).
% domain.pdivides_imp_roots_incl
thf(fact_383_domain_Oaxioms_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R2 )
=> ( cring_a_b @ R2 ) ) ).
% domain.axioms(1)
thf(fact_384_domain_Oaxioms_I1_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( cring_3148771470849435808t_unit @ R2 ) ) ).
% domain.axioms(1)
thf(fact_385_field_Ois__ring,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R2 )
=> ( ring_s8247141995668492373t_unit @ R2 ) ) ).
% field.is_ring
thf(fact_386_field_Ois__ring,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( ring_l6212528067271185461t_unit @ R2 ) ) ).
% field.is_ring
thf(fact_387_field_Ois__ring,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R2 )
=> ( ring_a_b @ R2 ) ) ).
% field.is_ring
thf(fact_388_ring_Ois__abelian__group,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R2 )
=> ( abelian_group_a_b @ R2 ) ) ).
% ring.is_abelian_group
thf(fact_389_ring_Ois__abelian__group,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( abelia3891852623213500406t_unit @ R2 ) ) ).
% ring.is_abelian_group
thf(fact_390_field_Oaxioms_I1_J,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R2 )
=> ( domain1617769409708967785t_unit @ R2 ) ) ).
% field.axioms(1)
thf(fact_391_field_Oaxioms_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R2 )
=> ( domain_a_b @ R2 ) ) ).
% field.axioms(1)
thf(fact_392_field_Oaxioms_I1_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( domain6553523120543210313t_unit @ R2 ) ) ).
% field.axioms(1)
thf(fact_393_fieldE_I1_J,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R2 )
=> ( cring_3470013030684506304t_unit @ R2 ) ) ).
% fieldE(1)
thf(fact_394_fieldE_I1_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( cring_3148771470849435808t_unit @ R2 ) ) ).
% fieldE(1)
thf(fact_395_fieldE_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R2 )
=> ( cring_a_b @ R2 ) ) ).
% fieldE(1)
thf(fact_396_abelian__group_Oaxioms_I1_J,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ G )
=> ( abelian_monoid_a_b @ G ) ) ).
% abelian_group.axioms(1)
thf(fact_397_abelian__group_Oaxioms_I1_J,axiom,
! [G: partia2670972154091845814t_unit] :
( ( abelia3891852623213500406t_unit @ G )
=> ( abelia226231641709521465t_unit @ G ) ) ).
% abelian_group.axioms(1)
thf(fact_398_ring_Ois__monoid,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R2 )
=> ( monoid8385113658579753027xt_a_b @ R2 ) ) ).
% ring.is_monoid
thf(fact_399_ring_Ois__monoid,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( monoid5589397312508706001t_unit @ R2 ) ) ).
% ring.is_monoid
thf(fact_400_semiring_Oaxioms_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R2 )
=> ( abelian_monoid_a_b @ R2 ) ) ).
% semiring.axioms(1)
thf(fact_401_semiring_Oaxioms_I1_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( abelia226231641709521465t_unit @ R2 ) ) ).
% semiring.axioms(1)
thf(fact_402_cring_Oaxioms_I2_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( comm_m1219397618491936389t_unit @ R2 ) ) ).
% cring.axioms(2)
thf(fact_403_cring_Oaxioms_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R2 )
=> ( comm_m952295370001973751xt_a_b @ R2 ) ) ).
% cring.axioms(2)
thf(fact_404_semiring_Oaxioms_I2_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( monoid5589397312508706001t_unit @ R2 ) ) ).
% semiring.axioms(2)
thf(fact_405_semiring_Oaxioms_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R2 )
=> ( monoid8385113658579753027xt_a_b @ R2 ) ) ).
% semiring.axioms(2)
thf(fact_406_ring_Oring__simprules_I2_J,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R2 ) @ ( partia141011252114345353t_unit @ R2 ) ) ) ).
% ring.ring_simprules(2)
thf(fact_407_ring_Oring__simprules_I2_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ).
% ring.ring_simprules(2)
thf(fact_408_ring_Oring__simprules_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R2 )
=> ( member_a @ ( zero_a_b @ R2 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ).
% ring.ring_simprules(2)
thf(fact_409_ring_Oring__simprules_I22_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ X3 @ ( add_li7652885771158616974t_unit @ R2 @ Y3 @ Z2 ) )
= ( add_li7652885771158616974t_unit @ R2 @ Y3 @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_410_ring_Oring__simprules_I22_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ X3 @ ( add_a_b @ R2 @ Y3 @ Z2 ) )
= ( add_a_b @ R2 @ Y3 @ ( add_a_b @ R2 @ X3 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_411_ring_Oring__simprules_I10_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 )
= ( add_li7652885771158616974t_unit @ R2 @ Y3 @ X3 ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_412_ring_Oring__simprules_I10_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ X3 @ Y3 )
= ( add_a_b @ R2 @ Y3 @ X3 ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_413_ring_Oring__simprules_I7_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R2 @ X3 @ ( add_li7652885771158616974t_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_414_ring_Oring__simprules_I7_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ ( add_a_b @ R2 @ X3 @ Y3 ) @ Z2 )
= ( add_a_b @ R2 @ X3 @ ( add_a_b @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_415_ring_Oring__simprules_I1_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_416_ring_Oring__simprules_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ ( add_a_b @ R2 @ X3 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_417_ring_Oring__simprules_I11_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ ( mult_l7073676228092353617t_unit @ R2 @ X3 @ Y3 ) @ Z2 )
= ( mult_l7073676228092353617t_unit @ R2 @ X3 @ ( mult_l7073676228092353617t_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_418_ring_Oring__simprules_I11_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ X3 @ Y3 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ R2 @ X3 @ ( mult_a_ring_ext_a_b @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_419_ring_Oring__simprules_I5_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R2 @ X3 @ Y3 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_420_ring_Oring__simprules_I5_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R2 @ X3 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_421_cring_Ocring__simprules_I2_J,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R2 )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R2 ) @ ( partia141011252114345353t_unit @ R2 ) ) ) ).
% cring.cring_simprules(2)
thf(fact_422_cring_Ocring__simprules_I2_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ).
% cring.cring_simprules(2)
thf(fact_423_cring_Ocring__simprules_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R2 )
=> ( member_a @ ( zero_a_b @ R2 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ).
% cring.cring_simprules(2)
thf(fact_424_cring_Ocring__simprules_I23_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ X3 @ ( add_li7652885771158616974t_unit @ R2 @ Y3 @ Z2 ) )
= ( add_li7652885771158616974t_unit @ R2 @ Y3 @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Z2 ) ) ) ) ) ) ) ).
% cring.cring_simprules(23)
thf(fact_425_cring_Ocring__simprules_I23_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ X3 @ ( add_a_b @ R2 @ Y3 @ Z2 ) )
= ( add_a_b @ R2 @ Y3 @ ( add_a_b @ R2 @ X3 @ Z2 ) ) ) ) ) ) ) ).
% cring.cring_simprules(23)
thf(fact_426_cring_Ocring__simprules_I10_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 )
= ( add_li7652885771158616974t_unit @ R2 @ Y3 @ X3 ) ) ) ) ) ).
% cring.cring_simprules(10)
thf(fact_427_cring_Ocring__simprules_I10_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ X3 @ Y3 )
= ( add_a_b @ R2 @ Y3 @ X3 ) ) ) ) ) ).
% cring.cring_simprules(10)
thf(fact_428_cring_Ocring__simprules_I7_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R2 @ X3 @ ( add_li7652885771158616974t_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% cring.cring_simprules(7)
thf(fact_429_cring_Ocring__simprules_I7_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ ( add_a_b @ R2 @ X3 @ Y3 ) @ Z2 )
= ( add_a_b @ R2 @ X3 @ ( add_a_b @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% cring.cring_simprules(7)
thf(fact_430_cring_Ocring__simprules_I1_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ).
% cring.cring_simprules(1)
thf(fact_431_cring_Ocring__simprules_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ ( add_a_b @ R2 @ X3 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).
% cring.cring_simprules(1)
thf(fact_432_cring_Ocring__simprules_I24_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ X3 @ ( mult_l7073676228092353617t_unit @ R2 @ Y3 @ Z2 ) )
= ( mult_l7073676228092353617t_unit @ R2 @ Y3 @ ( mult_l7073676228092353617t_unit @ R2 @ X3 @ Z2 ) ) ) ) ) ) ) ).
% cring.cring_simprules(24)
thf(fact_433_cring_Ocring__simprules_I24_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ X3 @ ( mult_a_ring_ext_a_b @ R2 @ Y3 @ Z2 ) )
= ( mult_a_ring_ext_a_b @ R2 @ Y3 @ ( mult_a_ring_ext_a_b @ R2 @ X3 @ Z2 ) ) ) ) ) ) ) ).
% cring.cring_simprules(24)
thf(fact_434_cring_Ocring__simprules_I14_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ X3 @ Y3 )
= ( mult_l7073676228092353617t_unit @ R2 @ Y3 @ X3 ) ) ) ) ) ).
% cring.cring_simprules(14)
thf(fact_435_cring_Ocring__simprules_I14_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ X3 @ Y3 )
= ( mult_a_ring_ext_a_b @ R2 @ Y3 @ X3 ) ) ) ) ) ).
% cring.cring_simprules(14)
thf(fact_436_cring_Ocring__simprules_I11_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ ( mult_l7073676228092353617t_unit @ R2 @ X3 @ Y3 ) @ Z2 )
= ( mult_l7073676228092353617t_unit @ R2 @ X3 @ ( mult_l7073676228092353617t_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% cring.cring_simprules(11)
thf(fact_437_cring_Ocring__simprules_I11_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ X3 @ Y3 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ R2 @ X3 @ ( mult_a_ring_ext_a_b @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% cring.cring_simprules(11)
thf(fact_438_cring_Ocring__simprules_I5_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R2 @ X3 @ Y3 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ).
% cring.cring_simprules(5)
thf(fact_439_cring_Ocring__simprules_I5_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R2 @ X3 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).
% cring.cring_simprules(5)
thf(fact_440_abelian__groupE_I2_J,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( abelia5304159692179083286t_unit @ R2 )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R2 ) @ ( partia141011252114345353t_unit @ R2 ) ) ) ).
% abelian_groupE(2)
thf(fact_441_abelian__groupE_I2_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( abelia3891852623213500406t_unit @ R2 )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ).
% abelian_groupE(2)
thf(fact_442_abelian__groupE_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ R2 )
=> ( member_a @ ( zero_a_b @ R2 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ).
% abelian_groupE(2)
thf(fact_443_abelian__groupE_I4_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( abelia3891852623213500406t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 )
= ( add_li7652885771158616974t_unit @ R2 @ Y3 @ X3 ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_444_abelian__groupE_I4_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( abelian_group_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ X3 @ Y3 )
= ( add_a_b @ R2 @ Y3 @ X3 ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_445_abelian__groupE_I3_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( abelia3891852623213500406t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R2 @ X3 @ ( add_li7652885771158616974t_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_446_abelian__groupE_I3_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( abelian_group_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ ( add_a_b @ R2 @ X3 @ Y3 ) @ Z2 )
= ( add_a_b @ R2 @ X3 @ ( add_a_b @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_447_abelian__groupE_I1_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( abelia3891852623213500406t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_448_abelian__groupE_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( abelian_group_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ ( add_a_b @ R2 @ X3 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_449_monoid_Oassoc__subst,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,F: list_a > list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ! [A2: list_a,B2: list_a] :
( ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
& ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ G ) )
& ( associ8407585678920448409t_unit @ G @ A2 @ B2 ) )
=> ( ( member_list_a @ ( F @ A2 ) @ ( partia5361259788508890537t_unit @ G ) )
& ( member_list_a @ ( F @ B2 ) @ ( partia5361259788508890537t_unit @ G ) )
& ( associ8407585678920448409t_unit @ G @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( associ8407585678920448409t_unit @ G @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ) ).
% monoid.assoc_subst
thf(fact_450_monoid_Oassoc__subst,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,F: a > a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ! [A2: a,B2: a] :
( ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
& ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
& ( associ5860276527279195403xt_a_b @ G @ A2 @ B2 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ G ) )
& ( member_a @ ( F @ B2 ) @ ( partia707051561876973205xt_a_b @ G ) )
& ( associ5860276527279195403xt_a_b @ G @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ) ).
% monoid.assoc_subst
thf(fact_451_monoid_Oassoc__subst,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,F: a > a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ( ! [A2: a,B2: a] :
( ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
& ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
& ( associ6879500422977059064t_unit @ G @ A2 @ B2 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia6735698275553448452t_unit @ G ) )
& ( member_a @ ( F @ B2 ) @ ( partia6735698275553448452t_unit @ G ) )
& ( associ6879500422977059064t_unit @ G @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ) ).
% monoid.assoc_subst
thf(fact_452_monoid_Oassociated__refl,axiom,
! [G: partia2670972154091845814t_unit,A: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( associ8407585678920448409t_unit @ G @ A @ A ) ) ) ).
% monoid.associated_refl
thf(fact_453_monoid_Oassociated__refl,axiom,
! [G: partia2175431115845679010xt_a_b,A: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A @ A ) ) ) ).
% monoid.associated_refl
thf(fact_454_monoid_Oassociated__refl,axiom,
! [G: partia8223610829204095565t_unit,A: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ A @ A ) ) ) ).
% monoid.associated_refl
thf(fact_455_monoid_Oassociated__trans,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ( associ8407585678920448409t_unit @ G @ B @ C )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( associ8407585678920448409t_unit @ G @ A @ C ) ) ) ) ) ) ).
% monoid.associated_trans
thf(fact_456_monoid_Oassociated__trans,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ( associ5860276527279195403xt_a_b @ G @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A @ C ) ) ) ) ) ) ).
% monoid.associated_trans
thf(fact_457_monoid_Oassociated__trans,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ( ( associ6879500422977059064t_unit @ G @ B @ C )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ A @ C ) ) ) ) ) ) ).
% monoid.associated_trans
thf(fact_458_abelian__monoidE_I2_J,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( abelia3322010900105369177t_unit @ R2 )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R2 ) @ ( partia141011252114345353t_unit @ R2 ) ) ) ).
% abelian_monoidE(2)
thf(fact_459_abelian__monoidE_I2_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( abelia226231641709521465t_unit @ R2 )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ).
% abelian_monoidE(2)
thf(fact_460_abelian__monoidE_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ R2 )
=> ( member_a @ ( zero_a_b @ R2 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ).
% abelian_monoidE(2)
thf(fact_461_abelian__monoid_Ozero__closed,axiom,
! [G: partia7496981018696276118t_unit] :
( ( abelia3322010900105369177t_unit @ G )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ G ) @ ( partia141011252114345353t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_462_abelian__monoid_Ozero__closed,axiom,
! [G: partia2670972154091845814t_unit] :
( ( abelia226231641709521465t_unit @ G )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_463_abelian__monoid_Ozero__closed,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ G )
=> ( member_a @ ( zero_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_464_abelian__monoidE_I5_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( abelia226231641709521465t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 )
= ( add_li7652885771158616974t_unit @ R2 @ Y3 @ X3 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_465_abelian__monoidE_I5_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( abelian_monoid_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ X3 @ Y3 )
= ( add_a_b @ R2 @ Y3 @ X3 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_466_abelian__monoidE_I3_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( abelia226231641709521465t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R2 @ X3 @ ( add_li7652885771158616974t_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_467_abelian__monoidE_I3_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( abelian_monoid_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ ( add_a_b @ R2 @ X3 @ Y3 ) @ Z2 )
= ( add_a_b @ R2 @ X3 @ ( add_a_b @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_468_abelian__monoidE_I1_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( abelia226231641709521465t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_469_abelian__monoidE_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( abelian_monoid_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ ( add_a_b @ R2 @ X3 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_470_abelian__monoid_Oa__comm,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X3 @ Y3 )
= ( add_li7652885771158616974t_unit @ G @ Y3 @ X3 ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_471_abelian__monoid_Oa__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X3 @ Y3 )
= ( add_a_b @ G @ Y3 @ X3 ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_472_abelian__monoid_Oa__assoc,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( add_li7652885771158616974t_unit @ G @ X3 @ Y3 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ G @ X3 @ ( add_li7652885771158616974t_unit @ G @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_473_abelian__monoid_Oa__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( add_a_b @ G @ X3 @ Y3 ) @ Z2 )
= ( add_a_b @ G @ X3 @ ( add_a_b @ G @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_474_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X3 @ ( add_li7652885771158616974t_unit @ G @ Y3 @ Z2 ) )
= ( add_li7652885771158616974t_unit @ G @ Y3 @ ( add_li7652885771158616974t_unit @ G @ X3 @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_475_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X3 @ ( add_a_b @ G @ Y3 @ Z2 ) )
= ( add_a_b @ G @ Y3 @ ( add_a_b @ G @ X3 @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_476_abelian__monoid_Oa__closed,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ G @ X3 @ Y3 ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_477_abelian__monoid_Oa__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( add_a_b @ G @ X3 @ Y3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_478_ring_Oring__simprules_I4_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ R2 @ X3 @ Y3 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_479_ring_Oring__simprules_I4_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ ( a_minus_a_b @ R2 @ X3 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_480_semiring_Osemiring__simprules_I2_J,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R2 )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R2 ) @ ( partia141011252114345353t_unit @ R2 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_481_semiring_Osemiring__simprules_I2_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_482_semiring_Osemiring__simprules_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R2 )
=> ( member_a @ ( zero_a_b @ R2 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_483_monoid_OUnits__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A @ B ) ) ) ) ).
% monoid.Units_assoc
thf(fact_484_monoid_OUnits__assoc,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) )
=> ( associ8407585678920448409t_unit @ G @ A @ B ) ) ) ) ).
% monoid.Units_assoc
thf(fact_485_monoid_OUnits__assoc,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ A @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ B @ ( units_a_Product_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ A @ B ) ) ) ) ).
% monoid.Units_assoc
thf(fact_486_semiring_Osemiring__simprules_I12_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ X3 @ ( add_li7652885771158616974t_unit @ R2 @ Y3 @ Z2 ) )
= ( add_li7652885771158616974t_unit @ R2 @ Y3 @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_487_semiring_Osemiring__simprules_I12_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( semiring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ X3 @ ( add_a_b @ R2 @ Y3 @ Z2 ) )
= ( add_a_b @ R2 @ Y3 @ ( add_a_b @ R2 @ X3 @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_488_semiring_Osemiring__simprules_I7_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 )
= ( add_li7652885771158616974t_unit @ R2 @ Y3 @ X3 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_489_semiring_Osemiring__simprules_I7_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( semiring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ X3 @ Y3 )
= ( add_a_b @ R2 @ Y3 @ X3 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_490_semiring_Osemiring__simprules_I5_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R2 @ X3 @ ( add_li7652885771158616974t_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_491_semiring_Osemiring__simprules_I5_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( semiring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ ( add_a_b @ R2 @ X3 @ Y3 ) @ Z2 )
= ( add_a_b @ R2 @ X3 @ ( add_a_b @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_492_semiring_Osemiring__simprules_I1_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_493_semiring_Osemiring__simprules_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( semiring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ ( add_a_b @ R2 @ X3 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_494_semiring_Osemiring__simprules_I8_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ ( mult_l7073676228092353617t_unit @ R2 @ X3 @ Y3 ) @ Z2 )
= ( mult_l7073676228092353617t_unit @ R2 @ X3 @ ( mult_l7073676228092353617t_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_495_semiring_Osemiring__simprules_I8_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( semiring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ X3 @ Y3 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ R2 @ X3 @ ( mult_a_ring_ext_a_b @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_496_semiring_Osemiring__simprules_I3_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R2 @ X3 @ Y3 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_497_semiring_Osemiring__simprules_I3_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( semiring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R2 @ X3 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_498_cring_Ocring__simprules_I4_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ R2 @ X3 @ Y3 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ).
% cring.cring_simprules(4)
thf(fact_499_cring_Ocring__simprules_I4_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ ( a_minus_a_b @ R2 @ X3 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).
% cring.cring_simprules(4)
thf(fact_500_abelian__group_Ominus__closed,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ G @ X3 @ Y3 ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_group.minus_closed
thf(fact_501_abelian__group_Ominus__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( a_minus_a_b @ G @ X3 @ Y3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_group.minus_closed
thf(fact_502_cring_Ointro,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( comm_m1219397618491936389t_unit @ R2 )
=> ( cring_3148771470849435808t_unit @ R2 ) ) ) ).
% cring.intro
thf(fact_503_cring_Ointro,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R2 )
=> ( ( comm_m952295370001973751xt_a_b @ R2 )
=> ( cring_a_b @ R2 ) ) ) ).
% cring.intro
thf(fact_504_cring__def,axiom,
( cring_3148771470849435808t_unit
= ( ^ [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
& ( comm_m1219397618491936389t_unit @ R ) ) ) ) ).
% cring_def
thf(fact_505_cring__def,axiom,
( cring_a_b
= ( ^ [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
& ( comm_m952295370001973751xt_a_b @ R ) ) ) ) ).
% cring_def
thf(fact_506_ring_Oring__simprules_I15_J,axiom,
! [R2: partia7496981018696276118t_unit,X3: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( add_se2486902527185523630t_unit @ R2 @ X3 @ ( zero_s2910681146719230829t_unit @ R2 ) )
= X3 ) ) ) ).
% ring.ring_simprules(15)
thf(fact_507_ring_Oring__simprules_I15_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ X3 @ ( zero_l4142658623432671053t_unit @ R2 ) )
= X3 ) ) ) ).
% ring.ring_simprules(15)
thf(fact_508_ring_Oring__simprules_I15_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ X3 @ ( zero_a_b @ R2 ) )
= X3 ) ) ) ).
% ring.ring_simprules(15)
thf(fact_509_ring_Oring__simprules_I8_J,axiom,
! [R2: partia7496981018696276118t_unit,X3: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( add_se2486902527185523630t_unit @ R2 @ ( zero_s2910681146719230829t_unit @ R2 ) @ X3 )
= X3 ) ) ) ).
% ring.ring_simprules(8)
thf(fact_510_ring_Oring__simprules_I8_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ ( zero_l4142658623432671053t_unit @ R2 ) @ X3 )
= X3 ) ) ) ).
% ring.ring_simprules(8)
thf(fact_511_ring_Oring__simprules_I8_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ ( zero_a_b @ R2 ) @ X3 )
= X3 ) ) ) ).
% ring.ring_simprules(8)
thf(fact_512_ring_Oring__simprules_I25_J,axiom,
! [R2: partia7496981018696276118t_unit,X3: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( mult_s7802724872828879953t_unit @ R2 @ X3 @ ( zero_s2910681146719230829t_unit @ R2 ) )
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_513_ring_Oring__simprules_I25_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ X3 @ ( zero_l4142658623432671053t_unit @ R2 ) )
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_514_ring_Oring__simprules_I25_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ X3 @ ( zero_a_b @ R2 ) )
= ( zero_a_b @ R2 ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_515_ring_Oring__simprules_I24_J,axiom,
! [R2: partia7496981018696276118t_unit,X3: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( mult_s7802724872828879953t_unit @ R2 @ ( zero_s2910681146719230829t_unit @ R2 ) @ X3 )
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_516_ring_Oring__simprules_I24_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ ( zero_l4142658623432671053t_unit @ R2 ) @ X3 )
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_517_ring_Oring__simprules_I24_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( zero_a_b @ R2 ) @ X3 )
= ( zero_a_b @ R2 ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_518_ring_Oring__simprules_I23_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ Z2 @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 ) )
= ( add_li7652885771158616974t_unit @ R2 @ ( mult_l7073676228092353617t_unit @ R2 @ Z2 @ X3 ) @ ( mult_l7073676228092353617t_unit @ R2 @ Z2 @ Y3 ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_519_ring_Oring__simprules_I23_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ Z2 @ ( add_a_b @ R2 @ X3 @ Y3 ) )
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ Z2 @ X3 ) @ ( mult_a_ring_ext_a_b @ R2 @ Z2 @ Y3 ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_520_ring_Oring__simprules_I13_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R2 @ ( mult_l7073676228092353617t_unit @ R2 @ X3 @ Z2 ) @ ( mult_l7073676228092353617t_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_521_ring_Oring__simprules_I13_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( add_a_b @ R2 @ X3 @ Y3 ) @ Z2 )
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ X3 @ Z2 ) @ ( mult_a_ring_ext_a_b @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_522_domain_Ointegral,axiom,
! [R2: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R2 )
=> ( ( ( mult_s7802724872828879953t_unit @ R2 @ A @ B )
= ( zero_s2910681146719230829t_unit @ R2 ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( A
= ( zero_s2910681146719230829t_unit @ R2 ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_523_domain_Ointegral,axiom,
! [R2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( ( mult_l7073676228092353617t_unit @ R2 @ A @ B )
= ( zero_l4142658623432671053t_unit @ R2 ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( A
= ( zero_l4142658623432671053t_unit @ R2 ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_524_domain_Ointegral,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R2 )
=> ( ( ( mult_a_ring_ext_a_b @ R2 @ A @ B )
= ( zero_a_b @ R2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( A
= ( zero_a_b @ R2 ) )
| ( B
= ( zero_a_b @ R2 ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_525_domain_Om__lcancel,axiom,
! [R2: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a,C: set_list_a] :
( ( domain1617769409708967785t_unit @ R2 )
=> ( ( A
!= ( zero_s2910681146719230829t_unit @ R2 ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R2 @ A @ B )
= ( mult_s7802724872828879953t_unit @ R2 @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_526_domain_Om__lcancel,axiom,
! [R2: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ R2 ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R2 @ A @ B )
= ( mult_l7073676228092353617t_unit @ R2 @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_527_domain_Om__lcancel,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R2 )
=> ( ( A
!= ( zero_a_b @ R2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ( mult_a_ring_ext_a_b @ R2 @ A @ B )
= ( mult_a_ring_ext_a_b @ R2 @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_528_domain_Om__rcancel,axiom,
! [R2: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a,C: set_list_a] :
( ( domain1617769409708967785t_unit @ R2 )
=> ( ( A
!= ( zero_s2910681146719230829t_unit @ R2 ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R2 @ B @ A )
= ( mult_s7802724872828879953t_unit @ R2 @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_529_domain_Om__rcancel,axiom,
! [R2: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ R2 ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R2 @ B @ A )
= ( mult_l7073676228092353617t_unit @ R2 @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_530_domain_Om__rcancel,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R2 )
=> ( ( A
!= ( zero_a_b @ R2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ( mult_a_ring_ext_a_b @ R2 @ B @ A )
= ( mult_a_ring_ext_a_b @ R2 @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_531_domain_Ointegral__iff,axiom,
! [R2: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R2 )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R2 @ A @ B )
= ( zero_s2910681146719230829t_unit @ R2 ) )
= ( ( A
= ( zero_s2910681146719230829t_unit @ R2 ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_532_domain_Ointegral__iff,axiom,
! [R2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R2 @ A @ B )
= ( zero_l4142658623432671053t_unit @ R2 ) )
= ( ( A
= ( zero_l4142658623432671053t_unit @ R2 ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_533_domain_Ointegral__iff,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ( mult_a_ring_ext_a_b @ R2 @ A @ B )
= ( zero_a_b @ R2 ) )
= ( ( A
= ( zero_a_b @ R2 ) )
| ( B
= ( zero_a_b @ R2 ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_534_cring_Ocring__simprules_I16_J,axiom,
! [R2: partia7496981018696276118t_unit,X3: set_list_a] :
( ( cring_3470013030684506304t_unit @ R2 )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( add_se2486902527185523630t_unit @ R2 @ X3 @ ( zero_s2910681146719230829t_unit @ R2 ) )
= X3 ) ) ) ).
% cring.cring_simprules(16)
thf(fact_535_cring_Ocring__simprules_I16_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ X3 @ ( zero_l4142658623432671053t_unit @ R2 ) )
= X3 ) ) ) ).
% cring.cring_simprules(16)
thf(fact_536_cring_Ocring__simprules_I16_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ X3 @ ( zero_a_b @ R2 ) )
= X3 ) ) ) ).
% cring.cring_simprules(16)
thf(fact_537_cring_Ocring__simprules_I8_J,axiom,
! [R2: partia7496981018696276118t_unit,X3: set_list_a] :
( ( cring_3470013030684506304t_unit @ R2 )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( add_se2486902527185523630t_unit @ R2 @ ( zero_s2910681146719230829t_unit @ R2 ) @ X3 )
= X3 ) ) ) ).
% cring.cring_simprules(8)
thf(fact_538_cring_Ocring__simprules_I8_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ ( zero_l4142658623432671053t_unit @ R2 ) @ X3 )
= X3 ) ) ) ).
% cring.cring_simprules(8)
thf(fact_539_cring_Ocring__simprules_I8_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ ( zero_a_b @ R2 ) @ X3 )
= X3 ) ) ) ).
% cring.cring_simprules(8)
thf(fact_540_cring_Ocring__simprules_I27_J,axiom,
! [R2: partia7496981018696276118t_unit,X3: set_list_a] :
( ( cring_3470013030684506304t_unit @ R2 )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( mult_s7802724872828879953t_unit @ R2 @ X3 @ ( zero_s2910681146719230829t_unit @ R2 ) )
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ).
% cring.cring_simprules(27)
thf(fact_541_cring_Ocring__simprules_I27_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ X3 @ ( zero_l4142658623432671053t_unit @ R2 ) )
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ).
% cring.cring_simprules(27)
thf(fact_542_cring_Ocring__simprules_I27_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ X3 @ ( zero_a_b @ R2 ) )
= ( zero_a_b @ R2 ) ) ) ) ).
% cring.cring_simprules(27)
thf(fact_543_cring_Ocring__simprules_I26_J,axiom,
! [R2: partia7496981018696276118t_unit,X3: set_list_a] :
( ( cring_3470013030684506304t_unit @ R2 )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( mult_s7802724872828879953t_unit @ R2 @ ( zero_s2910681146719230829t_unit @ R2 ) @ X3 )
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ).
% cring.cring_simprules(26)
thf(fact_544_cring_Ocring__simprules_I26_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ ( zero_l4142658623432671053t_unit @ R2 ) @ X3 )
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ).
% cring.cring_simprules(26)
thf(fact_545_cring_Ocring__simprules_I26_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( zero_a_b @ R2 ) @ X3 )
= ( zero_a_b @ R2 ) ) ) ) ).
% cring.cring_simprules(26)
thf(fact_546_cring_Ocring__simprules_I25_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ Z2 @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 ) )
= ( add_li7652885771158616974t_unit @ R2 @ ( mult_l7073676228092353617t_unit @ R2 @ Z2 @ X3 ) @ ( mult_l7073676228092353617t_unit @ R2 @ Z2 @ Y3 ) ) ) ) ) ) ) ).
% cring.cring_simprules(25)
thf(fact_547_cring_Ocring__simprules_I25_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ Z2 @ ( add_a_b @ R2 @ X3 @ Y3 ) )
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ Z2 @ X3 ) @ ( mult_a_ring_ext_a_b @ R2 @ Z2 @ Y3 ) ) ) ) ) ) ) ).
% cring.cring_simprules(25)
thf(fact_548_cring_Ocring__simprules_I13_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R2 @ ( mult_l7073676228092353617t_unit @ R2 @ X3 @ Z2 ) @ ( mult_l7073676228092353617t_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% cring.cring_simprules(13)
thf(fact_549_cring_Ocring__simprules_I13_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( add_a_b @ R2 @ X3 @ Y3 ) @ Z2 )
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ X3 @ Z2 ) @ ( mult_a_ring_ext_a_b @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% cring.cring_simprules(13)
thf(fact_550_abelian__groupE_I6_J,axiom,
! [R2: partia7496981018696276118t_unit,X3: set_list_a] :
( ( abelia5304159692179083286t_unit @ R2 )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ? [X: set_list_a] :
( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R2 ) )
& ( ( add_se2486902527185523630t_unit @ R2 @ X @ X3 )
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_551_abelian__groupE_I6_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia3891852623213500406t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ? [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
& ( ( add_li7652885771158616974t_unit @ R2 @ X @ X3 )
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_552_abelian__groupE_I6_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_group_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
& ( ( add_a_b @ R2 @ X @ X3 )
= ( zero_a_b @ R2 ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_553_abelian__groupE_I5_J,axiom,
! [R2: partia7496981018696276118t_unit,X3: set_list_a] :
( ( abelia5304159692179083286t_unit @ R2 )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( add_se2486902527185523630t_unit @ R2 @ ( zero_s2910681146719230829t_unit @ R2 ) @ X3 )
= X3 ) ) ) ).
% abelian_groupE(5)
thf(fact_554_abelian__groupE_I5_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia3891852623213500406t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ ( zero_l4142658623432671053t_unit @ R2 ) @ X3 )
= X3 ) ) ) ).
% abelian_groupE(5)
thf(fact_555_abelian__groupE_I5_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_group_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ ( zero_a_b @ R2 ) @ X3 )
= X3 ) ) ) ).
% abelian_groupE(5)
thf(fact_556_abelian__groupI,axiom,
! [R2: partia7496981018696276118t_unit] :
( ! [X: set_list_a] :
( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R2 ) )
=> ! [Y2: set_list_a] :
( ( member_set_list_a @ Y2 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( member_set_list_a @ ( add_se2486902527185523630t_unit @ R2 @ X @ Y2 ) @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R2 ) @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ! [X: set_list_a] :
( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R2 ) )
=> ! [Y2: set_list_a] :
( ( member_set_list_a @ Y2 @ ( partia141011252114345353t_unit @ R2 ) )
=> ! [Z: set_list_a] :
( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( add_se2486902527185523630t_unit @ R2 @ ( add_se2486902527185523630t_unit @ R2 @ X @ Y2 ) @ Z )
= ( add_se2486902527185523630t_unit @ R2 @ X @ ( add_se2486902527185523630t_unit @ R2 @ Y2 @ Z ) ) ) ) ) )
=> ( ! [X: set_list_a] :
( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R2 ) )
=> ! [Y2: set_list_a] :
( ( member_set_list_a @ Y2 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( add_se2486902527185523630t_unit @ R2 @ X @ Y2 )
= ( add_se2486902527185523630t_unit @ R2 @ Y2 @ X ) ) ) )
=> ( ! [X: set_list_a] :
( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( add_se2486902527185523630t_unit @ R2 @ ( zero_s2910681146719230829t_unit @ R2 ) @ X )
= X ) )
=> ( ! [X: set_list_a] :
( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R2 ) )
=> ? [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ ( partia141011252114345353t_unit @ R2 ) )
& ( ( add_se2486902527185523630t_unit @ R2 @ Xa @ X )
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) )
=> ( abelia5304159692179083286t_unit @ R2 ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_557_abelian__groupI,axiom,
! [R2: partia2670972154091845814t_unit] :
( ! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ! [Y2: list_a] :
( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R2 @ X @ Y2 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( member_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ! [Y2: list_a] :
( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ! [Z: list_a] :
( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ ( add_li7652885771158616974t_unit @ R2 @ X @ Y2 ) @ Z )
= ( add_li7652885771158616974t_unit @ R2 @ X @ ( add_li7652885771158616974t_unit @ R2 @ Y2 @ Z ) ) ) ) ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ! [Y2: list_a] :
( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ X @ Y2 )
= ( add_li7652885771158616974t_unit @ R2 @ Y2 @ X ) ) ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ ( zero_l4142658623432671053t_unit @ R2 ) @ X )
= X ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ ( partia5361259788508890537t_unit @ R2 ) )
& ( ( add_li7652885771158616974t_unit @ R2 @ Xa @ X )
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) )
=> ( abelia3891852623213500406t_unit @ R2 ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_558_abelian__groupI,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ! [Y2: a] :
( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ ( add_a_b @ R2 @ X @ Y2 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R2 ) @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ! [Y2: a] :
( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ! [Z: a] :
( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ ( add_a_b @ R2 @ X @ Y2 ) @ Z )
= ( add_a_b @ R2 @ X @ ( add_a_b @ R2 @ Y2 @ Z ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ! [Y2: a] :
( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ X @ Y2 )
= ( add_a_b @ R2 @ Y2 @ X ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ ( zero_a_b @ R2 ) @ X )
= X ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ R2 ) )
& ( ( add_a_b @ R2 @ Xa @ X )
= ( zero_a_b @ R2 ) ) ) )
=> ( abelian_group_a_b @ R2 ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_559_monoid_Oprod__unit__l,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ) ) ).
% monoid.prod_unit_l
thf(fact_560_monoid_Oprod__unit__l,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ) ).
% monoid.prod_unit_l
thf(fact_561_monoid_Oprod__unit__l,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ ( mult_a_Product_unit @ G @ A @ B ) @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ A @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ B @ ( units_a_Product_unit @ G ) ) ) ) ) ) ) ).
% monoid.prod_unit_l
thf(fact_562_monoid_Oprod__unit__r,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ) ) ).
% monoid.prod_unit_r
thf(fact_563_monoid_Oprod__unit__r,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ) ).
% monoid.prod_unit_r
thf(fact_564_monoid_Oprod__unit__r,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ ( mult_a_Product_unit @ G @ A @ B ) @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ B @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ A @ ( units_a_Product_unit @ G ) ) ) ) ) ) ) ).
% monoid.prod_unit_r
thf(fact_565_Ring_Ointegral,axiom,
! [R2: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( field_26233345952514695t_unit @ R2 )
=> ( ( ( mult_s7802724872828879953t_unit @ R2 @ A @ B )
= ( zero_s2910681146719230829t_unit @ R2 ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( A
= ( zero_s2910681146719230829t_unit @ R2 ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_566_Ring_Ointegral,axiom,
! [R2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( ( ( mult_l7073676228092353617t_unit @ R2 @ A @ B )
= ( zero_l4142658623432671053t_unit @ R2 ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( A
= ( zero_l4142658623432671053t_unit @ R2 ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_567_Ring_Ointegral,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( field_a_b @ R2 )
=> ( ( ( mult_a_ring_ext_a_b @ R2 @ A @ B )
= ( zero_a_b @ R2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( A
= ( zero_a_b @ R2 ) )
| ( B
= ( zero_a_b @ R2 ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_568_monoid_Omult__cong__r,axiom,
! [G: partia2670972154091845814t_unit,B: list_a,B3: list_a,A: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ B @ B3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( associ8407585678920448409t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) @ ( mult_l7073676228092353617t_unit @ G @ A @ B3 ) ) ) ) ) ) ) ).
% monoid.mult_cong_r
thf(fact_569_monoid_Omult__cong__r,axiom,
! [G: partia2175431115845679010xt_a_b,B: a,B3: a,A: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ B @ B3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( mult_a_ring_ext_a_b @ G @ A @ B3 ) ) ) ) ) ) ) ).
% monoid.mult_cong_r
thf(fact_570_monoid_Omult__cong__r,axiom,
! [G: partia8223610829204095565t_unit,B: a,B3: a,A: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ B @ B3 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ ( mult_a_Product_unit @ G @ A @ B ) @ ( mult_a_Product_unit @ G @ A @ B3 ) ) ) ) ) ) ) ).
% monoid.mult_cong_r
thf(fact_571_comm__monoid_Ounit__factor,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( comm_m1219397618491936389t_unit @ G )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ) ).
% comm_monoid.unit_factor
thf(fact_572_comm__monoid_Ounit__factor,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( comm_m952295370001973751xt_a_b @ G )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% comm_monoid.unit_factor
thf(fact_573_comm__monoid_Ounit__factor,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( member_a @ ( mult_a_Product_unit @ G @ A @ B ) @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ A @ ( units_a_Product_unit @ G ) ) ) ) ) ) ).
% comm_monoid.unit_factor
thf(fact_574_abelian__monoidE_I4_J,axiom,
! [R2: partia7496981018696276118t_unit,X3: set_list_a] :
( ( abelia3322010900105369177t_unit @ R2 )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( add_se2486902527185523630t_unit @ R2 @ ( zero_s2910681146719230829t_unit @ R2 ) @ X3 )
= X3 ) ) ) ).
% abelian_monoidE(4)
thf(fact_575_abelian__monoidE_I4_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia226231641709521465t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ ( zero_l4142658623432671053t_unit @ R2 ) @ X3 )
= X3 ) ) ) ).
% abelian_monoidE(4)
thf(fact_576_abelian__monoidE_I4_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_monoid_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ ( zero_a_b @ R2 ) @ X3 )
= X3 ) ) ) ).
% abelian_monoidE(4)
thf(fact_577_abelian__monoid_Ol__zero,axiom,
! [G: partia7496981018696276118t_unit,X3: set_list_a] :
( ( abelia3322010900105369177t_unit @ G )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ G ) )
=> ( ( add_se2486902527185523630t_unit @ G @ ( zero_s2910681146719230829t_unit @ G ) @ X3 )
= X3 ) ) ) ).
% abelian_monoid.l_zero
thf(fact_578_abelian__monoid_Ol__zero,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( zero_l4142658623432671053t_unit @ G ) @ X3 )
= X3 ) ) ) ).
% abelian_monoid.l_zero
thf(fact_579_abelian__monoid_Ol__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( zero_a_b @ G ) @ X3 )
= X3 ) ) ) ).
% abelian_monoid.l_zero
thf(fact_580_abelian__monoid_Or__zero,axiom,
! [G: partia7496981018696276118t_unit,X3: set_list_a] :
( ( abelia3322010900105369177t_unit @ G )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ G ) )
=> ( ( add_se2486902527185523630t_unit @ G @ X3 @ ( zero_s2910681146719230829t_unit @ G ) )
= X3 ) ) ) ).
% abelian_monoid.r_zero
thf(fact_581_abelian__monoid_Or__zero,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X3 @ ( zero_l4142658623432671053t_unit @ G ) )
= X3 ) ) ) ).
% abelian_monoid.r_zero
thf(fact_582_abelian__monoid_Or__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X3 @ ( zero_a_b @ G ) )
= X3 ) ) ) ).
% abelian_monoid.r_zero
thf(fact_583_abelian__monoid_Ominus__unique,axiom,
! [G: partia7496981018696276118t_unit,Y3: set_list_a,X3: set_list_a,Y4: set_list_a] :
( ( abelia3322010900105369177t_unit @ G )
=> ( ( ( add_se2486902527185523630t_unit @ G @ Y3 @ X3 )
= ( zero_s2910681146719230829t_unit @ G ) )
=> ( ( ( add_se2486902527185523630t_unit @ G @ X3 @ Y4 )
= ( zero_s2910681146719230829t_unit @ G ) )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ G ) )
=> ( ( member_set_list_a @ Y3 @ ( partia141011252114345353t_unit @ G ) )
=> ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ G ) )
=> ( Y3 = Y4 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_584_abelian__monoid_Ominus__unique,axiom,
! [G: partia2670972154091845814t_unit,Y3: list_a,X3: list_a,Y4: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( ( add_li7652885771158616974t_unit @ G @ Y3 @ X3 )
= ( zero_l4142658623432671053t_unit @ G ) )
=> ( ( ( add_li7652885771158616974t_unit @ G @ X3 @ Y4 )
= ( zero_l4142658623432671053t_unit @ G ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ G ) )
=> ( Y3 = Y4 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_585_abelian__monoid_Ominus__unique,axiom,
! [G: partia2175431115845679010xt_a_b,Y3: a,X3: a,Y4: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( ( add_a_b @ G @ Y3 @ X3 )
= ( zero_a_b @ G ) )
=> ( ( ( add_a_b @ G @ X3 @ Y4 )
= ( zero_a_b @ G ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( Y3 = Y4 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_586_abelian__monoidI,axiom,
! [R2: partia7496981018696276118t_unit] :
( ! [X: set_list_a,Y2: set_list_a] :
( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( member_set_list_a @ Y2 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( member_set_list_a @ ( add_se2486902527185523630t_unit @ R2 @ X @ Y2 ) @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R2 ) @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ! [X: set_list_a,Y2: set_list_a,Z: set_list_a] :
( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( member_set_list_a @ Y2 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( add_se2486902527185523630t_unit @ R2 @ ( add_se2486902527185523630t_unit @ R2 @ X @ Y2 ) @ Z )
= ( add_se2486902527185523630t_unit @ R2 @ X @ ( add_se2486902527185523630t_unit @ R2 @ Y2 @ Z ) ) ) ) ) )
=> ( ! [X: set_list_a] :
( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( add_se2486902527185523630t_unit @ R2 @ ( zero_s2910681146719230829t_unit @ R2 ) @ X )
= X ) )
=> ( ! [X: set_list_a,Y2: set_list_a] :
( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( member_set_list_a @ Y2 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( add_se2486902527185523630t_unit @ R2 @ X @ Y2 )
= ( add_se2486902527185523630t_unit @ R2 @ Y2 @ X ) ) ) )
=> ( abelia3322010900105369177t_unit @ R2 ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_587_abelian__monoidI,axiom,
! [R2: partia2670972154091845814t_unit] :
( ! [X: list_a,Y2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R2 @ X @ Y2 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( member_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ! [X: list_a,Y2: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ ( add_li7652885771158616974t_unit @ R2 @ X @ Y2 ) @ Z )
= ( add_li7652885771158616974t_unit @ R2 @ X @ ( add_li7652885771158616974t_unit @ R2 @ Y2 @ Z ) ) ) ) ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ ( zero_l4142658623432671053t_unit @ R2 ) @ X )
= X ) )
=> ( ! [X: list_a,Y2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ X @ Y2 )
= ( add_li7652885771158616974t_unit @ R2 @ Y2 @ X ) ) ) )
=> ( abelia226231641709521465t_unit @ R2 ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_588_abelian__monoidI,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ! [X: a,Y2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ ( add_a_b @ R2 @ X @ Y2 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R2 ) @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ ( add_a_b @ R2 @ X @ Y2 ) @ Z )
= ( add_a_b @ R2 @ X @ ( add_a_b @ R2 @ Y2 @ Z ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ ( zero_a_b @ R2 ) @ X )
= X ) )
=> ( ! [X: a,Y2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ X @ Y2 )
= ( add_a_b @ R2 @ Y2 @ X ) ) ) )
=> ( abelian_monoid_a_b @ R2 ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_589_comm__monoid_Omult__cong__l,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,A5: list_a,B: list_a] :
( ( comm_m1219397618491936389t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A @ A5 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( associ8407585678920448409t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) @ ( mult_l7073676228092353617t_unit @ G @ A5 @ B ) ) ) ) ) ) ) ).
% comm_monoid.mult_cong_l
thf(fact_590_comm__monoid_Omult__cong__l,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,A5: a,B: a] :
( ( comm_m952295370001973751xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ A5 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( mult_a_ring_ext_a_b @ G @ A5 @ B ) ) ) ) ) ) ) ).
% comm_monoid.mult_cong_l
thf(fact_591_comm__monoid_Omult__cong__l,axiom,
! [G: partia8223610829204095565t_unit,A: a,A5: a,B: a] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A @ A5 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A5 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( associ6879500422977059064t_unit @ G @ ( mult_a_Product_unit @ G @ A @ B ) @ ( mult_a_Product_unit @ G @ A5 @ B ) ) ) ) ) ) ) ).
% comm_monoid.mult_cong_l
thf(fact_592_semiring_Osemiring__simprules_I11_J,axiom,
! [R2: partia7496981018696276118t_unit,X3: set_list_a] :
( ( semiri4000464634269493571t_unit @ R2 )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( add_se2486902527185523630t_unit @ R2 @ X3 @ ( zero_s2910681146719230829t_unit @ R2 ) )
= X3 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_593_semiring_Osemiring__simprules_I11_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ X3 @ ( zero_l4142658623432671053t_unit @ R2 ) )
= X3 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_594_semiring_Osemiring__simprules_I11_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ X3 @ ( zero_a_b @ R2 ) )
= X3 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_595_semiring_Osemiring__simprules_I6_J,axiom,
! [R2: partia7496981018696276118t_unit,X3: set_list_a] :
( ( semiri4000464634269493571t_unit @ R2 )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( add_se2486902527185523630t_unit @ R2 @ ( zero_s2910681146719230829t_unit @ R2 ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_596_semiring_Osemiring__simprules_I6_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( add_li7652885771158616974t_unit @ R2 @ ( zero_l4142658623432671053t_unit @ R2 ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_597_semiring_Osemiring__simprules_I6_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( add_a_b @ R2 @ ( zero_a_b @ R2 ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_598_comm__monoid_OUnits__cong,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( comm_m1219397618491936389t_unit @ G )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ) ).
% comm_monoid.Units_cong
thf(fact_599_comm__monoid_OUnits__cong,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( comm_m952295370001973751xt_a_b @ G )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% comm_monoid.Units_cong
thf(fact_600_comm__monoid_OUnits__cong,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( member_a @ A @ ( units_a_Product_unit @ G ) )
=> ( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ B @ ( units_a_Product_unit @ G ) ) ) ) ) ) ).
% comm_monoid.Units_cong
thf(fact_601_semiring_Ol__null,axiom,
! [R2: partia7496981018696276118t_unit,X3: set_list_a] :
( ( semiri4000464634269493571t_unit @ R2 )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( mult_s7802724872828879953t_unit @ R2 @ ( zero_s2910681146719230829t_unit @ R2 ) @ X3 )
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ).
% semiring.l_null
thf(fact_602_semiring_Ol__null,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ ( zero_l4142658623432671053t_unit @ R2 ) @ X3 )
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ).
% semiring.l_null
thf(fact_603_semiring_Ol__null,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( zero_a_b @ R2 ) @ X3 )
= ( zero_a_b @ R2 ) ) ) ) ).
% semiring.l_null
thf(fact_604_semiring_Or__null,axiom,
! [R2: partia7496981018696276118t_unit,X3: set_list_a] :
( ( semiri4000464634269493571t_unit @ R2 )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( mult_s7802724872828879953t_unit @ R2 @ X3 @ ( zero_s2910681146719230829t_unit @ R2 ) )
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ).
% semiring.r_null
thf(fact_605_semiring_Or__null,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ X3 @ ( zero_l4142658623432671053t_unit @ R2 ) )
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ).
% semiring.r_null
thf(fact_606_semiring_Or__null,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ X3 @ ( zero_a_b @ R2 ) )
= ( zero_a_b @ R2 ) ) ) ) ).
% semiring.r_null
thf(fact_607_semiring_Ol__distr,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R2 @ ( mult_l7073676228092353617t_unit @ R2 @ X3 @ Z2 ) @ ( mult_l7073676228092353617t_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_608_semiring_Ol__distr,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( semiring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( add_a_b @ R2 @ X3 @ Y3 ) @ Z2 )
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ X3 @ Z2 ) @ ( mult_a_ring_ext_a_b @ R2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_609_semiring_Or__distr,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ Z2 @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y3 ) )
= ( add_li7652885771158616974t_unit @ R2 @ ( mult_l7073676228092353617t_unit @ R2 @ Z2 @ X3 ) @ ( mult_l7073676228092353617t_unit @ R2 @ Z2 @ Y3 ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_610_semiring_Or__distr,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( semiring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ Z2 @ ( add_a_b @ R2 @ X3 @ Y3 ) )
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ Z2 @ X3 ) @ ( mult_a_ring_ext_a_b @ R2 @ Z2 @ Y3 ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_611_const__term__simprules__shell_I3_J,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( const_term_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ Q ) )
= ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ) ).
% const_term_simprules_shell(3)
thf(fact_612_const__term__simprules__shell_I2_J,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( const_term_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ) ).
% const_term_simprules_shell(2)
thf(fact_613_subdomainI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subdomain_a_b @ H @ r ) ) ).
% subdomainI'
thf(fact_614_cring_Ocgenideal__is__principalideal,axiom,
! [R2: partia2670972154091845814t_unit,I: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ R2 @ I ) @ R2 ) ) ) ).
% cring.cgenideal_is_principalideal
thf(fact_615_cring_Ocgenideal__is__principalideal,axiom,
! [R2: partia2175431115845679010xt_a_b,I: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ R2 @ I ) @ R2 ) ) ) ).
% cring.cgenideal_is_principalideal
thf(fact_616_monoid__cancelI,axiom,
( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A2 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A2 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B2 @ C2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_617_pdivides__imp__root__sharing,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( eval_a_b @ r @ P @ A )
= ( zero_a_b @ r ) )
=> ( ( eval_a_b @ r @ Q @ A )
= ( zero_a_b @ r ) ) ) ) ) ) ).
% pdivides_imp_root_sharing
thf(fact_618_monoid_Omonoid__comm__monoidI,axiom,
! [G: partia2670972154091845814t_unit] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ! [X: list_a,Y2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ X @ Y2 )
= ( mult_l7073676228092353617t_unit @ G @ Y2 @ X ) ) ) )
=> ( comm_m1219397618491936389t_unit @ G ) ) ) ).
% monoid.monoid_comm_monoidI
thf(fact_619_monoid_Omonoid__comm__monoidI,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ! [X: a,Y2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X @ Y2 )
= ( mult_a_ring_ext_a_b @ G @ Y2 @ X ) ) ) )
=> ( comm_m952295370001973751xt_a_b @ G ) ) ) ).
% monoid.monoid_comm_monoidI
thf(fact_620_monoid_Omonoid__comm__monoidI,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ! [X: a,Y2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ X @ Y2 )
= ( mult_a_Product_unit @ G @ Y2 @ X ) ) ) )
=> ( comm_m7681468956318391052t_unit @ G ) ) ) ).
% monoid.monoid_comm_monoidI
thf(fact_621_ring_Or__right__minus__eq,axiom,
! [R2: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( ( a_minu2642007939804611572t_unit @ R2 @ A @ B )
= ( zero_s2910681146719230829t_unit @ R2 ) )
= ( A = B ) ) ) ) ) ).
% ring.r_right_minus_eq
thf(fact_622_ring_Or__right__minus__eq,axiom,
! [R2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ( a_minu3984020753470702548t_unit @ R2 @ A @ B )
= ( zero_l4142658623432671053t_unit @ R2 ) )
= ( A = B ) ) ) ) ) ).
% ring.r_right_minus_eq
thf(fact_623_ring_Or__right__minus__eq,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ( a_minus_a_b @ R2 @ A @ B )
= ( zero_a_b @ R2 ) )
= ( A = B ) ) ) ) ) ).
% ring.r_right_minus_eq
thf(fact_624_comm__monoid_Onat__pow__distrib,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,N: nat] :
( ( comm_m1219397618491936389t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( pow_li1142815632869257134it_nat @ G @ ( mult_l7073676228092353617t_unit @ G @ X3 @ Y3 ) @ N )
= ( mult_l7073676228092353617t_unit @ G @ ( pow_li1142815632869257134it_nat @ G @ X3 @ N ) @ ( pow_li1142815632869257134it_nat @ G @ Y3 @ N ) ) ) ) ) ) ).
% comm_monoid.nat_pow_distrib
thf(fact_625_comm__monoid_Onat__pow__distrib,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a,N: nat] :
( ( comm_m952295370001973751xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( pow_a_1026414303147256608_b_nat @ G @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y3 ) @ N )
= ( mult_a_ring_ext_a_b @ G @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ G @ Y3 @ N ) ) ) ) ) ) ).
% comm_monoid.nat_pow_distrib
thf(fact_626_comm__monoid_Onat__pow__distrib,axiom,
! [G: partia8223610829204095565t_unit,X3: a,Y3: a,N: nat] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( pow_a_1875594501834816709it_nat @ G @ ( mult_a_Product_unit @ G @ X3 @ Y3 ) @ N )
= ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X3 @ N ) @ ( pow_a_1875594501834816709it_nat @ G @ Y3 @ N ) ) ) ) ) ) ).
% comm_monoid.nat_pow_distrib
thf(fact_627_divides__trans,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( factor8216151070175719842xt_a_b @ r @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ C ) ) ) ) ).
% divides_trans
thf(fact_628_zero__divides,axiom,
! [A: a] :
( ( factor8216151070175719842xt_a_b @ r @ ( zero_a_b @ r ) @ A )
= ( A
= ( zero_a_b @ r ) ) ) ).
% zero_divides
thf(fact_629_divides__zero,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( zero_a_b @ r ) ) ) ).
% divides_zero
thf(fact_630_local_Odivides__mult,axiom,
! [A: a,C: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% local.divides_mult
thf(fact_631_divides__prod__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ) ).
% divides_prod_l
thf(fact_632_divides__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ).
% divides_prod_r
thf(fact_633_divides__unit,axiom,
! [A: a,U: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ U )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% divides_unit
thf(fact_634_unit__divides,axiom,
! [U: a,A: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ U @ A ) ) ) ).
% unit_divides
thf(fact_635_divides__cong__l,axiom,
! [X3: a,X4: a,Y3: a] :
( ( associ5860276527279195403xt_a_b @ r @ X3 @ X4 )
=> ( ( factor8216151070175719842xt_a_b @ r @ X4 @ Y3 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ X3 @ Y3 ) ) ) ) ).
% divides_cong_l
thf(fact_636_divides__cong__r,axiom,
! [X3: a,Y3: a,Y4: a] :
( ( factor8216151070175719842xt_a_b @ r @ X3 @ Y3 )
=> ( ( associ5860276527279195403xt_a_b @ r @ Y3 @ Y4 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ X3 @ Y4 ) ) ) ) ).
% divides_cong_r
thf(fact_637_univ__poly__not__field,axiom,
! [K2: set_a] :
( ( subring_a_b @ K2 @ r )
=> ~ ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% univ_poly_not_field
thf(fact_638_univ__poly__is__abelian__monoid,axiom,
! [K2: set_a] :
( ( subring_a_b @ K2 @ r )
=> ( abelia226231641709521465t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% univ_poly_is_abelian_monoid
thf(fact_639_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_640_const__term__not__zero,axiom,
! [P: list_a] :
( ( ( const_term_a_b @ r @ P )
!= ( zero_a_b @ r ) )
=> ( P != nil_a ) ) ).
% const_term_not_zero
thf(fact_641_eval__var,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X3 )
= X3 ) ) ).
% eval_var
thf(fact_642_const__term__def,axiom,
! [P: list_a] :
( ( const_term_a_b @ r @ P )
= ( eval_a_b @ r @ P @ ( zero_a_b @ r ) ) ) ).
% const_term_def
thf(fact_643_const__term__simprules__shell_I1_J,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( member_a @ ( const_term_a_b @ r @ P ) @ K2 ) ) ) ).
% const_term_simprules_shell(1)
thf(fact_644_is__root__def,axiom,
! [P: list_a,X3: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P @ X3 )
= ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P @ X3 )
= ( zero_a_b @ r ) )
& ( P != nil_a ) ) ) ).
% is_root_def
thf(fact_645_divides__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ A ) ) ).
% divides_refl
thf(fact_646_minus__closed,axiom,
! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X3 @ Y3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_647_r__right__minus__eq,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( A = B ) ) ) ) ).
% r_right_minus_eq
thf(fact_648_divides__mult__lI,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% divides_mult_lI
thf(fact_649_divides__mult__rI,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% divides_mult_rI
thf(fact_650_monoid__cancel_Ois__monoid__cancel,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( monoid5798828371819920185xt_a_b @ G ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_651_monoid__cancel_Ois__monoid__cancel,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid1999574367301118026t_unit @ G )
=> ( monoid1999574367301118026t_unit @ G ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_652_ring_Oeval_Ocong,axiom,
eval_a_b = eval_a_b ).
% ring.eval.cong
thf(fact_653_ring_Oconst__term_Ocong,axiom,
const_term_a_b = const_term_a_b ).
% ring.const_term.cong
thf(fact_654_ring_Oconst__term__def,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( const_6738166269504826821t_unit @ R2 @ P )
= ( eval_l34571156754992824t_unit @ R2 @ P @ ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ).
% ring.const_term_def
thf(fact_655_ring_Oconst__term__def,axiom,
! [R2: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( const_3308765751713425893t_unit @ R2 @ P )
= ( eval_s5133945360527818456t_unit @ R2 @ P @ ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ).
% ring.const_term_def
thf(fact_656_ring_Oconst__term__def,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R2 )
=> ( ( const_term_a_b @ R2 @ P )
= ( eval_a_b @ R2 @ P @ ( zero_a_b @ R2 ) ) ) ) ).
% ring.const_term_def
thf(fact_657_monoid__cancel_Oaxioms_I1_J,axiom,
! [G: partia2670972154091845814t_unit] :
( ( monoid4303264861975686087t_unit @ G )
=> ( monoid5589397312508706001t_unit @ G ) ) ).
% monoid_cancel.axioms(1)
thf(fact_658_monoid__cancel_Oaxioms_I1_J,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( monoid8385113658579753027xt_a_b @ G ) ) ).
% monoid_cancel.axioms(1)
thf(fact_659_monoid__cancel_Oaxioms_I1_J,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid1999574367301118026t_unit @ G )
=> ( monoid2746444814143937472t_unit @ G ) ) ).
% monoid_cancel.axioms(1)
thf(fact_660_monoid__cancel_Or__cancel,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,C: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ A @ C )
= ( mult_l7073676228092353617t_unit @ G @ B @ C ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_661_monoid__cancel_Or__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,C: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ A @ C )
= ( mult_a_ring_ext_a_b @ G @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_662_monoid__cancel_Or__cancel,axiom,
! [G: partia8223610829204095565t_unit,A: a,C: a,B: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ A @ C )
= ( mult_a_Product_unit @ G @ B @ C ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_663_monoid__cancel_Ol__cancel,axiom,
! [G: partia2670972154091845814t_unit,C: list_a,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ C @ A )
= ( mult_l7073676228092353617t_unit @ G @ C @ B ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_664_monoid__cancel_Ol__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,C: a,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ C @ A )
= ( mult_a_ring_ext_a_b @ G @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_665_monoid__cancel_Ol__cancel,axiom,
! [G: partia8223610829204095565t_unit,C: a,A: a,B: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ C @ A )
= ( mult_a_Product_unit @ G @ C @ B ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_666_ring_Oeval_Osimps_I1_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( eval_l34571156754992824t_unit @ R2 @ nil_list_a )
= ( ^ [Uu: list_a] : ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ).
% ring.eval.simps(1)
thf(fact_667_ring_Oeval_Osimps_I1_J,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( eval_s5133945360527818456t_unit @ R2 @ nil_set_list_a )
= ( ^ [Uu: set_list_a] : ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ).
% ring.eval.simps(1)
thf(fact_668_ring_Oeval_Osimps_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R2 )
=> ( ( eval_a_b @ R2 @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ R2 ) ) ) ) ).
% ring.eval.simps(1)
thf(fact_669_domain_Ouniv__poly__not__field,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ~ ( field_1861437471013600865t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) ) ) ).
% domain.univ_poly_not_field
thf(fact_670_domain_Ouniv__poly__not__field,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ~ ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ) ).
% domain.univ_poly_not_field
thf(fact_671_ring_Oeval__var,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( eval_l34571156754992824t_unit @ R2 @ ( var_li8453953174693405341t_unit @ R2 ) @ X3 )
= X3 ) ) ) ).
% ring.eval_var
thf(fact_672_ring_Oeval__var,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( eval_a_b @ R2 @ ( var_a_b @ R2 ) @ X3 )
= X3 ) ) ) ).
% ring.eval_var
thf(fact_673_domain_Ouniv__poly__is__abelian__monoid,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( abelia3641329199688042803t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) ) ) ).
% domain.univ_poly_is_abelian_monoid
thf(fact_674_domain_Ouniv__poly__is__abelian__monoid,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( abelia226231641709521465t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) ) ) ).
% domain.univ_poly_is_abelian_monoid
thf(fact_675_ring_Oconst__term__not__zero,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( ( const_6738166269504826821t_unit @ R2 @ P )
!= ( zero_l4142658623432671053t_unit @ R2 ) )
=> ( P != nil_list_a ) ) ) ).
% ring.const_term_not_zero
thf(fact_676_ring_Oconst__term__not__zero,axiom,
! [R2: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( ( const_3308765751713425893t_unit @ R2 @ P )
!= ( zero_s2910681146719230829t_unit @ R2 ) )
=> ( P != nil_set_list_a ) ) ) ).
% ring.const_term_not_zero
thf(fact_677_ring_Oconst__term__not__zero,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R2 )
=> ( ( ( const_term_a_b @ R2 @ P )
!= ( zero_a_b @ R2 ) )
=> ( P != nil_a ) ) ) ).
% ring.const_term_not_zero
thf(fact_678_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,B3: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( associ8407585678920448409t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) @ ( mult_l7073676228092353617t_unit @ G @ A @ B3 ) )
=> ( associ8407585678920448409t_unit @ G @ B @ B3 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_679_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,B3: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( mult_a_ring_ext_a_b @ G @ A @ B3 ) )
=> ( associ5860276527279195403xt_a_b @ G @ B @ B3 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_680_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,B3: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( associ6879500422977059064t_unit @ G @ ( mult_a_Product_unit @ G @ A @ B ) @ ( mult_a_Product_unit @ G @ A @ B3 ) )
=> ( associ6879500422977059064t_unit @ G @ B @ B3 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_681_monoid__cancel_Odivides__mult__l,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( factor1757716651909850160t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ C @ A ) @ ( mult_l7073676228092353617t_unit @ G @ C @ B ) )
= ( factor1757716651909850160t_unit @ G @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_682_monoid__cancel_Odivides__mult__l,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ C @ A ) @ ( mult_a_ring_ext_a_b @ G @ C @ B ) )
= ( factor8216151070175719842xt_a_b @ G @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_683_monoid__cancel_Odivides__mult__l,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ ( mult_a_Product_unit @ G @ C @ A ) @ ( mult_a_Product_unit @ G @ C @ B ) )
= ( factor3040189038382604065t_unit @ G @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_684_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_685_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_686_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( member_a @ A @ ( units_a_Product_unit @ G ) )
=> ( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ B @ ( units_a_Product_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_687_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_688_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_689_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ( ( member_a @ B @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ A @ ( units_a_Product_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_690_monoid_Omonoid__cancelI,axiom,
! [G: partia2670972154091845814t_unit] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ! [A2: list_a,B2: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ G @ C2 @ A2 )
= ( mult_l7073676228092353617t_unit @ G @ C2 @ B2 ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: list_a,B2: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ G @ A2 @ C2 )
= ( mult_l7073676228092353617t_unit @ G @ B2 @ C2 ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ G ) ) ) ) ).
% monoid.monoid_cancelI
thf(fact_691_monoid_Omonoid__cancelI,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ G @ C2 @ A2 )
= ( mult_a_ring_ext_a_b @ G @ C2 @ B2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ G @ A2 @ C2 )
= ( mult_a_ring_ext_a_b @ G @ B2 @ C2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ G ) ) ) ) ).
% monoid.monoid_cancelI
thf(fact_692_monoid_Omonoid__cancelI,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_Product_unit @ G @ C2 @ A2 )
= ( mult_a_Product_unit @ G @ C2 @ B2 ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_Product_unit @ G @ A2 @ C2 )
= ( mult_a_Product_unit @ G @ B2 @ C2 ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid1999574367301118026t_unit @ G ) ) ) ) ).
% monoid.monoid_cancelI
thf(fact_693_monoid__cancel_Oprime__cong,axiom,
! [G: partia2670972154091845814t_unit,P: list_a,P4: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( prime_2011924034616061926t_unit @ G @ P )
=> ( ( associ8407585678920448409t_unit @ G @ P @ P4 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ P4 @ ( partia5361259788508890537t_unit @ G ) )
=> ( prime_2011924034616061926t_unit @ G @ P4 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_694_monoid__cancel_Oprime__cong,axiom,
! [G: partia2175431115845679010xt_a_b,P: a,P4: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( prime_a_ring_ext_a_b @ G @ P )
=> ( ( associ5860276527279195403xt_a_b @ G @ P @ P4 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ P4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( prime_a_ring_ext_a_b @ G @ P4 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_695_monoid__cancel_Oprime__cong,axiom,
! [G: partia8223610829204095565t_unit,P: a,P4: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( prime_a_Product_unit @ G @ P )
=> ( ( associ6879500422977059064t_unit @ G @ P @ P4 )
=> ( ( member_a @ P @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ P4 @ ( partia6735698275553448452t_unit @ G ) )
=> ( prime_a_Product_unit @ G @ P4 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_696_principalideal_Ois__principalideal,axiom,
! [I2: set_a,R2: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R2 )
=> ( principalideal_a_b @ I2 @ R2 ) ) ).
% principalideal.is_principalideal
thf(fact_697_domain_Oconst__term__simprules__shell_I1_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ R2 @ P ) @ K2 ) ) ) ) ).
% domain.const_term_simprules_shell(1)
thf(fact_698_domain_Oconst__term__simprules__shell_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( member_a @ ( const_term_a_b @ R2 @ P ) @ K2 ) ) ) ) ).
% domain.const_term_simprules_shell(1)
thf(fact_699_monoid__cancel_Oassociated__iff,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ G ) )
& ( A
= ( mult_l7073676228092353617t_unit @ G @ B @ X2 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_700_monoid__cancel_Oassociated__iff,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ G ) )
& ( A
= ( mult_a_ring_ext_a_b @ G @ B @ X2 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_701_monoid__cancel_Oassociated__iff,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( associ6879500422977059064t_unit @ G @ A @ B )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_Product_unit @ G ) )
& ( A
= ( mult_a_Product_unit @ G @ B @ X2 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_702_monoid__cancel_OassociatedE2,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ! [U2: list_a] :
( ( A
= ( mult_l7073676228092353617t_unit @ G @ B @ U2 ) )
=> ~ ( member_list_a @ U2 @ ( units_2932844235741507942t_unit @ G ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ~ ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_703_monoid__cancel_OassociatedE2,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ! [U2: a] :
( ( A
= ( mult_a_ring_ext_a_b @ G @ B @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_ring_ext_a_b @ G ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ~ ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_704_monoid__cancel_OassociatedE2,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ( ! [U2: a] :
( ( A
= ( mult_a_Product_unit @ G @ B @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_Product_unit @ G ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ~ ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_705_monoid__cancel_OassociatedD2,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ? [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
& ( A
= ( mult_l7073676228092353617t_unit @ G @ B @ X ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_706_monoid__cancel_OassociatedD2,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
& ( A
= ( mult_a_ring_ext_a_b @ G @ B @ X ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_707_monoid__cancel_OassociatedD2,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ? [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ G ) )
& ( A
= ( mult_a_Product_unit @ G @ B @ X ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_708_ring_Ois__root__def,axiom,
! [R2: partia7496981018696276118t_unit,P: list_set_list_a,X3: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( polyno4320237611291262604t_unit @ R2 @ P @ X3 )
= ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R2 ) )
& ( ( eval_s5133945360527818456t_unit @ R2 @ P @ X3 )
= ( zero_s2910681146719230829t_unit @ R2 ) )
& ( P != nil_set_list_a ) ) ) ) ).
% ring.is_root_def
thf(fact_709_ring_Ois__root__def,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( polyno6951661231331188332t_unit @ R2 @ P @ X3 )
= ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
& ( ( eval_l34571156754992824t_unit @ R2 @ P @ X3 )
= ( zero_l4142658623432671053t_unit @ R2 ) )
& ( P != nil_list_a ) ) ) ) ).
% ring.is_root_def
thf(fact_710_ring_Ois__root__def,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a,X3: a] :
( ( ring_a_b @ R2 )
=> ( ( polyno4133073214067823460ot_a_b @ R2 @ P @ X3 )
= ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
& ( ( eval_a_b @ R2 @ P @ X3 )
= ( zero_a_b @ R2 ) )
& ( P != nil_a ) ) ) ) ).
% ring.is_root_def
thf(fact_711_domain_Opdivides__imp__root__sharing,axiom,
! [R2: partia7496981018696276118t_unit,P: list_set_list_a,Q: list_set_list_a,A: set_list_a] :
( ( domain1617769409708967785t_unit @ R2 )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
=> ( ( polyno9075941895896075626t_unit @ R2 @ P @ Q )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( ( eval_s5133945360527818456t_unit @ R2 @ P @ A )
= ( zero_s2910681146719230829t_unit @ R2 ) )
=> ( ( eval_s5133945360527818456t_unit @ R2 @ Q @ A )
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_712_domain_Opdivides__imp__root__sharing,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,A: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
=> ( ( polyno8016796738000020810t_unit @ R2 @ P @ Q )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ( eval_l34571156754992824t_unit @ R2 @ P @ A )
= ( zero_l4142658623432671053t_unit @ R2 ) )
=> ( ( eval_l34571156754992824t_unit @ R2 @ Q @ A )
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_713_domain_Opdivides__imp__root__sharing,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,A: a] :
( ( domain_a_b @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R2 @ P @ Q )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ( eval_a_b @ R2 @ P @ A )
= ( zero_a_b @ R2 ) )
=> ( ( eval_a_b @ R2 @ Q @ A )
= ( zero_a_b @ R2 ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_714_Group_Ocomm__monoid_Oaxioms_I1_J,axiom,
! [G: partia2670972154091845814t_unit] :
( ( comm_m1219397618491936389t_unit @ G )
=> ( monoid5589397312508706001t_unit @ G ) ) ).
% Group.comm_monoid.axioms(1)
thf(fact_715_Group_Ocomm__monoid_Oaxioms_I1_J,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( comm_m952295370001973751xt_a_b @ G )
=> ( monoid8385113658579753027xt_a_b @ G ) ) ).
% Group.comm_monoid.axioms(1)
thf(fact_716_Group_Ocomm__monoid_Oaxioms_I1_J,axiom,
! [G: partia8223610829204095565t_unit] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( monoid2746444814143937472t_unit @ G ) ) ).
% Group.comm_monoid.axioms(1)
thf(fact_717_domain_Oconst__term__simprules__shell_I3_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( const_6738166269504826821t_unit @ R2 @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P @ Q ) )
= ( add_li7652885771158616974t_unit @ R2 @ ( const_6738166269504826821t_unit @ R2 @ P ) @ ( const_6738166269504826821t_unit @ R2 @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(3)
thf(fact_718_domain_Oconst__term__simprules__shell_I3_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( const_term_a_b @ R2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P @ Q ) )
= ( add_a_b @ R2 @ ( const_term_a_b @ R2 @ P ) @ ( const_term_a_b @ R2 @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(3)
thf(fact_719_domain_Oconst__term__simprules__shell_I2_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( const_6738166269504826821t_unit @ R2 @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P @ Q ) )
= ( mult_l7073676228092353617t_unit @ R2 @ ( const_6738166269504826821t_unit @ R2 @ P ) @ ( const_6738166269504826821t_unit @ R2 @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(2)
thf(fact_720_domain_Oconst__term__simprules__shell_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( const_term_a_b @ R2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ R2 @ ( const_term_a_b @ R2 @ P ) @ ( const_term_a_b @ R2 @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(2)
thf(fact_721_monoid_Om__closed,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ X3 @ Y3 ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% monoid.m_closed
thf(fact_722_monoid_Om__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% monoid.m_closed
thf(fact_723_monoid_Om__closed,axiom,
! [G: partia8223610829204095565t_unit,X3: a,Y3: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ ( mult_a_Product_unit @ G @ X3 @ Y3 ) @ ( partia6735698275553448452t_unit @ G ) ) ) ) ) ).
% monoid.m_closed
thf(fact_724_monoid_Om__assoc,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ X3 @ Y3 ) @ Z2 )
= ( mult_l7073676228092353617t_unit @ G @ X3 @ ( mult_l7073676228092353617t_unit @ G @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% monoid.m_assoc
thf(fact_725_monoid_Om__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y3 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ G @ X3 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% monoid.m_assoc
thf(fact_726_monoid_Om__assoc,axiom,
! [G: partia8223610829204095565t_unit,X3: a,Y3: a,Z2: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X3 @ Y3 ) @ Z2 )
= ( mult_a_Product_unit @ G @ X3 @ ( mult_a_Product_unit @ G @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% monoid.m_assoc
thf(fact_727_comm__monoid_Om__lcomm,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( comm_m1219397618491936389t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ X3 @ ( mult_l7073676228092353617t_unit @ G @ Y3 @ Z2 ) )
= ( mult_l7073676228092353617t_unit @ G @ Y3 @ ( mult_l7073676228092353617t_unit @ G @ X3 @ Z2 ) ) ) ) ) ) ) ).
% comm_monoid.m_lcomm
thf(fact_728_comm__monoid_Om__lcomm,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( comm_m952295370001973751xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X3 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) )
= ( mult_a_ring_ext_a_b @ G @ Y3 @ ( mult_a_ring_ext_a_b @ G @ X3 @ Z2 ) ) ) ) ) ) ) ).
% comm_monoid.m_lcomm
thf(fact_729_comm__monoid_Om__lcomm,axiom,
! [G: partia8223610829204095565t_unit,X3: a,Y3: a,Z2: a] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ X3 @ ( mult_a_Product_unit @ G @ Y3 @ Z2 ) )
= ( mult_a_Product_unit @ G @ Y3 @ ( mult_a_Product_unit @ G @ X3 @ Z2 ) ) ) ) ) ) ) ).
% comm_monoid.m_lcomm
thf(fact_730_comm__monoid_Om__comm,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( comm_m1219397618491936389t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ X3 @ Y3 )
= ( mult_l7073676228092353617t_unit @ G @ Y3 @ X3 ) ) ) ) ) ).
% comm_monoid.m_comm
thf(fact_731_comm__monoid_Om__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( comm_m952295370001973751xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X3 @ Y3 )
= ( mult_a_ring_ext_a_b @ G @ Y3 @ X3 ) ) ) ) ) ).
% comm_monoid.m_comm
thf(fact_732_comm__monoid_Om__comm,axiom,
! [G: partia8223610829204095565t_unit,X3: a,Y3: a] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ X3 @ Y3 )
= ( mult_a_Product_unit @ G @ Y3 @ X3 ) ) ) ) ) ).
% comm_monoid.m_comm
thf(fact_733_comm__monoid_Om__ac_I1_J,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( comm_m1219397618491936389t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ X3 @ Y3 ) @ Z2 )
= ( mult_l7073676228092353617t_unit @ G @ X3 @ ( mult_l7073676228092353617t_unit @ G @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% comm_monoid.m_ac(1)
thf(fact_734_comm__monoid_Om__ac_I1_J,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( comm_m952295370001973751xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y3 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ G @ X3 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% comm_monoid.m_ac(1)
thf(fact_735_comm__monoid_Om__ac_I1_J,axiom,
! [G: partia8223610829204095565t_unit,X3: a,Y3: a,Z2: a] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X3 @ Y3 ) @ Z2 )
= ( mult_a_Product_unit @ G @ X3 @ ( mult_a_Product_unit @ G @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% comm_monoid.m_ac(1)
thf(fact_736_monoid_OUnits__closed,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ G ) )
=> ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) ) ) ) ).
% monoid.Units_closed
thf(fact_737_monoid_OUnits__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ G ) )
=> ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% monoid.Units_closed
thf(fact_738_monoid_OUnits__closed,axiom,
! [G: partia8223610829204095565t_unit,X3: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X3 @ ( units_a_Product_unit @ G ) )
=> ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) ) ) ) ).
% monoid.Units_closed
thf(fact_739_monoid_Onat__pow__closed,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,N: nat] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ G @ X3 @ N ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ).
% monoid.nat_pow_closed
thf(fact_740_monoid_Onat__pow__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,N: nat] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ N ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% monoid.nat_pow_closed
thf(fact_741_monoid_Onat__pow__closed,axiom,
! [G: partia8223610829204095565t_unit,X3: a,N: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ ( pow_a_1875594501834816709it_nat @ G @ X3 @ N ) @ ( partia6735698275553448452t_unit @ G ) ) ) ) ).
% monoid.nat_pow_closed
thf(fact_742_ring_Ocgenideal__self,axiom,
! [R2: partia2670972154091845814t_unit,I: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ I @ ( cgenid9131348535277946915t_unit @ R2 @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_743_ring_Ocgenideal__self,axiom,
! [R2: partia2175431115845679010xt_a_b,I: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ R2 @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_744_monoid_OUnits__m__closed,axiom,
! [G: partia8223610829204095565t_unit,X3: a,Y3: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X3 @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ Y3 @ ( units_a_Product_unit @ G ) )
=> ( member_a @ ( mult_a_Product_unit @ G @ X3 @ Y3 ) @ ( units_a_Product_unit @ G ) ) ) ) ) ).
% monoid.Units_m_closed
thf(fact_745_monoid_OUnits__m__closed,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( units_2932844235741507942t_unit @ G ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ X3 @ Y3 ) @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ).
% monoid.Units_m_closed
thf(fact_746_monoid_OUnits__m__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( units_a_ring_ext_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y3 ) @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.Units_m_closed
thf(fact_747_monoid_OUnits__pow__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,D: nat] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ G ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ D ) @ ( units_a_ring_ext_a_b @ G ) ) ) ) ).
% monoid.Units_pow_closed
thf(fact_748_monoid_OUnits__pow__closed,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,D: nat] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ G ) )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ G @ X3 @ D ) @ ( units_2932844235741507942t_unit @ G ) ) ) ) ).
% monoid.Units_pow_closed
thf(fact_749_monoid_OUnits__pow__closed,axiom,
! [G: partia8223610829204095565t_unit,X3: a,D: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X3 @ ( units_a_Product_unit @ G ) )
=> ( member_a @ ( pow_a_1875594501834816709it_nat @ G @ X3 @ D ) @ ( units_a_Product_unit @ G ) ) ) ) ).
% monoid.Units_pow_closed
thf(fact_750_ring_Oonepideal,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ R2 ) @ R2 ) ) ).
% ring.onepideal
thf(fact_751_ring_Oonepideal,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R2 )
=> ( principalideal_a_b @ ( partia707051561876973205xt_a_b @ R2 ) @ R2 ) ) ).
% ring.onepideal
thf(fact_752_monoid_OUnits__l__cancel,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,Z2: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ X3 @ Y3 )
= ( mult_l7073676228092353617t_unit @ G @ X3 @ Z2 ) )
= ( Y3 = Z2 ) ) ) ) ) ) ).
% monoid.Units_l_cancel
thf(fact_753_monoid_OUnits__l__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a,Z2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X3 @ Y3 )
= ( mult_a_ring_ext_a_b @ G @ X3 @ Z2 ) )
= ( Y3 = Z2 ) ) ) ) ) ) ).
% monoid.Units_l_cancel
thf(fact_754_monoid_OUnits__l__cancel,axiom,
! [G: partia8223610829204095565t_unit,X3: a,Y3: a,Z2: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X3 @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ X3 @ Y3 )
= ( mult_a_Product_unit @ G @ X3 @ Z2 ) )
= ( Y3 = Z2 ) ) ) ) ) ) ).
% monoid.Units_l_cancel
thf(fact_755_monoid_Ogroup__commutes__pow,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,N: nat] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ X3 @ Y3 )
= ( mult_l7073676228092353617t_unit @ G @ Y3 @ X3 ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( pow_li1142815632869257134it_nat @ G @ X3 @ N ) @ Y3 )
= ( mult_l7073676228092353617t_unit @ G @ Y3 @ ( pow_li1142815632869257134it_nat @ G @ X3 @ N ) ) ) ) ) ) ) ).
% monoid.group_commutes_pow
thf(fact_756_monoid_Ogroup__commutes__pow,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a,N: nat] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X3 @ Y3 )
= ( mult_a_ring_ext_a_b @ G @ Y3 @ X3 ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ N ) @ Y3 )
= ( mult_a_ring_ext_a_b @ G @ Y3 @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ N ) ) ) ) ) ) ) ).
% monoid.group_commutes_pow
thf(fact_757_monoid_Ogroup__commutes__pow,axiom,
! [G: partia8223610829204095565t_unit,X3: a,Y3: a,N: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ X3 @ Y3 )
= ( mult_a_Product_unit @ G @ Y3 @ X3 ) )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X3 @ N ) @ Y3 )
= ( mult_a_Product_unit @ G @ Y3 @ ( pow_a_1875594501834816709it_nat @ G @ X3 @ N ) ) ) ) ) ) ) ).
% monoid.group_commutes_pow
thf(fact_758_monoid_Opow__mult__distrib,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a,N: nat] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ X3 @ Y3 )
= ( mult_l7073676228092353617t_unit @ G @ Y3 @ X3 ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( pow_li1142815632869257134it_nat @ G @ ( mult_l7073676228092353617t_unit @ G @ X3 @ Y3 ) @ N )
= ( mult_l7073676228092353617t_unit @ G @ ( pow_li1142815632869257134it_nat @ G @ X3 @ N ) @ ( pow_li1142815632869257134it_nat @ G @ Y3 @ N ) ) ) ) ) ) ) ).
% monoid.pow_mult_distrib
thf(fact_759_monoid_Opow__mult__distrib,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a,N: nat] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X3 @ Y3 )
= ( mult_a_ring_ext_a_b @ G @ Y3 @ X3 ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( pow_a_1026414303147256608_b_nat @ G @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y3 ) @ N )
= ( mult_a_ring_ext_a_b @ G @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ G @ Y3 @ N ) ) ) ) ) ) ) ).
% monoid.pow_mult_distrib
thf(fact_760_monoid_Opow__mult__distrib,axiom,
! [G: partia8223610829204095565t_unit,X3: a,Y3: a,N: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ X3 @ Y3 )
= ( mult_a_Product_unit @ G @ Y3 @ X3 ) )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( pow_a_1875594501834816709it_nat @ G @ ( mult_a_Product_unit @ G @ X3 @ Y3 ) @ N )
= ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X3 @ N ) @ ( pow_a_1875594501834816709it_nat @ G @ Y3 @ N ) ) ) ) ) ) ) ).
% monoid.pow_mult_distrib
thf(fact_761_monoid_Onat__pow__comm,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,N: nat,M: nat] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( pow_li1142815632869257134it_nat @ G @ X3 @ N ) @ ( pow_li1142815632869257134it_nat @ G @ X3 @ M ) )
= ( mult_l7073676228092353617t_unit @ G @ ( pow_li1142815632869257134it_nat @ G @ X3 @ M ) @ ( pow_li1142815632869257134it_nat @ G @ X3 @ N ) ) ) ) ) ).
% monoid.nat_pow_comm
thf(fact_762_monoid_Onat__pow__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,N: nat,M: nat] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ M ) )
= ( mult_a_ring_ext_a_b @ G @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ M ) @ ( pow_a_1026414303147256608_b_nat @ G @ X3 @ N ) ) ) ) ) ).
% monoid.nat_pow_comm
thf(fact_763_monoid_Onat__pow__comm,axiom,
! [G: partia8223610829204095565t_unit,X3: a,N: nat,M: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X3 @ N ) @ ( pow_a_1875594501834816709it_nat @ G @ X3 @ M ) )
= ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X3 @ M ) @ ( pow_a_1875594501834816709it_nat @ G @ X3 @ N ) ) ) ) ) ).
% monoid.nat_pow_comm
thf(fact_764_units__of__pow,axiom,
! [X3: a,N: nat] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( units_8174867845824275201xt_a_b @ r ) @ X3 @ N )
= ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) ) ) ).
% units_of_pow
thf(fact_765_isgcd__divides__l,axiom,
! [A: a,B: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ A @ A @ B ) ) ) ) ).
% isgcd_divides_l
thf(fact_766_isgcd__divides__r,axiom,
! [B: a,A: a] :
( ( factor8216151070175719842xt_a_b @ r @ B @ A )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ B @ A @ B ) ) ) ) ).
% isgcd_divides_r
thf(fact_767_eval__monom,axiom,
! [B: a,A: a,N: nat] :
( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( monom_a_b @ r @ B @ N ) @ A )
= ( mult_a_ring_ext_a_b @ r @ B @ ( pow_a_1026414303147256608_b_nat @ r @ A @ N ) ) ) ) ) ).
% eval_monom
thf(fact_768_coeff__add,axiom,
! [K2: set_a,F: list_a,G3: list_a,I: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ G3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( coeff_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ F @ G3 ) @ I )
= ( add_a_b @ r @ ( coeff_a_b @ r @ F @ I ) @ ( coeff_a_b @ r @ G3 @ I ) ) ) ) ) ) ).
% coeff_add
thf(fact_769_pdivides__iff,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( polynomial_a_b @ r @ K2 @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ Q ) ) ) ) ) ).
% pdivides_iff
thf(fact_770_coeff__iff__length__cond,axiom,
! [P12: list_a,P23: list_a] :
( ( ( size_size_list_a @ P12 )
= ( size_size_list_a @ P23 ) )
=> ( ( P12 = P23 )
= ( ( coeff_a_b @ r @ P12 )
= ( coeff_a_b @ r @ P23 ) ) ) ) ).
% coeff_iff_length_cond
thf(fact_771_coeff__iff__polynomial__cond,axiom,
! [K2: set_a,P12: list_a,P23: list_a] :
( ( polynomial_a_b @ r @ K2 @ P12 )
=> ( ( polynomial_a_b @ r @ K2 @ P23 )
=> ( ( P12 = P23 )
= ( ( coeff_a_b @ r @ P12 )
= ( coeff_a_b @ r @ P23 ) ) ) ) ) ).
% coeff_iff_polynomial_cond
thf(fact_772_coeff_Osimps_I1_J,axiom,
( ( coeff_a_b @ r @ nil_a )
= ( ^ [Uu: nat] : ( zero_a_b @ r ) ) ) ).
% coeff.simps(1)
thf(fact_773_monom__coeff,axiom,
! [A: a,N: nat] :
( ( coeff_a_b @ r @ ( monom_a_b @ r @ A @ N ) )
= ( ^ [I3: nat] : ( if_a @ ( I3 = N ) @ A @ ( zero_a_b @ r ) ) ) ) ).
% monom_coeff
thf(fact_774_var__closed_I2_J,axiom,
! [K2: set_a] :
( ( subring_a_b @ K2 @ r )
=> ( polynomial_a_b @ r @ K2 @ ( var_a_b @ r ) ) ) ).
% var_closed(2)
thf(fact_775_coeff__length,axiom,
! [P: list_a,I: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ P ) @ I )
=> ( ( coeff_a_b @ r @ P @ I )
= ( zero_a_b @ r ) ) ) ).
% coeff_length
thf(fact_776_poly__coeff__in__carrier,axiom,
! [K2: set_a,P: list_a,I: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( member_a @ ( coeff_a_b @ r @ P @ I ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_coeff_in_carrier
thf(fact_777_eval__poly__in__carrier,axiom,
! [K2: set_a,P: list_a,X3: a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% eval_poly_in_carrier
thf(fact_778_units__of__units,axiom,
! [G: partia2670972154091845814t_unit] :
( ( units_8735880885477018085t_unit @ ( units_6477118173342999439t_unit @ G ) )
= ( units_2932844235741507942t_unit @ G ) ) ).
% units_of_units
thf(fact_779_units__of__units,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( units_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G ) )
= ( units_a_ring_ext_a_b @ G ) ) ).
% units_of_units
thf(fact_780_units__of__units,axiom,
! [G: partia8223610829204095565t_unit] :
( ( units_a_Product_unit @ ( units_7501539392726747778t_unit @ G ) )
= ( units_a_Product_unit @ G ) ) ).
% units_of_units
thf(fact_781_zero__is__polynomial,axiom,
! [K2: set_a] : ( polynomial_a_b @ r @ K2 @ nil_a ) ).
% zero_is_polynomial
thf(fact_782_carrier__polynomial,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( polynomial_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ P ) ) ) ).
% carrier_polynomial
thf(fact_783_ring_Ocoeff__iff__polynomial__cond,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P12: list_list_a,P23: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( polyno1315193887021588240t_unit @ R2 @ K2 @ P12 )
=> ( ( polyno1315193887021588240t_unit @ R2 @ K2 @ P23 )
=> ( ( P12 = P23 )
= ( ( coeff_6360649920519955023t_unit @ R2 @ P12 )
= ( coeff_6360649920519955023t_unit @ R2 @ P23 ) ) ) ) ) ) ).
% ring.coeff_iff_polynomial_cond
thf(fact_784_ring_Ocoeff__iff__polynomial__cond,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P12: list_a,P23: list_a] :
( ( ring_a_b @ R2 )
=> ( ( polynomial_a_b @ R2 @ K2 @ P12 )
=> ( ( polynomial_a_b @ R2 @ K2 @ P23 )
=> ( ( P12 = P23 )
= ( ( coeff_a_b @ R2 @ P12 )
= ( coeff_a_b @ R2 @ P23 ) ) ) ) ) ) ).
% ring.coeff_iff_polynomial_cond
thf(fact_785_ring_Omonom_Ocong,axiom,
monom_a_b = monom_a_b ).
% ring.monom.cong
thf(fact_786_ring_Ocoeff_Ocong,axiom,
coeff_a_b = coeff_a_b ).
% ring.coeff.cong
thf(fact_787_ring_Omonom__coeff,axiom,
! [R2: partia2670972154091845814t_unit,A: list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( coeff_6360649920519955023t_unit @ R2 @ ( monom_7446464087056152608t_unit @ R2 @ A @ N ) )
= ( ^ [I3: nat] : ( if_list_a @ ( I3 = N ) @ A @ ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ) ).
% ring.monom_coeff
thf(fact_788_ring_Omonom__coeff,axiom,
! [R2: partia7496981018696276118t_unit,A: set_list_a,N: nat] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( coeff_5603115904260830831t_unit @ R2 @ ( monom_317758005976320064t_unit @ R2 @ A @ N ) )
= ( ^ [I3: nat] : ( if_set_list_a @ ( I3 = N ) @ A @ ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ) ).
% ring.monom_coeff
thf(fact_789_ring_Omonom__coeff,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,N: nat] :
( ( ring_a_b @ R2 )
=> ( ( coeff_a_b @ R2 @ ( monom_a_b @ R2 @ A @ N ) )
= ( ^ [I3: nat] : ( if_a @ ( I3 = N ) @ A @ ( zero_a_b @ R2 ) ) ) ) ) ).
% ring.monom_coeff
thf(fact_790_ring_Opoly__coeff__in__carrier,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,I: nat] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( polyno1315193887021588240t_unit @ R2 @ K2 @ P )
=> ( member_list_a @ ( coeff_6360649920519955023t_unit @ R2 @ P @ I ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ).
% ring.poly_coeff_in_carrier
thf(fact_791_ring_Opoly__coeff__in__carrier,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,I: nat] :
( ( ring_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( polynomial_a_b @ R2 @ K2 @ P )
=> ( member_a @ ( coeff_a_b @ R2 @ P @ I ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).
% ring.poly_coeff_in_carrier
thf(fact_792_ring_Ozero__is__polynomial,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( polyno1315193887021588240t_unit @ R2 @ K2 @ nil_list_a ) ) ).
% ring.zero_is_polynomial
thf(fact_793_ring_Ozero__is__polynomial,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R2 )
=> ( polynomial_a_b @ R2 @ K2 @ nil_a ) ) ).
% ring.zero_is_polynomial
thf(fact_794_univ__poly__carrier,axiom,
( polynomial_a_b
= ( ^ [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] : ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ).
% univ_poly_carrier
thf(fact_795_units__of__mult,axiom,
! [G: partia8223610829204095565t_unit] :
( ( mult_a_Product_unit @ ( units_7501539392726747778t_unit @ G ) )
= ( mult_a_Product_unit @ G ) ) ).
% units_of_mult
thf(fact_796_units__of__mult,axiom,
! [G: partia2670972154091845814t_unit] :
( ( mult_l6995149843440949818t_unit @ ( units_6477118173342999439t_unit @ G ) )
= ( mult_l7073676228092353617t_unit @ G ) ) ).
% units_of_mult
thf(fact_797_units__of__mult,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( mult_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G ) )
= ( mult_a_ring_ext_a_b @ G ) ) ).
% units_of_mult
thf(fact_798_ring_Ocoeff__iff__length__cond,axiom,
! [R2: partia2670972154091845814t_unit,P12: list_list_a,P23: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( ( size_s349497388124573686list_a @ P12 )
= ( size_s349497388124573686list_a @ P23 ) )
=> ( ( P12 = P23 )
= ( ( coeff_6360649920519955023t_unit @ R2 @ P12 )
= ( coeff_6360649920519955023t_unit @ R2 @ P23 ) ) ) ) ) ).
% ring.coeff_iff_length_cond
thf(fact_799_ring_Ocoeff__iff__length__cond,axiom,
! [R2: partia2175431115845679010xt_a_b,P12: list_a,P23: list_a] :
( ( ring_a_b @ R2 )
=> ( ( ( size_size_list_a @ P12 )
= ( size_size_list_a @ P23 ) )
=> ( ( P12 = P23 )
= ( ( coeff_a_b @ R2 @ P12 )
= ( coeff_a_b @ R2 @ P23 ) ) ) ) ) ).
% ring.coeff_iff_length_cond
thf(fact_800_units__of__carrier,axiom,
! [G: partia2670972154091845814t_unit] :
( ( partia7074150537345710456t_unit @ ( units_6477118173342999439t_unit @ G ) )
= ( units_2932844235741507942t_unit @ G ) ) ).
% units_of_carrier
thf(fact_801_units__of__carrier,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( partia6735698275553448452t_unit @ ( units_8174867845824275201xt_a_b @ G ) )
= ( units_a_ring_ext_a_b @ G ) ) ).
% units_of_carrier
thf(fact_802_units__of__carrier,axiom,
! [G: partia8223610829204095565t_unit] :
( ( partia6735698275553448452t_unit @ ( units_7501539392726747778t_unit @ G ) )
= ( units_a_Product_unit @ G ) ) ).
% units_of_carrier
thf(fact_803_ring_Ocarrier__polynomial,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( polyno1315193887021588240t_unit @ R2 @ K2 @ P )
=> ( polyno1315193887021588240t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) @ P ) ) ) ) ).
% ring.carrier_polynomial
thf(fact_804_ring_Ocarrier__polynomial,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a] :
( ( ring_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( polynomial_a_b @ R2 @ K2 @ P )
=> ( polynomial_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) @ P ) ) ) ) ).
% ring.carrier_polynomial
thf(fact_805_domain_Ovar__closed_I2_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( polyno1315193887021588240t_unit @ R2 @ K2 @ ( var_li8453953174693405341t_unit @ R2 ) ) ) ) ).
% domain.var_closed(2)
thf(fact_806_domain_Ovar__closed_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( polynomial_a_b @ R2 @ K2 @ ( var_a_b @ R2 ) ) ) ) ).
% domain.var_closed(2)
thf(fact_807_ring_Ocoeff_Osimps_I1_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( coeff_6360649920519955023t_unit @ R2 @ nil_list_a )
= ( ^ [Uu: nat] : ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ).
% ring.coeff.simps(1)
thf(fact_808_ring_Ocoeff_Osimps_I1_J,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( coeff_5603115904260830831t_unit @ R2 @ nil_set_list_a )
= ( ^ [Uu: nat] : ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ).
% ring.coeff.simps(1)
thf(fact_809_ring_Ocoeff_Osimps_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R2 )
=> ( ( coeff_a_b @ R2 @ nil_a )
= ( ^ [Uu: nat] : ( zero_a_b @ R2 ) ) ) ) ).
% ring.coeff.simps(1)
thf(fact_810_isgcd__def,axiom,
( isgcd_1118609098697428327t_unit
= ( ^ [G2: partia2670972154091845814t_unit,X2: list_a,A4: list_a,B4: list_a] :
( ( factor1757716651909850160t_unit @ G2 @ X2 @ A4 )
& ( factor1757716651909850160t_unit @ G2 @ X2 @ B4 )
& ! [Y5: list_a] :
( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( ( factor1757716651909850160t_unit @ G2 @ Y5 @ A4 )
& ( factor1757716651909850160t_unit @ G2 @ Y5 @ B4 ) )
=> ( factor1757716651909850160t_unit @ G2 @ Y5 @ X2 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_811_isgcd__def,axiom,
( isgcd_a_ring_ext_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b,X2: a,A4: a,B4: a] :
( ( factor8216151070175719842xt_a_b @ G2 @ X2 @ A4 )
& ( factor8216151070175719842xt_a_b @ G2 @ X2 @ B4 )
& ! [Y5: a] :
( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( ( factor8216151070175719842xt_a_b @ G2 @ Y5 @ A4 )
& ( factor8216151070175719842xt_a_b @ G2 @ Y5 @ B4 ) )
=> ( factor8216151070175719842xt_a_b @ G2 @ Y5 @ X2 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_812_isgcd__def,axiom,
( isgcd_a_Product_unit
= ( ^ [G2: partia8223610829204095565t_unit,X2: a,A4: a,B4: a] :
( ( factor3040189038382604065t_unit @ G2 @ X2 @ A4 )
& ( factor3040189038382604065t_unit @ G2 @ X2 @ B4 )
& ! [Y5: a] :
( ( member_a @ Y5 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( ( factor3040189038382604065t_unit @ G2 @ Y5 @ A4 )
& ( factor3040189038382604065t_unit @ G2 @ Y5 @ B4 ) )
=> ( factor3040189038382604065t_unit @ G2 @ Y5 @ X2 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_813_ring_Oeval__poly__in__carrier,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( polyno1315193887021588240t_unit @ R2 @ K2 @ P )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ R2 @ P @ X3 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ) ).
% ring.eval_poly_in_carrier
thf(fact_814_ring_Oeval__poly__in__carrier,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,X3: a] :
( ( ring_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( polynomial_a_b @ R2 @ K2 @ P )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ ( eval_a_b @ R2 @ P @ X3 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ) ).
% ring.eval_poly_in_carrier
thf(fact_815_ring_Ocoeff__length,axiom,
! [R2: partia2670972154091845814t_unit,P: list_list_a,I: nat] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ P ) @ I )
=> ( ( coeff_6360649920519955023t_unit @ R2 @ P @ I )
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ).
% ring.coeff_length
thf(fact_816_ring_Ocoeff__length,axiom,
! [R2: partia7496981018696276118t_unit,P: list_set_list_a,I: nat] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( ord_less_eq_nat @ ( size_s1991367317912710102list_a @ P ) @ I )
=> ( ( coeff_5603115904260830831t_unit @ R2 @ P @ I )
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ).
% ring.coeff_length
thf(fact_817_ring_Ocoeff__length,axiom,
! [R2: partia2175431115845679010xt_a_b,P: list_a,I: nat] :
( ( ring_a_b @ R2 )
=> ( ( ord_less_eq_nat @ ( size_size_list_a @ P ) @ I )
=> ( ( coeff_a_b @ R2 @ P @ I )
= ( zero_a_b @ R2 ) ) ) ) ).
% ring.coeff_length
thf(fact_818_monoid_Oisgcd__divides__l,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( factor1757716651909850160t_unit @ G @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( isgcd_1118609098697428327t_unit @ G @ A @ A @ B ) ) ) ) ) ).
% monoid.isgcd_divides_l
thf(fact_819_monoid_Oisgcd__divides__l,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( factor8216151070175719842xt_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( isgcd_a_ring_ext_a_b @ G @ A @ A @ B ) ) ) ) ) ).
% monoid.isgcd_divides_l
thf(fact_820_monoid_Oisgcd__divides__l,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( isgcd_a_Product_unit @ G @ A @ A @ B ) ) ) ) ) ).
% monoid.isgcd_divides_l
thf(fact_821_monoid_Oisgcd__divides__r,axiom,
! [G: partia2670972154091845814t_unit,B: list_a,A: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( factor1757716651909850160t_unit @ G @ B @ A )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( isgcd_1118609098697428327t_unit @ G @ B @ A @ B ) ) ) ) ) ).
% monoid.isgcd_divides_r
thf(fact_822_monoid_Oisgcd__divides__r,axiom,
! [G: partia2175431115845679010xt_a_b,B: a,A: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( factor8216151070175719842xt_a_b @ G @ B @ A )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( isgcd_a_ring_ext_a_b @ G @ B @ A @ B ) ) ) ) ) ).
% monoid.isgcd_divides_r
thf(fact_823_monoid_Oisgcd__divides__r,axiom,
! [G: partia8223610829204095565t_unit,B: a,A: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ B @ A )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( isgcd_a_Product_unit @ G @ B @ A @ B ) ) ) ) ) ).
% monoid.isgcd_divides_r
thf(fact_824_domain_Opdivides__iff,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( polyno1315193887021588240t_unit @ R2 @ K2 @ P )
=> ( ( polyno1315193887021588240t_unit @ R2 @ K2 @ Q )
=> ( ( polyno8016796738000020810t_unit @ R2 @ P @ Q )
= ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ P @ Q ) ) ) ) ) ) ).
% domain.pdivides_iff
thf(fact_825_domain_Opdivides__iff,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( polynomial_a_b @ R2 @ K2 @ P )
=> ( ( polynomial_a_b @ R2 @ K2 @ Q )
=> ( ( polyno5814909790663948098es_a_b @ R2 @ P @ Q )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ P @ Q ) ) ) ) ) ) ).
% domain.pdivides_iff
thf(fact_826_monoid_Ounits__of__pow,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,N: nat] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( pow_li8657086744513738943it_nat @ ( units_6477118173342999439t_unit @ G ) @ X3 @ N )
= ( pow_li1142815632869257134it_nat @ G @ X3 @ N ) ) ) ) ).
% monoid.units_of_pow
thf(fact_827_monoid_Ounits__of__pow,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,N: nat] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( units_8174867845824275201xt_a_b @ G ) @ X3 @ N )
= ( pow_a_1026414303147256608_b_nat @ G @ X3 @ N ) ) ) ) ).
% monoid.units_of_pow
thf(fact_828_monoid_Ounits__of__pow,axiom,
! [G: partia8223610829204095565t_unit,X3: a,N: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X3 @ ( units_a_Product_unit @ G ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( units_7501539392726747778t_unit @ G ) @ X3 @ N )
= ( pow_a_1875594501834816709it_nat @ G @ X3 @ N ) ) ) ) ).
% monoid.units_of_pow
thf(fact_829_ring_Ocoeff__add,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,F: list_list_a,G3: list_list_a,I: nat] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( member_list_list_a @ G3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) ) )
=> ( ( coeff_6360649920519955023t_unit @ R2 @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ F @ G3 ) @ I )
= ( add_li7652885771158616974t_unit @ R2 @ ( coeff_6360649920519955023t_unit @ R2 @ F @ I ) @ ( coeff_6360649920519955023t_unit @ R2 @ G3 @ I ) ) ) ) ) ) ) ).
% ring.coeff_add
thf(fact_830_ring_Ocoeff__add,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,F: list_a,G3: list_a,I: nat] :
( ( ring_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( member_list_a @ G3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K2 ) ) )
=> ( ( coeff_a_b @ R2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ F @ G3 ) @ I )
= ( add_a_b @ R2 @ ( coeff_a_b @ R2 @ F @ I ) @ ( coeff_a_b @ R2 @ G3 @ I ) ) ) ) ) ) ) ).
% ring.coeff_add
thf(fact_831_ring_Oeval__monom,axiom,
! [R2: partia2670972154091845814t_unit,B: list_a,A: list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( eval_l34571156754992824t_unit @ R2 @ ( monom_7446464087056152608t_unit @ R2 @ B @ N ) @ A )
= ( mult_l7073676228092353617t_unit @ R2 @ B @ ( pow_li1142815632869257134it_nat @ R2 @ A @ N ) ) ) ) ) ) ).
% ring.eval_monom
thf(fact_832_ring_Oeval__monom,axiom,
! [R2: partia2175431115845679010xt_a_b,B: a,A: a,N: nat] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( eval_a_b @ R2 @ ( monom_a_b @ R2 @ B @ N ) @ A )
= ( mult_a_ring_ext_a_b @ R2 @ B @ ( pow_a_1026414303147256608_b_nat @ R2 @ A @ N ) ) ) ) ) ) ).
% ring.eval_monom
thf(fact_833_unitary__monom__eq__var__pow,axiom,
! [K2: set_a,N: nat] :
( ( subring_a_b @ K2 @ r )
=> ( ( monom_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K2 ) @ ( var_a_b @ r ) @ N ) ) ) ).
% unitary_monom_eq_var_pow
thf(fact_834_to__contain__is__to__divide,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( cgenid547466209912283029xt_a_b @ r @ B ) @ ( cgenid547466209912283029xt_a_b @ r @ A ) )
= ( factor8216151070175719842xt_a_b @ r @ A @ B ) ) ) ) ).
% to_contain_is_to_divide
thf(fact_835_subcringI,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( ! [H12: a,H22: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( mult_a_ring_ext_a_b @ r @ H12 @ H22 )
= ( mult_a_ring_ext_a_b @ r @ H22 @ H12 ) ) ) )
=> ( subcring_a_b @ H @ r ) ) ) ).
% subcringI
thf(fact_836_irreducible__prod__rI,axiom,
! [A: a,B: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_rI
thf(fact_837_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_838_subring__props_I3_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K2 ) ) ).
% subring_props(3)
thf(fact_839_zero__is__irreducible__iff__field,axiom,
( ( irredu6211895646901577903xt_a_b @ r @ ( zero_a_b @ r ) )
= ( field_a_b @ r ) ) ).
% zero_is_irreducible_iff_field
thf(fact_840_carrier__is__subcring,axiom,
subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subcring
thf(fact_841_subcringI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subcring_a_b @ H @ r ) ) ).
% subcringI'
thf(fact_842_subring__props_I1_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_843_inv__unique,axiom,
! [Y3: a,X3: a,Y4: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y3 @ X3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y4 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y3 = Y4 ) ) ) ) ) ) ).
% inv_unique
thf(fact_844_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X )
= X ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_845_one__divides,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ A ) ) ).
% one_divides
thf(fact_846_Units__inv__comm,axiom,
! [X3: a,Y3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y3 @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_inv_comm
thf(fact_847_ring__irreducibleE_I2_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( irredu6211895646901577903xt_a_b @ r @ R3 ) ) ) ).
% ring_irreducibleE(2)
thf(fact_848_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A2
!= ( zero_a_b @ r ) )
=> ? [X5: a] :
( ( member_a @ X5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A2 @ X5 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_849_Units__l__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_850_Units__r__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_851_Unit__eq__dividesone,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
= ( factor8216151070175719842xt_a_b @ r @ U @ ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Unit_eq_dividesone
thf(fact_852_divides__one,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ ( one_a_ring_ext_a_b @ r ) )
= ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% divides_one
thf(fact_853_irreducible__prod__lI,axiom,
! [B: a,A: a] :
( ( irredu6211895646901577903xt_a_b @ r @ B )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_lI
thf(fact_854_subdomainI,axiom,
! [H: set_a] :
( ( subcring_a_b @ H @ r )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ! [H12: a,H22: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ r @ H12 @ H22 )
= ( zero_a_b @ r ) )
=> ( ( H12
= ( zero_a_b @ r ) )
| ( H22
= ( zero_a_b @ r ) ) ) ) ) )
=> ( subdomain_a_b @ H @ r ) ) ) ) ).
% subdomainI
thf(fact_855_monoid_Oone__closed,axiom,
! [G: partia2670972154091845814t_unit] :
( ( monoid5589397312508706001t_unit @ G )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) ) ) ).
% monoid.one_closed
thf(fact_856_monoid_Oone__closed,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ).
% monoid.one_closed
thf(fact_857_monoid_Oone__closed,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid2746444814143937472t_unit @ G )
=> ( member_a @ ( one_a_Product_unit @ G ) @ ( partia6735698275553448452t_unit @ G ) ) ) ).
% monoid.one_closed
thf(fact_858_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_859_Units__one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_one_closed
thf(fact_860_nat__pow__one,axiom,
! [N: nat] :
( ( pow_a_1026414303147256608_b_nat @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( one_a_ring_ext_a_b @ r ) ) ).
% nat_pow_one
thf(fact_861_monoid_Or__one,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ X3 @ ( one_li8328186300101108157t_unit @ G ) )
= X3 ) ) ) ).
% monoid.r_one
thf(fact_862_monoid_Or__one,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X3 @ ( one_a_ring_ext_a_b @ G ) )
= X3 ) ) ) ).
% monoid.r_one
thf(fact_863_monoid_Or__one,axiom,
! [G: partia8223610829204095565t_unit,X3: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ X3 @ ( one_a_Product_unit @ G ) )
= X3 ) ) ) ).
% monoid.r_one
thf(fact_864_monoid_Ol__one,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( one_li8328186300101108157t_unit @ G ) @ X3 )
= X3 ) ) ) ).
% monoid.l_one
thf(fact_865_monoid_Ol__one,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X3 )
= X3 ) ) ) ).
% monoid.l_one
thf(fact_866_monoid_Ol__one,axiom,
! [G: partia8223610829204095565t_unit,X3: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( one_a_Product_unit @ G ) @ X3 )
= X3 ) ) ) ).
% monoid.l_one
thf(fact_867_l__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X3 )
= X3 ) ) ).
% l_one
thf(fact_868_r__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( one_a_ring_ext_a_b @ r ) )
= X3 ) ) ).
% r_one
thf(fact_869_ring_Oring__simprules_I6_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R2 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ).
% ring.ring_simprules(6)
thf(fact_870_ring_Oring__simprules_I6_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R2 )
=> ( member_a @ ( one_a_ring_ext_a_b @ R2 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ).
% ring.ring_simprules(6)
thf(fact_871_domain_Ozero__not__one,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( zero_l4142658623432671053t_unit @ R2 )
!= ( one_li8328186300101108157t_unit @ R2 ) ) ) ).
% domain.zero_not_one
thf(fact_872_domain_Ozero__not__one,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R2 )
=> ( ( zero_s2910681146719230829t_unit @ R2 )
!= ( one_se1127990129394575805t_unit @ R2 ) ) ) ).
% domain.zero_not_one
thf(fact_873_domain_Ozero__not__one,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R2 )
=> ( ( zero_a_b @ R2 )
!= ( one_a_ring_ext_a_b @ R2 ) ) ) ).
% domain.zero_not_one
thf(fact_874_domain_Oone__not__zero,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( one_li8328186300101108157t_unit @ R2 )
!= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ).
% domain.one_not_zero
thf(fact_875_domain_Oone__not__zero,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R2 )
=> ( ( one_se1127990129394575805t_unit @ R2 )
!= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ).
% domain.one_not_zero
thf(fact_876_domain_Oone__not__zero,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R2 )
=> ( ( one_a_ring_ext_a_b @ R2 )
!= ( zero_a_b @ R2 ) ) ) ).
% domain.one_not_zero
thf(fact_877_monoid_OUnits__one__closed,axiom,
! [G: partia2670972154091845814t_unit] :
( ( monoid5589397312508706001t_unit @ G )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ G ) @ ( units_2932844235741507942t_unit @ G ) ) ) ).
% monoid.Units_one_closed
thf(fact_878_monoid_OUnits__one__closed,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid2746444814143937472t_unit @ G )
=> ( member_a @ ( one_a_Product_unit @ G ) @ ( units_a_Product_unit @ G ) ) ) ).
% monoid.Units_one_closed
thf(fact_879_monoid_OUnits__one__closed,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( units_a_ring_ext_a_b @ G ) ) ) ).
% monoid.Units_one_closed
thf(fact_880_cring_Ocring__simprules_I6_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R2 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ).
% cring.cring_simprules(6)
thf(fact_881_cring_Ocring__simprules_I6_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R2 )
=> ( member_a @ ( one_a_ring_ext_a_b @ R2 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ).
% cring.cring_simprules(6)
thf(fact_882_monoid_Onat__pow__one,axiom,
! [G: partia2670972154091845814t_unit,N: nat] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( pow_li1142815632869257134it_nat @ G @ ( one_li8328186300101108157t_unit @ G ) @ N )
= ( one_li8328186300101108157t_unit @ G ) ) ) ).
% monoid.nat_pow_one
thf(fact_883_monoid_Onat__pow__one,axiom,
! [G: partia8223610829204095565t_unit,N: nat] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( pow_a_1875594501834816709it_nat @ G @ ( one_a_Product_unit @ G ) @ N )
= ( one_a_Product_unit @ G ) ) ) ).
% monoid.nat_pow_one
thf(fact_884_monoid_Onat__pow__one,axiom,
! [G: partia2175431115845679010xt_a_b,N: nat] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( pow_a_1026414303147256608_b_nat @ G @ ( one_a_ring_ext_a_b @ G ) @ N )
= ( one_a_ring_ext_a_b @ G ) ) ) ).
% monoid.nat_pow_one
thf(fact_885_Ring_Oone__not__zero,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( ( one_li8328186300101108157t_unit @ R2 )
!= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ).
% Ring.one_not_zero
thf(fact_886_Ring_Oone__not__zero,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R2 )
=> ( ( one_se1127990129394575805t_unit @ R2 )
!= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ).
% Ring.one_not_zero
thf(fact_887_Ring_Oone__not__zero,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R2 )
=> ( ( one_a_ring_ext_a_b @ R2 )
!= ( zero_a_b @ R2 ) ) ) ).
% Ring.one_not_zero
thf(fact_888_semiring_Osemiring__simprules_I4_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R2 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_889_semiring_Osemiring__simprules_I4_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R2 )
=> ( member_a @ ( one_a_ring_ext_a_b @ R2 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_890_ring__irreducible__def,axiom,
( ring_r5115406448772830318t_unit
= ( ^ [R: partia7496981018696276118t_unit,A4: set_list_a] :
( ( A4
!= ( zero_s2910681146719230829t_unit @ R ) )
& ( irredu943254396193320253t_unit @ R @ A4 ) ) ) ) ).
% ring_irreducible_def
thf(fact_891_ring__irreducible__def,axiom,
( ring_r932985474545269838t_unit
= ( ^ [R: partia2670972154091845814t_unit,A4: list_a] :
( ( A4
!= ( zero_l4142658623432671053t_unit @ R ) )
& ( irredu4230924414530676029t_unit @ R @ A4 ) ) ) ) ).
% ring_irreducible_def
thf(fact_892_ring__irreducible__def,axiom,
( ring_r999134135267193926le_a_b
= ( ^ [R: partia2175431115845679010xt_a_b,A4: a] :
( ( A4
!= ( zero_a_b @ R ) )
& ( irredu6211895646901577903xt_a_b @ R @ A4 ) ) ) ) ).
% ring_irreducible_def
thf(fact_893_Group_Omonoid__def,axiom,
( monoid5589397312508706001t_unit
= ( ^ [G2: partia2670972154091845814t_unit] :
( ! [X2: list_a,Y5: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ G2 @ X2 @ Y5 ) @ ( partia5361259788508890537t_unit @ G2 ) ) ) )
& ! [X2: list_a,Y5: list_a,Z3: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( mult_l7073676228092353617t_unit @ G2 @ ( mult_l7073676228092353617t_unit @ G2 @ X2 @ Y5 ) @ Z3 )
= ( mult_l7073676228092353617t_unit @ G2 @ X2 @ ( mult_l7073676228092353617t_unit @ G2 @ Y5 @ Z3 ) ) ) ) ) )
& ( member_list_a @ ( one_li8328186300101108157t_unit @ G2 ) @ ( partia5361259788508890537t_unit @ G2 ) )
& ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( mult_l7073676228092353617t_unit @ G2 @ ( one_li8328186300101108157t_unit @ G2 ) @ X2 )
= X2 ) )
& ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( mult_l7073676228092353617t_unit @ G2 @ X2 @ ( one_li8328186300101108157t_unit @ G2 ) )
= X2 ) ) ) ) ) ).
% Group.monoid_def
thf(fact_894_Group_Omonoid__def,axiom,
( monoid8385113658579753027xt_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b] :
( ! [X2: a,Y5: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G2 @ X2 @ Y5 ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) )
& ! [X2: a,Y5: a,Z3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( mult_a_ring_ext_a_b @ G2 @ ( mult_a_ring_ext_a_b @ G2 @ X2 @ Y5 ) @ Z3 )
= ( mult_a_ring_ext_a_b @ G2 @ X2 @ ( mult_a_ring_ext_a_b @ G2 @ Y5 @ Z3 ) ) ) ) ) )
& ( member_a @ ( one_a_ring_ext_a_b @ G2 ) @ ( partia707051561876973205xt_a_b @ G2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( mult_a_ring_ext_a_b @ G2 @ ( one_a_ring_ext_a_b @ G2 ) @ X2 )
= X2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( mult_a_ring_ext_a_b @ G2 @ X2 @ ( one_a_ring_ext_a_b @ G2 ) )
= X2 ) ) ) ) ) ).
% Group.monoid_def
thf(fact_895_Group_Omonoid__def,axiom,
( monoid2746444814143937472t_unit
= ( ^ [G2: partia8223610829204095565t_unit] :
( ! [X2: a,Y5: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ Y5 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( member_a @ ( mult_a_Product_unit @ G2 @ X2 @ Y5 ) @ ( partia6735698275553448452t_unit @ G2 ) ) ) )
& ! [X2: a,Y5: a,Z3: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ Y5 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ Z3 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( mult_a_Product_unit @ G2 @ ( mult_a_Product_unit @ G2 @ X2 @ Y5 ) @ Z3 )
= ( mult_a_Product_unit @ G2 @ X2 @ ( mult_a_Product_unit @ G2 @ Y5 @ Z3 ) ) ) ) ) )
& ( member_a @ ( one_a_Product_unit @ G2 ) @ ( partia6735698275553448452t_unit @ G2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( mult_a_Product_unit @ G2 @ ( one_a_Product_unit @ G2 ) @ X2 )
= X2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( mult_a_Product_unit @ G2 @ X2 @ ( one_a_Product_unit @ G2 ) )
= X2 ) ) ) ) ) ).
% Group.monoid_def
thf(fact_896_monoidI,axiom,
! [G: partia2670972154091845814t_unit] :
( ! [X: list_a,Y2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ X @ Y2 ) @ ( partia5361259788508890537t_unit @ G ) ) ) )
=> ( ( member_list_a @ ( one_li8328186300101108157t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) )
=> ( ! [X: list_a,Y2: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ X @ Y2 ) @ Z )
= ( mult_l7073676228092353617t_unit @ G @ X @ ( mult_l7073676228092353617t_unit @ G @ Y2 @ Z ) ) ) ) ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( one_li8328186300101108157t_unit @ G ) @ X )
= X ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ X @ ( one_li8328186300101108157t_unit @ G ) )
= X ) )
=> ( monoid5589397312508706001t_unit @ G ) ) ) ) ) ) ).
% monoidI
thf(fact_897_monoidI,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ! [X: a,Y2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y2 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X @ Y2 ) @ Z )
= ( mult_a_ring_ext_a_b @ G @ X @ ( mult_a_ring_ext_a_b @ G @ Y2 @ Z ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X )
= X ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X @ ( one_a_ring_ext_a_b @ G ) )
= X ) )
=> ( monoid8385113658579753027xt_a_b @ G ) ) ) ) ) ) ).
% monoidI
thf(fact_898_monoidI,axiom,
! [G: partia8223610829204095565t_unit] :
( ! [X: a,Y2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ ( mult_a_Product_unit @ G @ X @ Y2 ) @ ( partia6735698275553448452t_unit @ G ) ) ) )
=> ( ( member_a @ ( one_a_Product_unit @ G ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X @ Y2 ) @ Z )
= ( mult_a_Product_unit @ G @ X @ ( mult_a_Product_unit @ G @ Y2 @ Z ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( one_a_Product_unit @ G ) @ X )
= X ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ X @ ( one_a_Product_unit @ G ) )
= X ) )
=> ( monoid2746444814143937472t_unit @ G ) ) ) ) ) ) ).
% monoidI
thf(fact_899_monoid_Oone__unique,axiom,
! [G: partia2670972154091845814t_unit,U: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ G ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ U @ X )
= X ) )
=> ( U
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ).
% monoid.one_unique
thf(fact_900_monoid_Oone__unique,axiom,
! [G: partia2175431115845679010xt_a_b,U: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ U @ X )
= X ) )
=> ( U
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.one_unique
thf(fact_901_monoid_Oone__unique,axiom,
! [G: partia8223610829204095565t_unit,U: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ U @ ( partia6735698275553448452t_unit @ G ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ U @ X )
= X ) )
=> ( U
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% monoid.one_unique
thf(fact_902_monoid_Oinv__unique,axiom,
! [G: partia2670972154091845814t_unit,Y3: list_a,X3: list_a,Y4: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ Y3 @ X3 )
= ( one_li8328186300101108157t_unit @ G ) )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ X3 @ Y4 )
= ( one_li8328186300101108157t_unit @ G ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ G ) )
=> ( Y3 = Y4 ) ) ) ) ) ) ) ).
% monoid.inv_unique
thf(fact_903_monoid_Oinv__unique,axiom,
! [G: partia2175431115845679010xt_a_b,Y3: a,X3: a,Y4: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ Y3 @ X3 )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X3 @ Y4 )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( Y3 = Y4 ) ) ) ) ) ) ) ).
% monoid.inv_unique
thf(fact_904_monoid_Oinv__unique,axiom,
! [G: partia8223610829204095565t_unit,Y3: a,X3: a,Y4: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ Y3 @ X3 )
= ( one_a_Product_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ X3 @ Y4 )
= ( one_a_Product_unit @ G ) )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y4 @ ( partia6735698275553448452t_unit @ G ) )
=> ( Y3 = Y4 ) ) ) ) ) ) ) ).
% monoid.inv_unique
thf(fact_905_Group_Omonoid_Ointro,axiom,
! [G: partia2670972154091845814t_unit] :
( ! [X: list_a,Y2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ X @ Y2 ) @ ( partia5361259788508890537t_unit @ G ) ) ) )
=> ( ! [X: list_a,Y2: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ X @ Y2 ) @ Z )
= ( mult_l7073676228092353617t_unit @ G @ X @ ( mult_l7073676228092353617t_unit @ G @ Y2 @ Z ) ) ) ) ) )
=> ( ( member_list_a @ ( one_li8328186300101108157t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( one_li8328186300101108157t_unit @ G ) @ X )
= X ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ X @ ( one_li8328186300101108157t_unit @ G ) )
= X ) )
=> ( monoid5589397312508706001t_unit @ G ) ) ) ) ) ) ).
% Group.monoid.intro
thf(fact_906_Group_Omonoid_Ointro,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ! [X: a,Y2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y2 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) )
=> ( ! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X @ Y2 ) @ Z )
= ( mult_a_ring_ext_a_b @ G @ X @ ( mult_a_ring_ext_a_b @ G @ Y2 @ Z ) ) ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X )
= X ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X @ ( one_a_ring_ext_a_b @ G ) )
= X ) )
=> ( monoid8385113658579753027xt_a_b @ G ) ) ) ) ) ) ).
% Group.monoid.intro
thf(fact_907_Group_Omonoid_Ointro,axiom,
! [G: partia8223610829204095565t_unit] :
( ! [X: a,Y2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ ( mult_a_Product_unit @ G @ X @ Y2 ) @ ( partia6735698275553448452t_unit @ G ) ) ) )
=> ( ! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X @ Y2 ) @ Z )
= ( mult_a_Product_unit @ G @ X @ ( mult_a_Product_unit @ G @ Y2 @ Z ) ) ) ) ) )
=> ( ( member_a @ ( one_a_Product_unit @ G ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( one_a_Product_unit @ G ) @ X )
= X ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ X @ ( one_a_Product_unit @ G ) )
= X ) )
=> ( monoid2746444814143937472t_unit @ G ) ) ) ) ) ) ).
% Group.monoid.intro
thf(fact_908_comm__monoidI,axiom,
! [G: partia2670972154091845814t_unit] :
( ! [X: list_a,Y2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ X @ Y2 ) @ ( partia5361259788508890537t_unit @ G ) ) ) )
=> ( ( member_list_a @ ( one_li8328186300101108157t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) )
=> ( ! [X: list_a,Y2: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ X @ Y2 ) @ Z )
= ( mult_l7073676228092353617t_unit @ G @ X @ ( mult_l7073676228092353617t_unit @ G @ Y2 @ Z ) ) ) ) ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( one_li8328186300101108157t_unit @ G ) @ X )
= X ) )
=> ( ! [X: list_a,Y2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ X @ Y2 )
= ( mult_l7073676228092353617t_unit @ G @ Y2 @ X ) ) ) )
=> ( comm_m1219397618491936389t_unit @ G ) ) ) ) ) ) ).
% comm_monoidI
thf(fact_909_comm__monoidI,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ! [X: a,Y2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y2 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X @ Y2 ) @ Z )
= ( mult_a_ring_ext_a_b @ G @ X @ ( mult_a_ring_ext_a_b @ G @ Y2 @ Z ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X )
= X ) )
=> ( ! [X: a,Y2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X @ Y2 )
= ( mult_a_ring_ext_a_b @ G @ Y2 @ X ) ) ) )
=> ( comm_m952295370001973751xt_a_b @ G ) ) ) ) ) ) ).
% comm_monoidI
thf(fact_910_comm__monoidI,axiom,
! [G: partia8223610829204095565t_unit] :
( ! [X: a,Y2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ ( mult_a_Product_unit @ G @ X @ Y2 ) @ ( partia6735698275553448452t_unit @ G ) ) ) )
=> ( ( member_a @ ( one_a_Product_unit @ G ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X @ Y2 ) @ Z )
= ( mult_a_Product_unit @ G @ X @ ( mult_a_Product_unit @ G @ Y2 @ Z ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( one_a_Product_unit @ G ) @ X )
= X ) )
=> ( ! [X: a,Y2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ X @ Y2 )
= ( mult_a_Product_unit @ G @ Y2 @ X ) ) ) )
=> ( comm_m7681468956318391052t_unit @ G ) ) ) ) ) ) ).
% comm_monoidI
thf(fact_911_ring_Oring__simprules_I12_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ ( one_li8328186300101108157t_unit @ R2 ) @ X3 )
= X3 ) ) ) ).
% ring.ring_simprules(12)
thf(fact_912_ring_Oring__simprules_I12_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( one_a_ring_ext_a_b @ R2 ) @ X3 )
= X3 ) ) ) ).
% ring.ring_simprules(12)
thf(fact_913_monoid_OUnits__inv__comm,axiom,
! [G: partia8223610829204095565t_unit,X3: a,Y3: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ X3 @ Y3 )
= ( one_a_Product_unit @ G ) )
=> ( ( member_a @ X3 @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ Y3 @ ( units_a_Product_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ Y3 @ X3 )
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% monoid.Units_inv_comm
thf(fact_914_monoid_OUnits__inv__comm,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ X3 @ Y3 )
= ( one_li8328186300101108157t_unit @ G ) )
=> ( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ Y3 @ X3 )
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ) ).
% monoid.Units_inv_comm
thf(fact_915_monoid_OUnits__inv__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y3: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X3 @ Y3 )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ Y3 @ X3 )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% monoid.Units_inv_comm
thf(fact_916_cring_Ocring__simprules_I12_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ ( one_li8328186300101108157t_unit @ R2 ) @ X3 )
= X3 ) ) ) ).
% cring.cring_simprules(12)
thf(fact_917_cring_Ocring__simprules_I12_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( one_a_ring_ext_a_b @ R2 ) @ X3 )
= X3 ) ) ) ).
% cring.cring_simprules(12)
thf(fact_918_ring_Oone__divides,axiom,
! [R2: partia2670972154091845814t_unit,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( factor1757716651909850160t_unit @ R2 @ ( one_li8328186300101108157t_unit @ R2 ) @ A ) ) ) ).
% ring.one_divides
thf(fact_919_ring_Oone__divides,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( factor8216151070175719842xt_a_b @ R2 @ ( one_a_ring_ext_a_b @ R2 ) @ A ) ) ) ).
% ring.one_divides
thf(fact_920_semiring_Osemiring__simprules_I9_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ ( one_li8328186300101108157t_unit @ R2 ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_921_semiring_Osemiring__simprules_I9_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( one_a_ring_ext_a_b @ R2 ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_922_domain_Ozero__is__irreducible__iff__field,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( irredu4230924414530676029t_unit @ R2 @ ( zero_l4142658623432671053t_unit @ R2 ) )
= ( field_6388047844668329575t_unit @ R2 ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_923_domain_Ozero__is__irreducible__iff__field,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R2 )
=> ( ( irredu943254396193320253t_unit @ R2 @ ( zero_s2910681146719230829t_unit @ R2 ) )
= ( field_26233345952514695t_unit @ R2 ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_924_domain_Ozero__is__irreducible__iff__field,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R2 )
=> ( ( irredu6211895646901577903xt_a_b @ R2 @ ( zero_a_b @ R2 ) )
= ( field_a_b @ R2 ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_925_domain_Oring__irreducibleE_I2_J,axiom,
! [R2: partia2670972154091845814t_unit,R3: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ring_r932985474545269838t_unit @ R2 @ R3 )
=> ( irredu4230924414530676029t_unit @ R2 @ R3 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_926_domain_Oring__irreducibleE_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b,R3: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_r999134135267193926le_a_b @ R2 @ R3 )
=> ( irredu6211895646901577903xt_a_b @ R2 @ R3 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_927_monoid_OUnits__r__inv__ex,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ G ) )
=> ? [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
& ( ( mult_l7073676228092353617t_unit @ G @ X3 @ X )
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ).
% monoid.Units_r_inv_ex
thf(fact_928_monoid_OUnits__r__inv__ex,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ G ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ X3 @ X )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.Units_r_inv_ex
thf(fact_929_monoid_OUnits__r__inv__ex,axiom,
! [G: partia8223610829204095565t_unit,X3: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X3 @ ( units_a_Product_unit @ G ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ X3 @ X )
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% monoid.Units_r_inv_ex
thf(fact_930_monoid_OUnits__l__inv__ex,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ G ) )
=> ? [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
& ( ( mult_l7073676228092353617t_unit @ G @ X @ X3 )
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ).
% monoid.Units_l_inv_ex
thf(fact_931_monoid_OUnits__l__inv__ex,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ G ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ X @ X3 )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.Units_l_inv_ex
thf(fact_932_monoid_OUnits__l__inv__ex,axiom,
! [G: partia8223610829204095565t_unit,X3: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X3 @ ( units_a_Product_unit @ G ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ X @ X3 )
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% monoid.Units_l_inv_ex
thf(fact_933_monoid__cancel_Oirreducible__cong,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,A5: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( irredu4230924414530676029t_unit @ G @ A )
=> ( ( associ8407585678920448409t_unit @ G @ A @ A5 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ G ) )
=> ( irredu4230924414530676029t_unit @ G @ A5 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_934_monoid__cancel_Oirreducible__cong,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,A5: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( irredu6211895646901577903xt_a_b @ G @ A )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ A5 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( irredu6211895646901577903xt_a_b @ G @ A5 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_935_monoid__cancel_Oirreducible__cong,axiom,
! [G: partia8223610829204095565t_unit,A: a,A5: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( irredu4023057619401689684t_unit @ G @ A )
=> ( ( associ6879500422977059064t_unit @ G @ A @ A5 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A5 @ ( partia6735698275553448452t_unit @ G ) )
=> ( irredu4023057619401689684t_unit @ G @ A5 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_936_comm__monoid_OUnit__eq__dividesone,axiom,
! [G: partia2670972154091845814t_unit,U: list_a] :
( ( comm_m1219397618491936389t_unit @ G )
=> ( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ G ) )
= ( factor1757716651909850160t_unit @ G @ U @ ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ).
% comm_monoid.Unit_eq_dividesone
thf(fact_937_comm__monoid_OUnit__eq__dividesone,axiom,
! [G: partia2175431115845679010xt_a_b,U: a] :
( ( comm_m952295370001973751xt_a_b @ G )
=> ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ G ) )
= ( factor8216151070175719842xt_a_b @ G @ U @ ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% comm_monoid.Unit_eq_dividesone
thf(fact_938_comm__monoid_OUnit__eq__dividesone,axiom,
! [G: partia8223610829204095565t_unit,U: a] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( member_a @ U @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ G ) )
= ( factor3040189038382604065t_unit @ G @ U @ ( one_a_Product_unit @ G ) ) ) ) ) ).
% comm_monoid.Unit_eq_dividesone
thf(fact_939_cring_Odivides__one,axiom,
! [R2: partia2670972154091845814t_unit,A: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( factor1757716651909850160t_unit @ R2 @ A @ ( one_li8328186300101108157t_unit @ R2 ) )
= ( member_list_a @ A @ ( units_2932844235741507942t_unit @ R2 ) ) ) ) ) ).
% cring.divides_one
thf(fact_940_cring_Odivides__one,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( factor8216151070175719842xt_a_b @ R2 @ A @ ( one_a_ring_ext_a_b @ R2 ) )
= ( member_a @ A @ ( units_a_ring_ext_a_b @ R2 ) ) ) ) ) ).
% cring.divides_one
thf(fact_941_cring_Oto__contain__is__to__divide,axiom,
! [R2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ord_le8861187494160871172list_a @ ( cgenid9131348535277946915t_unit @ R2 @ B ) @ ( cgenid9131348535277946915t_unit @ R2 @ A ) )
= ( factor1757716651909850160t_unit @ R2 @ A @ B ) ) ) ) ) ).
% cring.to_contain_is_to_divide
thf(fact_942_cring_Oto__contain__is__to__divide,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ord_less_eq_set_a @ ( cgenid547466209912283029xt_a_b @ R2 @ B ) @ ( cgenid547466209912283029xt_a_b @ R2 @ A ) )
= ( factor8216151070175719842xt_a_b @ R2 @ A @ B ) ) ) ) ) ).
% cring.to_contain_is_to_divide
thf(fact_943_monoid_Oirreducible__prod__rI,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( irredu4230924414530676029t_unit @ G @ A )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( irredu4230924414530676029t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) ) ) ) ) ) ) ).
% monoid.irreducible_prod_rI
thf(fact_944_monoid_Oirreducible__prod__rI,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( irredu6211895646901577903xt_a_b @ G @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( irredu6211895646901577903xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A @ B ) ) ) ) ) ) ) ).
% monoid.irreducible_prod_rI
thf(fact_945_monoid_Oirreducible__prod__rI,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( irredu4023057619401689684t_unit @ G @ A )
=> ( ( member_a @ B @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( irredu4023057619401689684t_unit @ G @ ( mult_a_Product_unit @ G @ A @ B ) ) ) ) ) ) ) ).
% monoid.irreducible_prod_rI
thf(fact_946_divides__irreducible__condition,axiom,
! [G: partia2670972154091845814t_unit,R3: list_a,A: list_a] :
( ( irredu4230924414530676029t_unit @ G @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( factor1757716651909850160t_unit @ G @ A @ R3 )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) )
| ( associ8407585678920448409t_unit @ G @ A @ R3 ) ) ) ) ) ).
% divides_irreducible_condition
thf(fact_947_divides__irreducible__condition,axiom,
! [G: partia2175431115845679010xt_a_b,R3: a,A: a] :
( ( irredu6211895646901577903xt_a_b @ G @ R3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ A @ R3 )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
| ( associ5860276527279195403xt_a_b @ G @ A @ R3 ) ) ) ) ) ).
% divides_irreducible_condition
thf(fact_948_divides__irreducible__condition,axiom,
! [G: partia8223610829204095565t_unit,R3: a,A: a] :
( ( irredu4023057619401689684t_unit @ G @ R3 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ A @ R3 )
=> ( ( member_a @ A @ ( units_a_Product_unit @ G ) )
| ( associ6879500422977059064t_unit @ G @ A @ R3 ) ) ) ) ) ).
% divides_irreducible_condition
thf(fact_949_comm__monoid_Oirreducible__prod__lI,axiom,
! [G: partia2670972154091845814t_unit,B: list_a,A: list_a] :
( ( comm_m1219397618491936389t_unit @ G )
=> ( ( irredu4230924414530676029t_unit @ G @ B )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( irredu4230924414530676029t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) ) ) ) ) ) ) ).
% comm_monoid.irreducible_prod_lI
thf(fact_950_comm__monoid_Oirreducible__prod__lI,axiom,
! [G: partia2175431115845679010xt_a_b,B: a,A: a] :
( ( comm_m952295370001973751xt_a_b @ G )
=> ( ( irredu6211895646901577903xt_a_b @ G @ B )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( irredu6211895646901577903xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A @ B ) ) ) ) ) ) ) ).
% comm_monoid.irreducible_prod_lI
thf(fact_951_comm__monoid_Oirreducible__prod__lI,axiom,
! [G: partia8223610829204095565t_unit,B: a,A: a] :
( ( comm_m7681468956318391052t_unit @ G )
=> ( ( irredu4023057619401689684t_unit @ G @ B )
=> ( ( member_a @ A @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( irredu4023057619401689684t_unit @ G @ ( mult_a_Product_unit @ G @ A @ B ) ) ) ) ) ) ) ).
% comm_monoid.irreducible_prod_lI
thf(fact_952_cring_Ocring__fieldI2,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R2 )
=> ( ( ( zero_s2910681146719230829t_unit @ R2 )
!= ( one_se1127990129394575805t_unit @ R2 ) )
=> ( ! [A2: set_list_a] :
( ( member_set_list_a @ A2 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( A2
!= ( zero_s2910681146719230829t_unit @ R2 ) )
=> ? [X5: set_list_a] :
( ( member_set_list_a @ X5 @ ( partia141011252114345353t_unit @ R2 ) )
& ( ( mult_s7802724872828879953t_unit @ R2 @ A2 @ X5 )
= ( one_se1127990129394575805t_unit @ R2 ) ) ) ) )
=> ( field_26233345952514695t_unit @ R2 ) ) ) ) ).
% cring.cring_fieldI2
thf(fact_953_cring_Ocring__fieldI2,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( ( zero_l4142658623432671053t_unit @ R2 )
!= ( one_li8328186300101108157t_unit @ R2 ) )
=> ( ! [A2: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( A2
!= ( zero_l4142658623432671053t_unit @ R2 ) )
=> ? [X5: list_a] :
( ( member_list_a @ X5 @ ( partia5361259788508890537t_unit @ R2 ) )
& ( ( mult_l7073676228092353617t_unit @ R2 @ A2 @ X5 )
= ( one_li8328186300101108157t_unit @ R2 ) ) ) ) )
=> ( field_6388047844668329575t_unit @ R2 ) ) ) ) ).
% cring.cring_fieldI2
thf(fact_954_cring_Ocring__fieldI2,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R2 )
=> ( ( ( zero_a_b @ R2 )
!= ( one_a_ring_ext_a_b @ R2 ) )
=> ( ! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( A2
!= ( zero_a_b @ R2 ) )
=> ? [X5: a] :
( ( member_a @ X5 @ ( partia707051561876973205xt_a_b @ R2 ) )
& ( ( mult_a_ring_ext_a_b @ R2 @ A2 @ X5 )
= ( one_a_ring_ext_a_b @ R2 ) ) ) ) )
=> ( field_a_b @ R2 ) ) ) ) ).
% cring.cring_fieldI2
thf(fact_955_domain_Ounitary__monom__eq__var__pow,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( monom_7446464087056152608t_unit @ R2 @ ( one_li8328186300101108157t_unit @ R2 ) @ N )
= ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ ( var_li8453953174693405341t_unit @ R2 ) @ N ) ) ) ) ).
% domain.unitary_monom_eq_var_pow
thf(fact_956_domain_Ounitary__monom__eq__var__pow,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,N: nat] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( monom_a_b @ R2 @ ( one_a_ring_ext_a_b @ R2 ) @ N )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R2 @ K2 ) @ ( var_a_b @ R2 ) @ N ) ) ) ) ).
% domain.unitary_monom_eq_var_pow
thf(fact_957_ring_OsubdomainI,axiom,
! [R2: partia7496981018696276118t_unit,H: set_set_list_a] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( subcri7783154434480317835t_unit @ H @ R2 )
=> ( ( ( one_se1127990129394575805t_unit @ R2 )
!= ( zero_s2910681146719230829t_unit @ R2 ) )
=> ( ! [H12: set_list_a,H22: set_list_a] :
( ( member_set_list_a @ H12 @ H )
=> ( ( member_set_list_a @ H22 @ H )
=> ( ( ( mult_s7802724872828879953t_unit @ R2 @ H12 @ H22 )
= ( zero_s2910681146719230829t_unit @ R2 ) )
=> ( ( H12
= ( zero_s2910681146719230829t_unit @ R2 ) )
| ( H22
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ) )
=> ( subdom3220114454046903646t_unit @ H @ R2 ) ) ) ) ) ).
% ring.subdomainI
thf(fact_958_ring_OsubdomainI,axiom,
! [R2: partia2670972154091845814t_unit,H: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( subcri7763218559781929323t_unit @ H @ R2 )
=> ( ( ( one_li8328186300101108157t_unit @ R2 )
!= ( zero_l4142658623432671053t_unit @ R2 ) )
=> ( ! [H12: list_a,H22: list_a] :
( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( ( ( mult_l7073676228092353617t_unit @ R2 @ H12 @ H22 )
= ( zero_l4142658623432671053t_unit @ R2 ) )
=> ( ( H12
= ( zero_l4142658623432671053t_unit @ R2 ) )
| ( H22
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ) )
=> ( subdom7821232466298058046t_unit @ H @ R2 ) ) ) ) ) ).
% ring.subdomainI
thf(fact_959_ring_OsubdomainI,axiom,
! [R2: partia2175431115845679010xt_a_b,H: set_a] :
( ( ring_a_b @ R2 )
=> ( ( subcring_a_b @ H @ R2 )
=> ( ( ( one_a_ring_ext_a_b @ R2 )
!= ( zero_a_b @ R2 ) )
=> ( ! [H12: a,H22: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ R2 @ H12 @ H22 )
= ( zero_a_b @ R2 ) )
=> ( ( H12
= ( zero_a_b @ R2 ) )
| ( H22
= ( zero_a_b @ R2 ) ) ) ) ) )
=> ( subdomain_a_b @ H @ R2 ) ) ) ) ) ).
% ring.subdomainI
thf(fact_960_subring__degree__one__associatedI,axiom,
! [K2: set_a,A: a,A5: a,B: a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_a @ A @ K2 )
=> ( ( member_a @ A5 @ K2 )
=> ( ( member_a @ B @ K2 )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ A5 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( mult_a_ring_ext_a_b @ r @ A5 @ B ) @ nil_a ) ) ) ) ) ) ) ) ).
% subring_degree_one_associatedI
thf(fact_961_domainI,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R2 )
=> ( ( ( one_se1127990129394575805t_unit @ R2 )
!= ( zero_s2910681146719230829t_unit @ R2 ) )
=> ( ! [A2: set_list_a,B2: set_list_a] :
( ( ( mult_s7802724872828879953t_unit @ R2 @ A2 @ B2 )
= ( zero_s2910681146719230829t_unit @ R2 ) )
=> ( ( member_set_list_a @ A2 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( member_set_list_a @ B2 @ ( partia141011252114345353t_unit @ R2 ) )
=> ( ( A2
= ( zero_s2910681146719230829t_unit @ R2 ) )
| ( B2
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ) )
=> ( domain1617769409708967785t_unit @ R2 ) ) ) ) ).
% domainI
thf(fact_962_domainI,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( ( one_li8328186300101108157t_unit @ R2 )
!= ( zero_l4142658623432671053t_unit @ R2 ) )
=> ( ! [A2: list_a,B2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ R2 @ A2 @ B2 )
= ( zero_l4142658623432671053t_unit @ R2 ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( A2
= ( zero_l4142658623432671053t_unit @ R2 ) )
| ( B2
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ) )
=> ( domain6553523120543210313t_unit @ R2 ) ) ) ) ).
% domainI
thf(fact_963_domainI,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R2 )
=> ( ( ( one_a_ring_ext_a_b @ R2 )
!= ( zero_a_b @ R2 ) )
=> ( ! [A2: a,B2: a] :
( ( ( mult_a_ring_ext_a_b @ R2 @ A2 @ B2 )
= ( zero_a_b @ R2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( A2
= ( zero_a_b @ R2 ) )
| ( B2
= ( zero_a_b @ R2 ) ) ) ) ) )
=> ( domain_a_b @ R2 ) ) ) ) ).
% domainI
thf(fact_964_line__extension__smult__closed,axiom,
! [K2: set_a,E: set_a,A: a,K3: a,U: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ! [K4: a,V: a] :
( ( member_a @ K4 @ K2 )
=> ( ( member_a @ V @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K4 @ V ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K3 @ K2 )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ U ) @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_965_normalize_Ocases,axiom,
! [X3: list_a] :
( ( X3 != nil_a )
=> ~ ! [V: a,Va: list_a] :
( X3
!= ( cons_a @ V @ Va ) ) ) ).
% normalize.cases
thf(fact_966_combine_Ocases,axiom,
! [X3: produc9164743771328383783list_a] :
( ! [K4: a,Ks: list_a,U2: a,Us: list_a] :
( X3
!= ( produc6837034575241423639list_a @ ( cons_a @ K4 @ Ks ) @ ( cons_a @ U2 @ Us ) ) )
=> ( ! [Us: list_a] :
( X3
!= ( produc6837034575241423639list_a @ nil_a @ Us ) )
=> ~ ! [Ks: list_a] :
( X3
!= ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).
% combine.cases
thf(fact_967_poly__mult_Ocases,axiom,
! [X3: produc9164743771328383783list_a] :
( ! [P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ nil_a @ P22 ) )
=> ~ ! [V: a,Va: list_a,P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ Va ) @ P22 ) ) ) ).
% poly_mult.cases
thf(fact_968_line__extension__in__carrier,axiom,
! [K2: set_a,A: a,E: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_969_line__extension__mem__iff,axiom,
! [U: a,K2: set_a,A: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ K2 )
& ? [Y5: a] :
( ( member_a @ Y5 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ A ) @ Y5 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_970_one__is__polynomial,axiom,
! [K2: set_a] :
( ( subring_a_b @ K2 @ r )
=> ( polynomial_a_b @ r @ K2 @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) ) ) ).
% one_is_polynomial
thf(fact_971_ring_Odense__repr_Ocases,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( X3 != nil_list_a )
=> ~ ! [V: list_a,Va: list_list_a] :
( X3
!= ( cons_list_a @ V @ Va ) ) ) ) ).
% ring.dense_repr.cases
thf(fact_972_ring_Odense__repr_Ocases,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: list_a] :
( ( ring_a_b @ R2 )
=> ( ( X3 != nil_a )
=> ~ ! [V: a,Va: list_a] :
( X3
!= ( cons_a @ V @ Va ) ) ) ) ).
% ring.dense_repr.cases
thf(fact_973_univ__poly__one,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a] :
( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R2 @ K2 ) )
= ( cons_a @ ( one_a_ring_ext_a_b @ R2 ) @ nil_a ) ) ).
% univ_poly_one
thf(fact_974_ring_Opoly__mult_Ocases,axiom,
! [R2: partia2670972154091845814t_unit,X3: produc7709606177366032167list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ! [P22: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ nil_list_a @ P22 ) )
=> ~ ! [V: list_a,Va: list_list_a,P22: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ ( cons_list_a @ V @ Va ) @ P22 ) ) ) ) ).
% ring.poly_mult.cases
thf(fact_975_ring_Opoly__mult_Ocases,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: produc9164743771328383783list_a] :
( ( ring_a_b @ R2 )
=> ( ! [P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ nil_a @ P22 ) )
=> ~ ! [V: a,Va: list_a,P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ Va ) @ P22 ) ) ) ) ).
% ring.poly_mult.cases
thf(fact_976_var__def,axiom,
( var_se6008125447796440765t_unit
= ( ^ [R: partia7496981018696276118t_unit] : ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ nil_set_list_a ) ) ) ) ).
% var_def
thf(fact_977_var__def,axiom,
( var_a_b
= ( ^ [R: partia2175431115845679010xt_a_b] : ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) ) ) ).
% var_def
thf(fact_978_domain_Oone__is__polynomial,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( polyno1315193887021588240t_unit @ R2 @ K2 @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R2 ) @ nil_list_a ) ) ) ) ).
% domain.one_is_polynomial
thf(fact_979_domain_Oone__is__polynomial,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( polynomial_a_b @ R2 @ K2 @ ( cons_a @ ( one_a_ring_ext_a_b @ R2 ) @ nil_a ) ) ) ) ).
% domain.one_is_polynomial
thf(fact_980_subringE_I2_J,axiom,
! [H: set_set_list_a,R2: partia7496981018696276118t_unit] :
( ( subrin5643252653130547402t_unit @ H @ R2 )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R2 ) @ H ) ) ).
% subringE(2)
thf(fact_981_subringE_I2_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R2 )
=> ( member_a @ ( zero_a_b @ R2 ) @ H ) ) ).
% subringE(2)
thf(fact_982_subringE_I7_J,axiom,
! [H: set_list_a,R2: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R2 )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R2 @ H1 @ H2 ) @ H ) ) ) ) ).
% subringE(7)
thf(fact_983_subringE_I7_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subring_a_b @ H @ R2 )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( add_a_b @ R2 @ H1 @ H2 ) @ H ) ) ) ) ).
% subringE(7)
thf(fact_984_subfieldE_I4_J,axiom,
! [K2: set_list_a,R2: partia2670972154091845814t_unit,K1: list_a,K22: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_a @ K1 @ K2 )
=> ( ( member_list_a @ K22 @ K2 )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ K1 @ K22 )
= ( mult_l7073676228092353617t_unit @ R2 @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_985_subfieldE_I4_J,axiom,
! [K2: set_a,R2: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_a @ K1 @ K2 )
=> ( ( member_a @ K22 @ K2 )
=> ( ( mult_a_ring_ext_a_b @ R2 @ K1 @ K22 )
= ( mult_a_ring_ext_a_b @ R2 @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_986_subringE_I6_J,axiom,
! [H: set_list_a,R2: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R2 )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R2 @ H1 @ H2 ) @ H ) ) ) ) ).
% subringE(6)
thf(fact_987_subringE_I6_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subring_a_b @ H @ R2 )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R2 @ H1 @ H2 ) @ H ) ) ) ) ).
% subringE(6)
thf(fact_988_subfieldE_I1_J,axiom,
! [K2: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K2 @ R2 )
=> ( subring_a_b @ K2 @ R2 ) ) ).
% subfieldE(1)
thf(fact_989_subringE_I3_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R2 )
=> ( member_a @ ( one_a_ring_ext_a_b @ R2 ) @ H ) ) ).
% subringE(3)
thf(fact_990_subcringE_I2_J,axiom,
! [H: set_set_list_a,R2: partia7496981018696276118t_unit] :
( ( subcri7783154434480317835t_unit @ H @ R2 )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R2 ) @ H ) ) ).
% subcringE(2)
thf(fact_991_subcringE_I2_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R2 )
=> ( member_a @ ( zero_a_b @ R2 ) @ H ) ) ).
% subcringE(2)
thf(fact_992_subcringE_I7_J,axiom,
! [H: set_list_a,R2: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R2 )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R2 @ H1 @ H2 ) @ H ) ) ) ) ).
% subcringE(7)
thf(fact_993_subcringE_I7_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subcring_a_b @ H @ R2 )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( add_a_b @ R2 @ H1 @ H2 ) @ H ) ) ) ) ).
% subcringE(7)
thf(fact_994_subcring_Osub__m__comm,axiom,
! [H: set_list_a,R2: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R2 )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ H1 @ H2 )
= ( mult_l7073676228092353617t_unit @ R2 @ H2 @ H1 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_995_subcring_Osub__m__comm,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subcring_a_b @ H @ R2 )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( ( mult_a_ring_ext_a_b @ R2 @ H1 @ H2 )
= ( mult_a_ring_ext_a_b @ R2 @ H2 @ H1 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_996_subcringE_I6_J,axiom,
! [H: set_list_a,R2: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R2 )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R2 @ H1 @ H2 ) @ H ) ) ) ) ).
% subcringE(6)
thf(fact_997_subcringE_I6_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subcring_a_b @ H @ R2 )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R2 @ H1 @ H2 ) @ H ) ) ) ) ).
% subcringE(6)
thf(fact_998_domain_Osubring__degree__one__associatedI,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,A: list_a,A5: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R2 )
=> ( ( member_list_a @ A @ K2 )
=> ( ( member_list_a @ A5 @ K2 )
=> ( ( member_list_a @ B @ K2 )
=> ( ( ( mult_l7073676228092353617t_unit @ R2 @ A @ A5 )
= ( one_li8328186300101108157t_unit @ R2 ) )
=> ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K2 ) @ ( cons_list_a @ A @ ( cons_list_a @ B @ nil_list_a ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R2 ) @ ( cons_list_a @ ( mult_l7073676228092353617t_unit @ R2 @ A5 @ B ) @ nil_list_a ) ) ) ) ) ) ) ) ) ).
% domain.subring_degree_one_associatedI
thf(fact_999_domain_Osubring__degree__one__associatedI,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,A: a,A5: a,B: a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ K2 @ R2 )
=> ( ( member_a @ A @ K2 )
=> ( ( member_a @ A5 @ K2 )
=> ( ( member_a @ B @ K2 )
=> ( ( ( mult_a_ring_ext_a_b @ R2 @ A @ A5 )
= ( one_a_ring_ext_a_b @ R2 ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R2 @ K2 ) @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ R2 ) @ ( cons_a @ ( mult_a_ring_ext_a_b @ R2 @ A5 @ B ) @ nil_a ) ) ) ) ) ) ) ) ) ).
% domain.subring_degree_one_associatedI
thf(fact_1000_subfieldE_I2_J,axiom,
! [K2: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K2 @ R2 )
=> ( subcring_a_b @ K2 @ R2 ) ) ).
% subfieldE(2)
thf(fact_1001_subcring_Oaxioms_I1_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R2 )
=> ( subring_a_b @ H @ R2 ) ) ).
% subcring.axioms(1)
thf(fact_1002_subdomainE_I2_J,axiom,
! [H: set_set_list_a,R2: partia7496981018696276118t_unit] :
( ( subdom3220114454046903646t_unit @ H @ R2 )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R2 ) @ H ) ) ).
% subdomainE(2)
thf(fact_1003_subdomainE_I2_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R2 )
=> ( member_a @ ( zero_a_b @ R2 ) @ H ) ) ).
% subdomainE(2)
thf(fact_1004_subdomainE_I7_J,axiom,
! [H: set_list_a,R2: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R2 )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R2 @ H1 @ H2 ) @ H ) ) ) ) ).
% subdomainE(7)
thf(fact_1005_subdomainE_I7_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subdomain_a_b @ H @ R2 )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( add_a_b @ R2 @ H1 @ H2 ) @ H ) ) ) ) ).
% subdomainE(7)
thf(fact_1006_subdomainE_I8_J,axiom,
! [H: set_list_a,R2: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R2 )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ H1 @ H2 )
= ( mult_l7073676228092353617t_unit @ R2 @ H2 @ H1 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_1007_subdomainE_I8_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subdomain_a_b @ H @ R2 )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( ( mult_a_ring_ext_a_b @ R2 @ H1 @ H2 )
= ( mult_a_ring_ext_a_b @ R2 @ H2 @ H1 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_1008_subdomainE_I6_J,axiom,
! [H: set_list_a,R2: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R2 )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R2 @ H1 @ H2 ) @ H ) ) ) ) ).
% subdomainE(6)
thf(fact_1009_subdomainE_I6_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subdomain_a_b @ H @ R2 )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R2 @ H1 @ H2 ) @ H ) ) ) ) ).
% subdomainE(6)
thf(fact_1010_subfield_Oaxioms_I1_J,axiom,
! [K2: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K2 @ R2 )
=> ( subdomain_a_b @ K2 @ R2 ) ) ).
% subfield.axioms(1)
thf(fact_1011_subfieldE_I3_J,axiom,
! [K2: set_list_a,R2: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) ).
% subfieldE(3)
thf(fact_1012_subfieldE_I3_J,axiom,
! [K2: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K2 @ R2 )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ).
% subfieldE(3)
thf(fact_1013_subringE_I1_J,axiom,
! [H: set_list_a,R2: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R2 )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R2 ) ) ) ).
% subringE(1)
thf(fact_1014_subringE_I1_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R2 )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ).
% subringE(1)
thf(fact_1015_subfieldE_I5_J,axiom,
! [K2: set_set_list_a,R2: partia7496981018696276118t_unit,K1: set_list_a,K22: set_list_a] :
( ( subfie4339374749748326226t_unit @ K2 @ R2 )
=> ( ( member_set_list_a @ K1 @ K2 )
=> ( ( member_set_list_a @ K22 @ K2 )
=> ( ( ( mult_s7802724872828879953t_unit @ R2 @ K1 @ K22 )
= ( zero_s2910681146719230829t_unit @ R2 ) )
=> ( ( K1
= ( zero_s2910681146719230829t_unit @ R2 ) )
| ( K22
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_1016_subfieldE_I5_J,axiom,
! [K2: set_list_a,R2: partia2670972154091845814t_unit,K1: list_a,K22: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_a @ K1 @ K2 )
=> ( ( member_list_a @ K22 @ K2 )
=> ( ( ( mult_l7073676228092353617t_unit @ R2 @ K1 @ K22 )
= ( zero_l4142658623432671053t_unit @ R2 ) )
=> ( ( K1
= ( zero_l4142658623432671053t_unit @ R2 ) )
| ( K22
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_1017_subfieldE_I5_J,axiom,
! [K2: set_a,R2: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_a @ K1 @ K2 )
=> ( ( member_a @ K22 @ K2 )
=> ( ( ( mult_a_ring_ext_a_b @ R2 @ K1 @ K22 )
= ( zero_a_b @ R2 ) )
=> ( ( K1
= ( zero_a_b @ R2 ) )
| ( K22
= ( zero_a_b @ R2 ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_1018_subfieldE_I6_J,axiom,
! [K2: set_set_list_a,R2: partia7496981018696276118t_unit] :
( ( subfie4339374749748326226t_unit @ K2 @ R2 )
=> ( ( one_se1127990129394575805t_unit @ R2 )
!= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ).
% subfieldE(6)
thf(fact_1019_subfieldE_I6_J,axiom,
! [K2: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K2 @ R2 )
=> ( ( one_a_ring_ext_a_b @ R2 )
!= ( zero_a_b @ R2 ) ) ) ).
% subfieldE(6)
thf(fact_1020_ring_Ocarrier__is__subring,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ R2 ) @ R2 ) ) ).
% ring.carrier_is_subring
thf(fact_1021_ring_Ocarrier__is__subring,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R2 )
=> ( subring_a_b @ ( partia707051561876973205xt_a_b @ R2 ) @ R2 ) ) ).
% ring.carrier_is_subring
thf(fact_1022_subcringE_I1_J,axiom,
! [H: set_list_a,R2: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R2 )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R2 ) ) ) ).
% subcringE(1)
thf(fact_1023_subcringE_I1_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R2 )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ).
% subcringE(1)
thf(fact_1024_field_Ocarrier__is__subfield,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R2 )
=> ( subfie4339374749748326226t_unit @ ( partia141011252114345353t_unit @ R2 ) @ R2 ) ) ).
% field.carrier_is_subfield
thf(fact_1025_field_Ocarrier__is__subfield,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( subfie1779122896746047282t_unit @ ( partia5361259788508890537t_unit @ R2 ) @ R2 ) ) ).
% field.carrier_is_subfield
thf(fact_1026_field_Ocarrier__is__subfield,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R2 )
=> ( subfield_a_b @ ( partia707051561876973205xt_a_b @ R2 ) @ R2 ) ) ).
% field.carrier_is_subfield
thf(fact_1027_subdomainE_I1_J,axiom,
! [H: set_list_a,R2: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R2 )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R2 ) ) ) ).
% subdomainE(1)
thf(fact_1028_subdomainE_I1_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R2 )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ).
% subdomainE(1)
thf(fact_1029_subdomain_Osubintegral,axiom,
! [H: set_set_list_a,R2: partia7496981018696276118t_unit,H1: set_list_a,H2: set_list_a] :
( ( subdom3220114454046903646t_unit @ H @ R2 )
=> ( ( member_set_list_a @ H1 @ H )
=> ( ( member_set_list_a @ H2 @ H )
=> ( ( ( mult_s7802724872828879953t_unit @ R2 @ H1 @ H2 )
= ( zero_s2910681146719230829t_unit @ R2 ) )
=> ( ( H1
= ( zero_s2910681146719230829t_unit @ R2 ) )
| ( H2
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_1030_subdomain_Osubintegral,axiom,
! [H: set_list_a,R2: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R2 )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( ( ( mult_l7073676228092353617t_unit @ R2 @ H1 @ H2 )
= ( zero_l4142658623432671053t_unit @ R2 ) )
=> ( ( H1
= ( zero_l4142658623432671053t_unit @ R2 ) )
| ( H2
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_1031_subdomain_Osubintegral,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subdomain_a_b @ H @ R2 )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ R2 @ H1 @ H2 )
= ( zero_a_b @ R2 ) )
=> ( ( H1
= ( zero_a_b @ R2 ) )
| ( H2
= ( zero_a_b @ R2 ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_1032_cring_Ocarrier__is__subcring,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( subcri7763218559781929323t_unit @ ( partia5361259788508890537t_unit @ R2 ) @ R2 ) ) ).
% cring.carrier_is_subcring
thf(fact_1033_cring_Ocarrier__is__subcring,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R2 )
=> ( subcring_a_b @ ( partia707051561876973205xt_a_b @ R2 ) @ R2 ) ) ).
% cring.carrier_is_subcring
thf(fact_1034_subdomain_Osub__one__not__zero,axiom,
! [H: set_set_list_a,R2: partia7496981018696276118t_unit] :
( ( subdom3220114454046903646t_unit @ H @ R2 )
=> ( ( one_se1127990129394575805t_unit @ R2 )
!= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_1035_subdomain_Osub__one__not__zero,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R2 )
=> ( ( one_a_ring_ext_a_b @ R2 )
!= ( zero_a_b @ R2 ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_1036_cring_OsubcringI_H,axiom,
! [R2: partia2670972154091845814t_unit,H: set_list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ H @ R2 )
=> ( subcri7763218559781929323t_unit @ H @ R2 ) ) ) ).
% cring.subcringI'
thf(fact_1037_cring_OsubcringI_H,axiom,
! [R2: partia2175431115845679010xt_a_b,H: set_a] :
( ( cring_a_b @ R2 )
=> ( ( subring_a_b @ H @ R2 )
=> ( subcring_a_b @ H @ R2 ) ) ) ).
% cring.subcringI'
thf(fact_1038_domain_OsubdomainI_H,axiom,
! [R2: partia2670972154091845814t_unit,H: set_list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ H @ R2 )
=> ( subdom7821232466298058046t_unit @ H @ R2 ) ) ) ).
% domain.subdomainI'
thf(fact_1039_domain_OsubdomainI_H,axiom,
! [R2: partia2175431115845679010xt_a_b,H: set_a] :
( ( domain_a_b @ R2 )
=> ( ( subring_a_b @ H @ R2 )
=> ( subdomain_a_b @ H @ R2 ) ) ) ).
% domain.subdomainI'
thf(fact_1040_ring_OsubcringI,axiom,
! [R2: partia2670972154091845814t_unit,H: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( subrin6918843898125473962t_unit @ H @ R2 )
=> ( ! [H12: list_a,H22: list_a] :
( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ H12 @ H22 )
= ( mult_l7073676228092353617t_unit @ R2 @ H22 @ H12 ) ) ) )
=> ( subcri7763218559781929323t_unit @ H @ R2 ) ) ) ) ).
% ring.subcringI
thf(fact_1041_ring_OsubcringI,axiom,
! [R2: partia2175431115845679010xt_a_b,H: set_a] :
( ( ring_a_b @ R2 )
=> ( ( subring_a_b @ H @ R2 )
=> ( ! [H12: a,H22: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( mult_a_ring_ext_a_b @ R2 @ H12 @ H22 )
= ( mult_a_ring_ext_a_b @ R2 @ H22 @ H12 ) ) ) )
=> ( subcring_a_b @ H @ R2 ) ) ) ) ).
% ring.subcringI
thf(fact_1042_poly__mult__var,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ( P = nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ ( var_a_b @ r ) )
= nil_a ) )
& ( ( P != nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P @ ( var_a_b @ r ) )
= ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ).
% poly_mult_var
thf(fact_1043_is__root__imp__pdivides,axiom,
! [P: list_a,X3: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X3 )
=> ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X3 ) @ nil_a ) ) @ P ) ) ) ).
% is_root_imp_pdivides
thf(fact_1044_ring_Oline__extension__smult__closed,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,E: set_list_a,A: list_a,K3: list_a,U: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ! [K4: list_a,V: list_a] :
( ( member_list_a @ K4 @ K2 )
=> ( ( member_list_a @ V @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R2 @ K4 @ V ) @ E ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ K3 @ K2 )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ R2 @ K2 @ A @ E ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R2 @ K3 @ U ) @ ( embedd5150658419831591667t_unit @ R2 @ K2 @ A @ E ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_1045_ring_Oline__extension__smult__closed,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,E: set_a,A: a,K3: a,U: a] :
( ( ring_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ! [K4: a,V: a] :
( ( member_a @ K4 @ K2 )
=> ( ( member_a @ V @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R2 @ K4 @ V ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ K3 @ K2 )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ R2 @ K2 @ A @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R2 @ K3 @ U ) @ ( embedd971793762689825387on_a_b @ R2 @ K2 @ A @ E ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_1046_eval__as__unique__hom,axiom,
! [K2: set_a,X3: a,H3: list_a > a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K2 ) @ r @ H3 )
=> ( ! [K4: a] :
( ( member_a @ K4 @ K2 )
=> ( ( H3 @ ( cons_a @ K4 @ nil_a ) )
= K4 ) )
=> ( ( ( H3 @ ( var_a_b @ r ) )
= X3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( H3 @ P )
= ( eval_a_b @ r @ P @ X3 ) ) ) ) ) ) ) ) ).
% eval_as_unique_hom
thf(fact_1047_subring__props_I5_J,axiom,
! [K2: set_a,H3: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H3 @ K2 )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ K2 ) ) ) ).
% subring_props(5)
thf(fact_1048_add_Oinv__mult__group,axiom,
! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X3 @ Y3 ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y3 ) @ ( a_inv_a_b @ r @ X3 ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_1049_add_Oinv__solve__left,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left
thf(fact_1050_add_Oinv__solve__left_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
= A )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left'
thf(fact_1051_add_Oinv__solve__right,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right
thf(fact_1052_add_Oinv__solve__right_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
= A )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right'
thf(fact_1053_a__transpose__inv,axiom,
! [X3: a,Y3: a,Z2: a] :
( ( ( add_a_b @ r @ X3 @ Y3 )
= Z2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ Z2 )
= Y3 ) ) ) ) ) ).
% a_transpose_inv
thf(fact_1054_local_Ominus__add,axiom,
! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X3 @ Y3 ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ ( a_inv_a_b @ r @ Y3 ) ) ) ) ) ).
% local.minus_add
thf(fact_1055_r__neg1,axiom,
! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ ( add_a_b @ r @ X3 @ Y3 ) )
= Y3 ) ) ) ).
% r_neg1
thf(fact_1056_r__neg2,axiom,
! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ Y3 ) )
= Y3 ) ) ) ).
% r_neg2
thf(fact_1057_l__minus,axiom,
! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ Y3 )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 ) ) ) ) ) ).
% l_minus
thf(fact_1058_r__minus,axiom,
! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( a_inv_a_b @ r @ Y3 ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 ) ) ) ) ) ).
% r_minus
thf(fact_1059_minus__eq,axiom,
! [X3: a,Y3: a] :
( ( a_minus_a_b @ r @ X3 @ Y3 )
= ( add_a_b @ r @ X3 @ ( a_inv_a_b @ r @ Y3 ) ) ) ).
% minus_eq
thf(fact_1060_l__neg,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ X3 )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_1061_minus__equality,axiom,
! [Y3: a,X3: a] :
( ( ( add_a_b @ r @ Y3 @ X3 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X3 )
= Y3 ) ) ) ) ).
% minus_equality
thf(fact_1062_r__neg,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( a_inv_a_b @ r @ X3 ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_1063_sum__zero__eq__neg,axiom,
! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X3 @ Y3 )
= ( zero_a_b @ r ) )
=> ( X3
= ( a_inv_a_b @ r @ Y3 ) ) ) ) ) ).
% sum_zero_eq_neg
thf(fact_1064_square__eq__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X3 @ X3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( X3
= ( one_a_ring_ext_a_b @ r ) )
| ( X3
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% square_eq_one
thf(fact_1065_monic__degree__one__root__condition,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A ) @ nil_a ) ) @ B )
= ( A = B ) ) ) ).
% monic_degree_one_root_condition
thf(fact_1066_subringI,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ H )
=> ( ! [H4: a] :
( ( member_a @ H4 @ H )
=> ( member_a @ ( a_inv_a_b @ r @ H4 ) @ H ) )
=> ( ! [H12: a,H22: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H22 ) @ H ) ) )
=> ( ! [H12: a,H22: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( add_a_b @ r @ H12 @ H22 ) @ H ) ) )
=> ( subring_a_b @ H @ r ) ) ) ) ) ) ).
% subringI
thf(fact_1067_const__term__zero,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= ( zero_a_b @ r ) )
=> ~ ! [P5: list_a] :
( ( polynomial_a_b @ r @ K2 @ P5 )
=> ( ( P5 != nil_a )
=> ( P
!= ( append_a @ P5 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ) ) ).
% const_term_zero
thf(fact_1068_pdivides__imp__is__root,axiom,
! [P: list_a,X3: a] :
( ( P != nil_a )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X3 ) @ nil_a ) ) @ P )
=> ( polyno4133073214067823460ot_a_b @ r @ P @ X3 ) ) ) ) ).
% pdivides_imp_is_root
thf(fact_1069_determination__of__hom,axiom,
! [K2: set_a,A3: partia2175431115845679010xt_a_b,H3: list_a > a,G3: list_a > a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K2 ) @ A3 @ H3 )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K2 ) @ A3 @ G3 )
=> ( ! [K4: a] :
( ( member_a @ K4 @ K2 )
=> ( ( H3 @ ( cons_a @ K4 @ nil_a ) )
= ( G3 @ ( cons_a @ K4 @ nil_a ) ) ) )
=> ( ( ( H3 @ ( var_a_b @ r ) )
= ( G3 @ ( var_a_b @ r ) ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( H3 @ P )
= ( G3 @ P ) ) ) ) ) ) ) ) ).
% determination_of_hom
thf(fact_1070_a__inv__closed,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_1071_local_Ominus__minus,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X3 ) )
= X3 ) ) ).
% local.minus_minus
thf(fact_1072_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_1073_add_Oinv__eq__1__iff,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X3 )
= ( zero_a_b @ r ) )
= ( X3
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_1074_Units__minus__one__closed,axiom,
member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_minus_one_closed
thf(fact_1075_subringE_I5_J,axiom,
! [H: set_list_a,R2: partia2670972154091845814t_unit,H3: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R2 )
=> ( ( member_list_a @ H3 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R2 @ H3 ) @ H ) ) ) ).
% subringE(5)
thf(fact_1076_subringE_I5_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b,H3: a] :
( ( subring_a_b @ H @ R2 )
=> ( ( member_a @ H3 @ H )
=> ( member_a @ ( a_inv_a_b @ R2 @ H3 ) @ H ) ) ) ).
% subringE(5)
thf(fact_1077_ring_Osubring__props_I5_J,axiom,
! [R2: partia2670972154091845814t_unit,K2: set_list_a,H3: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R2 )
=> ( ( member_list_a @ H3 @ K2 )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R2 @ H3 ) @ K2 ) ) ) ) ).
% ring.subring_props(5)
thf(fact_1078_ring_Osubring__props_I5_J,axiom,
! [R2: partia2175431115845679010xt_a_b,K2: set_a,H3: a] :
( ( ring_a_b @ R2 )
=> ( ( subfield_a_b @ K2 @ R2 )
=> ( ( member_a @ H3 @ K2 )
=> ( member_a @ ( a_inv_a_b @ R2 @ H3 ) @ K2 ) ) ) ) ).
% ring.subring_props(5)
thf(fact_1079_ring_Oring__simprules_I3_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R2 @ X3 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_1080_ring_Oring__simprules_I3_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ ( a_inv_a_b @ R2 @ X3 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_1081_ring_Oring__simprules_I20_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( a_inv_8944721093294617173t_unit @ R2 @ ( a_inv_8944721093294617173t_unit @ R2 @ X3 ) )
= X3 ) ) ) ).
% ring.ring_simprules(20)
thf(fact_1082_ring_Oring__simprules_I20_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( a_inv_a_b @ R2 @ ( a_inv_a_b @ R2 @ X3 ) )
= X3 ) ) ) ).
% ring.ring_simprules(20)
thf(fact_1083_ring_Ominus__zero,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R2 )
=> ( ( a_inv_8944721093294617173t_unit @ R2 @ ( zero_l4142658623432671053t_unit @ R2 ) )
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ).
% ring.minus_zero
thf(fact_1084_ring_Ominus__zero,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( ring_s8247141995668492373t_unit @ R2 )
=> ( ( a_inv_5715216516650856053t_unit @ R2 @ ( zero_s2910681146719230829t_unit @ R2 ) )
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ).
% ring.minus_zero
thf(fact_1085_ring_Ominus__zero,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R2 )
=> ( ( a_inv_a_b @ R2 @ ( zero_a_b @ R2 ) )
= ( zero_a_b @ R2 ) ) ) ).
% ring.minus_zero
thf(fact_1086_cring_Ocring__simprules_I3_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R2 @ X3 ) @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ).
% cring.cring_simprules(3)
thf(fact_1087_cring_Ocring__simprules_I3_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ ( a_inv_a_b @ R2 @ X3 ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ).
% cring.cring_simprules(3)
thf(fact_1088_cring_Ocring__simprules_I21_J,axiom,
! [R2: partia2670972154091845814t_unit,X3: list_a] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( a_inv_8944721093294617173t_unit @ R2 @ ( a_inv_8944721093294617173t_unit @ R2 @ X3 ) )
= X3 ) ) ) ).
% cring.cring_simprules(21)
thf(fact_1089_cring_Ocring__simprules_I21_J,axiom,
! [R2: partia2175431115845679010xt_a_b,X3: a] :
( ( cring_a_b @ R2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( a_inv_a_b @ R2 @ ( a_inv_a_b @ R2 @ X3 ) )
= X3 ) ) ) ).
% cring.cring_simprules(21)
thf(fact_1090_cring_Ocring__simprules_I22_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R2 )
=> ( ( a_inv_8944721093294617173t_unit @ R2 @ ( zero_l4142658623432671053t_unit @ R2 ) )
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ).
% cring.cring_simprules(22)
thf(fact_1091_cring_Ocring__simprules_I22_J,axiom,
! [R2: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R2 )
=> ( ( a_inv_5715216516650856053t_unit @ R2 @ ( zero_s2910681146719230829t_unit @ R2 ) )
= ( zero_s2910681146719230829t_unit @ R2 ) ) ) ).
% cring.cring_simprules(22)
thf(fact_1092_cring_Ocring__simprules_I22_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R2 )
=> ( ( a_inv_a_b @ R2 @ ( zero_a_b @ R2 ) )
= ( zero_a_b @ R2 ) ) ) ).
% cring.cring_simprules(22)
thf(fact_1093_abelian__group_Oa__inv__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( a_inv_a_b @ G @ X3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% abelian_group.a_inv_closed
thf(fact_1094_roots__inclI,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A2 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P @ A2 ) ) @ Q ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).
% roots_inclI
thf(fact_1095_le__alg__mult__imp__pdivides,axiom,
! [X3: a,P: list_a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P @ X3 ) )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X3 ) @ nil_a ) ) @ N ) @ P ) ) ) ) ).
% le_alg_mult_imp_pdivides
thf(fact_1096_alg__multE_I2_J,axiom,
! [X3: a,P: list_a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X3 ) @ nil_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P @ X3 ) ) ) ) ) ) ).
% alg_multE(2)
thf(fact_1097_alg__multE_I1_J,axiom,
! [X3: a,P: list_a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X3 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P @ X3 ) ) @ P ) ) ) ) ).
% alg_multE(1)
thf(fact_1098_univ__poly__a__inv__consistent,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ) ).
% univ_poly_a_inv_consistent
thf(fact_1099_univ__poly__a__inv__length,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P ) )
= ( size_size_list_a @ P ) ) ) ) ).
% univ_poly_a_inv_length
thf(fact_1100_long__division__a__inv_I2_J,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% long_division_a_inv(2)
thf(fact_1101_long__division__a__inv_I1_J,axiom,
! [K2: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% long_division_a_inv(1)
thf(fact_1102_const__term__simprules__shell_I4_J,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( const_term_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P ) )
= ( a_inv_a_b @ r @ ( const_term_a_b @ r @ P ) ) ) ) ) ).
% const_term_simprules_shell(4)
thf(fact_1103_not__empty__rootsE,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( polynomial_roots_a_b @ r @ P )
!= zero_zero_multiset_a )
=> ~ ! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( set_mset_a @ ( polynomial_roots_a_b @ r @ P ) ) )
=> ( ( member_list_a @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A2 ) @ nil_a ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ~ ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A2 ) @ nil_a ) ) @ P ) ) ) ) ) ) ).
% not_empty_rootsE
thf(fact_1104_eval__append,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P @ Q ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( pow_a_1026414303147256608_b_nat @ r @ A @ ( size_size_list_a @ Q ) ) ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_append
thf(fact_1105_eval__append__aux,axiom,
! [P: list_a,B: a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P @ ( cons_a @ B @ nil_a ) ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ A ) @ B ) ) ) ) ) ).
% eval_append_aux
thf(fact_1106_polynomial__incl,axiom,
! [K2: set_a,P: list_a] :
( ( polynomial_a_b @ r @ K2 @ P )
=> ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K2 ) ) ).
% polynomial_incl
thf(fact_1107_coeff__in__carrier,axiom,
! [P: list_a,I: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( coeff_a_b @ r @ P @ I ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% coeff_in_carrier
thf(fact_1108_eval__in__carrier,axiom,
! [P: list_a,X3: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier
thf(fact_1109_const__term__simprules_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( const_term_a_b @ r @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% const_term_simprules(1)
thf(fact_1110_const__term__eq__last,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( append_a @ P @ ( cons_a @ A @ nil_a ) ) )
= A ) ) ) ).
% const_term_eq_last
thf(fact_1111_const__term__explicit,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= A )
=> ~ ! [P5: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P5 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( P
!= ( append_a @ P5 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ).
% const_term_explicit
thf(fact_1112_monom__in__carrier,axiom,
! [A: a,N: nat] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( monom_a_b @ r @ A @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% monom_in_carrier
thf(fact_1113_polynomial__in__carrier,axiom,
! [K2: set_a,P: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P )
=> ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% polynomial_in_carrier
thf(fact_1114_exp__base__closed,axiom,
! [X3: a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X3 @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% exp_base_closed
thf(fact_1115_degree__one__roots,axiom,
! [A: a,A5: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ A5 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( polynomial_roots_a_b @ r @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) )
= ( add_mset_a @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A5 @ B ) ) @ zero_zero_multiset_a ) ) ) ) ) ) ).
% degree_one_roots
thf(fact_1116_monic__degree__one__roots,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polynomial_roots_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A ) @ nil_a ) ) )
= ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ).
% monic_degree_one_roots
thf(fact_1117_roots__inclI_H,axiom,
! [P: list_a,M: multiset_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( ord_less_eq_nat @ ( polyno4422430861927485590lt_a_b @ r @ P @ A2 ) @ ( count_a @ M @ A2 ) ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ M ) ) ) ).
% roots_inclI'
thf(fact_1118_alg__mult__eq__count__roots,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4422430861927485590lt_a_b @ r @ P )
= ( count_a @ ( polynomial_roots_a_b @ r @ P ) ) ) ) ).
% alg_mult_eq_count_roots
thf(fact_1119_factors__mult,axiom,
! [Fa: list_a,A: a,Fb: list_a,B: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fa @ A )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% factors_mult
thf(fact_1120_a__r__coset__subset__G,axiom,
! [H: set_a,X3: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_r_coset_a_b @ r @ H @ X3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_r_coset_subset_G
thf(fact_1121_a__rcosI,axiom,
! [H3: a,H: set_a,X3: a] :
( ( member_a @ H3 @ H )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ H3 @ X3 ) @ ( a_r_coset_a_b @ r @ H @ X3 ) ) ) ) ) ).
% a_rcosI
thf(fact_1122_a__coset__add__assoc,axiom,
! [M2: set_a,G3: a,H3: a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ ( a_r_coset_a_b @ r @ M2 @ G3 ) @ H3 )
= ( a_r_coset_a_b @ r @ M2 @ ( add_a_b @ r @ G3 @ H3 ) ) ) ) ) ) ).
% a_coset_add_assoc
thf(fact_1123_a__coset__add__inv1,axiom,
! [M2: set_a,X3: a,Y3: a] :
( ( ( a_r_coset_a_b @ r @ M2 @ ( add_a_b @ r @ X3 @ ( a_inv_a_b @ r @ Y3 ) ) )
= M2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ M2 @ X3 )
= ( a_r_coset_a_b @ r @ M2 @ Y3 ) ) ) ) ) ) ).
% a_coset_add_inv1
thf(fact_1124_a__coset__add__inv2,axiom,
! [M2: set_a,X3: a,Y3: a] :
( ( ( a_r_coset_a_b @ r @ M2 @ X3 )
= ( a_r_coset_a_b @ r @ M2 @ Y3 ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ M2 @ ( add_a_b @ r @ X3 @ ( a_inv_a_b @ r @ Y3 ) ) )
= M2 ) ) ) ) ) ).
% a_coset_add_inv2
thf(fact_1125_factors__closed,axiom,
! [Fs: list_a,A: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% factors_closed
thf(fact_1126_factors__dividesI,axiom,
! [Fs: list_a,A: a,F: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fs @ A )
=> ( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ F @ A ) ) ) ) ).
% factors_dividesI
thf(fact_1127_factors__mult__single,axiom,
! [A: a,Fb: list_a,B: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( cons_a @ A @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% factors_mult_single
thf(fact_1128_a__coset__add__zero,axiom,
! [M2: set_a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ M2 @ ( zero_a_b @ r ) )
= M2 ) ) ).
% a_coset_add_zero
thf(fact_1129_add_Oone__in__subset,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ H )
=> ( member_a @ ( a_inv_a_b @ r @ X ) @ H ) )
=> ( ! [X: a] :
( ( member_a @ X @ H )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ H )
=> ( member_a @ ( add_a_b @ r @ X @ Xa2 ) @ H ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H ) ) ) ) ) ).
% add.one_in_subset
thf(fact_1130_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_1131_subring__props_I4_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( K2 != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_1132_a__rcosetsI,axiom,
! [H: set_a,X3: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_set_a @ ( a_r_coset_a_b @ r @ H @ X3 ) @ ( a_RCOSETS_a_b @ r @ H ) ) ) ) ).
% a_rcosetsI
thf(fact_1133_ee__trans,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
=> ( ( essent8953798148185448568xt_a_b @ r @ Bs @ Cs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( essent8953798148185448568xt_a_b @ r @ As @ Cs ) ) ) ) ) ) ).
% ee_trans
thf(fact_1134_ee__length,axiom,
! [As: list_a,Bs: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
=> ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) ) ) ).
% ee_length
thf(fact_1135_ee__sym,axiom,
! [As: list_a,Bs: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( essent8953798148185448568xt_a_b @ r @ Bs @ As ) ) ) ) ).
% ee_sym
thf(fact_1136_ee__refl,axiom,
! [As: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( essent8953798148185448568xt_a_b @ r @ As @ As ) ) ).
% ee_refl
thf(fact_1137_combine__append__zero,axiom,
! [Us2: list_a,Ks2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( append_a @ Ks2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Us2 )
= ( embedded_combine_a_b @ r @ Ks2 @ Us2 ) ) ) ).
% combine_append_zero
thf(fact_1138_a__lcos__m__assoc,axiom,
! [M2: set_a,G3: a,H3: a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G3 @ ( a_l_coset_a_b @ r @ H3 @ M2 ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G3 @ H3 ) @ M2 ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_1139_combine_Osimps_I3_J,axiom,
! [Ks2: list_a] :
( ( embedded_combine_a_b @ r @ Ks2 @ nil_a )
= ( zero_a_b @ r ) ) ).
% combine.simps(3)
thf(fact_1140_combine_Osimps_I2_J,axiom,
! [Us2: list_a] :
( ( embedded_combine_a_b @ r @ nil_a @ Us2 )
= ( zero_a_b @ r ) ) ).
% combine.simps(2)
thf(fact_1141_a__l__coset__subset__G,axiom,
! [H: set_a,X3: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X3 @ H ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_1142_combine_Osimps_I1_J,axiom,
! [K3: a,Ks2: list_a,U: a,Us2: list_a] :
( ( embedded_combine_a_b @ r @ ( cons_a @ K3 @ Ks2 ) @ ( cons_a @ U @ Us2 ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K3 @ U ) @ ( embedded_combine_a_b @ r @ Ks2 @ Us2 ) ) ) ).
% combine.simps(1)
thf(fact_1143_a__lcos__mult__one,axiom,
! [M2: set_a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M2 )
= M2 ) ) ).
% a_lcos_mult_one
thf(fact_1144_combine__eq__eval,axiom,
! [Ks2: list_a,X3: a] :
( ( embedded_combine_a_b @ r @ Ks2 @ ( polyno2922411391617481336se_a_b @ r @ X3 @ ( size_size_list_a @ Ks2 ) ) )
= ( eval_a_b @ r @ Ks2 @ X3 ) ) ).
% combine_eq_eval
thf(fact_1145_combine_Oelims,axiom,
! [X3: list_a,Xa3: list_a,Y3: a] :
( ( ( embedded_combine_a_b @ r @ X3 @ Xa3 )
= Y3 )
=> ( ! [K4: a,Ks: list_a] :
( ( X3
= ( cons_a @ K4 @ Ks ) )
=> ! [U2: a,Us: list_a] :
( ( Xa3
= ( cons_a @ U2 @ Us ) )
=> ( Y3
!= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K4 @ U2 ) @ ( embedded_combine_a_b @ r @ Ks @ Us ) ) ) ) )
=> ( ( ( X3 = nil_a )
=> ( Y3
!= ( zero_a_b @ r ) ) )
=> ~ ( ( Xa3 = nil_a )
=> ( Y3
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% combine.elims
thf(fact_1146_combine__append,axiom,
! [Ks2: list_a,Us2: list_a,Ks3: list_a,Vs: list_a] :
( ( ( size_size_list_a @ Ks2 )
= ( size_size_list_a @ Us2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( embedded_combine_a_b @ r @ Ks2 @ Us2 ) @ ( embedded_combine_a_b @ r @ Ks3 @ Vs ) )
= ( embedded_combine_a_b @ r @ ( append_a @ Ks2 @ Ks3 ) @ ( append_a @ Us2 @ Vs ) ) ) ) ) ) ) ) ).
% combine_append
thf(fact_1147_combine__in__carrier,axiom,
! [Ks2: list_a,Us2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( embedded_combine_a_b @ r @ Ks2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% combine_in_carrier
thf(fact_1148_multlist__dividesI,axiom,
! [F: a,Fs: list_a] :
( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ F @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% multlist_dividesI
thf(fact_1149_a__rcos__assoc__lcos,axiom,
! [H: set_a,K2: set_a,X3: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ ( a_r_coset_a_b @ r @ H @ X3 ) @ K2 )
= ( set_add_a_b @ r @ H @ ( a_l_coset_a_b @ r @ X3 @ K2 ) ) ) ) ) ) ).
% a_rcos_assoc_lcos
thf(fact_1150_setadd__subset__G,axiom,
! [H: set_a,K2: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H @ K2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% setadd_subset_G
thf(fact_1151_set__add__comm,axiom,
! [I2: set_a,J: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ J @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ I2 @ J )
= ( set_add_a_b @ r @ J @ I2 ) ) ) ) ).
% set_add_comm
thf(fact_1152_set__add__closed,axiom,
! [A3: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A3 @ B5 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_add_closed
thf(fact_1153_a__setmult__rcos__assoc,axiom,
! [H: set_a,K2: set_a,X3: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ H @ ( a_r_coset_a_b @ r @ K2 @ X3 ) )
= ( a_r_coset_a_b @ r @ ( set_add_a_b @ r @ H @ K2 ) @ X3 ) ) ) ) ) ).
% a_setmult_rcos_assoc
thf(fact_1154_factorsI,axiom,
! [Fs: list_a,A: a] :
( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Fs ) )
=> ( irredu6211895646901577903xt_a_b @ r @ X ) )
=> ( ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) )
= A )
=> ( factor5638265376665762323xt_a_b @ r @ Fs @ A ) ) ) ).
% factorsI
thf(fact_1155_add_Omultlist__closed,axiom,
! [Fs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( foldr_a_a @ ( add_a_b @ r ) @ Fs @ ( zero_a_b @ r ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.multlist_closed
thf(fact_1156_multlist__closed,axiom,
! [Fs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% multlist_closed
thf(fact_1157_multlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ As @ ( one_a_ring_ext_a_b @ r ) )
= ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Bs @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% multlist_perm_cong
thf(fact_1158_add_Omultlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( foldr_a_a @ ( add_a_b @ r ) @ As @ ( zero_a_b @ r ) )
= ( foldr_a_a @ ( add_a_b @ r ) @ Bs @ ( zero_a_b @ r ) ) ) ) ) ).
% add.multlist_perm_cong
thf(fact_1159_irrlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ As ) )
=> ( irredu6211895646901577903xt_a_b @ r @ X ) )
=> ! [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Bs ) )
=> ( irredu6211895646901577903xt_a_b @ r @ X5 ) ) ) ) ).
% irrlist_perm_cong
thf(fact_1160_perm__closed,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% perm_closed
thf(fact_1161_mult__of_Omultlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ As @ ( one_a_ring_ext_a_b @ r ) )
= ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Bs @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.multlist_perm_cong
thf(fact_1162_mult__of_Omultlist__dividesI,axiom,
! [F: a,Fs: list_a] :
( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ F @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.multlist_dividesI
thf(fact_1163_mult__of_Ocomm__monoid__axioms,axiom,
comm_m7681468956318391052t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.comm_monoid_axioms
thf(fact_1164_mult__of_Oassociated__sym,axiom,
! [A: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ A ) ) ).
% mult_of.associated_sym
thf(fact_1165_mult__of_Omonoid__axioms,axiom,
monoid2746444814143937472t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.monoid_axioms
thf(fact_1166_mult__of_OUnits__assoc,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ).
% mult_of.Units_assoc
thf(fact_1167_mult__of_Omonoid__cancel__axioms,axiom,
monoid1999574367301118026t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.monoid_cancel_axioms
thf(fact_1168_mult__of_Oprime__irreducible,axiom,
! [P: a] :
( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ).
% mult_of.prime_irreducible
thf(fact_1169_zero__is__prime_I2_J,axiom,
prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ ( zero_a_b @ r ) ).
% zero_is_prime(2)
thf(fact_1170_zero__is__irreducible__mult,axiom,
irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( zero_a_b @ r ) ).
% zero_is_irreducible_mult
thf(fact_1171_mult__of_Oprime__cong,axiom,
! [P: a,P4: a] :
( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ P @ P4 )
=> ( ( member_a @ P @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ P4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P4 ) ) ) ) ) ).
% mult_of.prime_cong
thf(fact_1172_mult__of_Oassoc__subst,axiom,
! [A: a,B: a,F: a > a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ! [A2: a,B2: a] :
( ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B2 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( member_a @ ( F @ B2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% mult_of.assoc_subst
thf(fact_1173_mult__of_Oassociated__trans,axiom,
! [A: a,B: a,C: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ C ) ) ) ) ) ).
% mult_of.associated_trans
thf(fact_1174_mult__of_OUnits__closed,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.Units_closed
thf(fact_1175_mult__of_Oassoc__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.assoc_unit_r
thf(fact_1176_mult__of_Oassoc__unit__l,axiom,
! [A: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.assoc_unit_l
thf(fact_1177_mult__of_Oirreducible__cong,axiom,
! [A: a,A5: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A5 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A5 ) ) ) ) ) ).
% mult_of.irreducible_cong
thf(fact_1178_mult__of_Oee__length,axiom,
! [As: list_a,Bs: list_a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) ) ) ).
% mult_of.ee_length
thf(fact_1179_mult__of_Ounits__of__pow,axiom,
! [X3: a,N: nat] :
( ( member_a @ X3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( units_7501539392726747778t_unit @ ( ring_mult_of_a_b @ r ) ) @ X3 @ N )
= ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X3 @ N ) ) ) ).
% mult_of.units_of_pow
thf(fact_1180_mult__of_OUnits__pow__closed,axiom,
! [X3: a,D: nat] :
( ( member_a @ X3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X3 @ D ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.Units_pow_closed
thf(fact_1181_mult__of_Omonoid__comm__monoidI,axiom,
( ! [X: a,Y2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X ) ) ) )
=> ( comm_m7681468956318391052t_unit @ ( ring_mult_of_a_b @ r ) ) ) ).
% mult_of.monoid_comm_monoidI
thf(fact_1182_mult__of_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C @ A )
= ( mult_a_ring_ext_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A = B ) ) ) ) ) ).
% mult_of.l_cancel
thf(fact_1183_mult__of_Om__assoc,axiom,
! [X3: a,Y3: a,Z2: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ r @ X3 @ ( mult_a_ring_ext_a_b @ r @ Y3 @ Z2 ) ) ) ) ) ) ).
% mult_of.m_assoc
thf(fact_1184_mult__of_Om__comm,axiom,
! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 )
= ( mult_a_ring_ext_a_b @ r @ Y3 @ X3 ) ) ) ) ).
% mult_of.m_comm
thf(fact_1185_mult__of_Om__lcomm,axiom,
! [X3: a,Y3: a,Z2: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( mult_a_ring_ext_a_b @ r @ Y3 @ Z2 ) )
= ( mult_a_ring_ext_a_b @ r @ Y3 @ ( mult_a_ring_ext_a_b @ r @ X3 @ Z2 ) ) ) ) ) ) ).
% mult_of.m_lcomm
thf(fact_1186_mult__of_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ C )
= ( mult_a_ring_ext_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A = B ) ) ) ) ) ).
% mult_of.r_cancel
thf(fact_1187_mult__of_Ounit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.unit_factor
thf(fact_1188_mult__of_Oprod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.prod_unit_r
thf(fact_1189_mult__of_Oprod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.prod_unit_l
thf(fact_1190_mult__of_Omult__cong__r,axiom,
! [B: a,B3: a,A: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ B3 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B3 ) ) ) ) ) ) ).
% mult_of.mult_cong_r
thf(fact_1191_mult__of_Omult__cong__l,axiom,
! [A: a,A5: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A5 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A5 @ B ) ) ) ) ) ) ).
% mult_of.mult_cong_l
thf(fact_1192_mult__of_Oassociated__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( A
= ( mult_a_ring_ext_a_b @ r @ B @ X2 ) ) ) ) ) ) ) ).
% mult_of.associated_iff
thf(fact_1193_mult__of_OassociatedI2_H,axiom,
! [A: a,B: a,U: a] :
( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ).
% mult_of.associatedI2'
thf(fact_1194_mult__of_OassociatedI2,axiom,
! [U: a,A: a,B: a] :
( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ).
% mult_of.associatedI2
thf(fact_1195_mult__of_OassociatedE2,axiom,
! [A: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ! [U2: a] :
( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ~ ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.associatedE2
thf(fact_1196_mult__of_OassociatedD2,axiom,
! [A: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( A
= ( mult_a_ring_ext_a_b @ r @ B @ X ) ) ) ) ) ) ).
% mult_of.associatedD2
thf(fact_1197_mult__of_Oassoc__r__cancel,axiom,
! [A: a,B: a,A5: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A5 @ B ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A5 ) ) ) ) ) ).
% mult_of.assoc_r_cancel
thf(fact_1198_mult__of_Oassoc__l__cancel,axiom,
! [A: a,B: a,B3: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B3 ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ B3 ) ) ) ) ) ).
% mult_of.assoc_l_cancel
thf(fact_1199_mult__of_Omonoid__cancelI,axiom,
( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A2 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B2 ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A2 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B2 @ C2 ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid1999574367301118026t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.monoid_cancelI
thf(fact_1200_mult__of_Oirreducible__prod__rI,axiom,
! [A: a,B: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% mult_of.irreducible_prod_rI
thf(fact_1201_mult__of_Oirreducible__prod__lI,axiom,
! [B: a,A: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ B )
=> ( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% mult_of.irreducible_prod_lI
thf(fact_1202_mult__of_Oirreducible__prodE,axiom,
! [A: a,B: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A )
=> ~ ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) )
=> ~ ( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ~ ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ B ) ) ) ) ) ) ).
% mult_of.irreducible_prodE
thf(fact_1203_mult__of_Ocarrier__not__empty,axiom,
( ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) )
!= bot_bot_set_a ) ).
% mult_of.carrier_not_empty
thf(fact_1204_mult__of_OUnits__inv__comm,axiom,
! [X3: a,Y3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y3 @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.Units_inv_comm
thf(fact_1205_assoc__iff__assoc__mult,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
= ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ).
% assoc_iff_assoc_mult
thf(fact_1206_mult__of_Oirrlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ As ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ X ) )
=> ! [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Bs ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ X5 ) ) ) ) ).
% mult_of.irrlist_perm_cong
thf(fact_1207_mult__of_Oisgcd__divides__r,axiom,
! [B: a,A: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ B @ A )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ B @ A @ B ) ) ) ) ).
% mult_of.isgcd_divides_r
thf(fact_1208_mult__of_Oisgcd__divides__l,axiom,
! [A: a,B: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A @ A @ B ) ) ) ) ).
% mult_of.isgcd_divides_l
thf(fact_1209_mult__of_Odivides__trans,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ C ) ) ) ) ).
% mult_of.divides_trans
thf(fact_1210_mult__of_Ounit__divides,axiom,
! [U: a,A: a] :
( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ U @ A ) ) ) ).
% mult_of.unit_divides
thf(fact_1211_mult__of_Odivides__unit,axiom,
! [A: a,U: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ U )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.divides_unit
thf(fact_1212_mult__of_Odivides__cong__r,axiom,
! [X3: a,Y3: a,Y4: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X3 @ Y3 )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ Y3 @ Y4 )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X3 @ Y4 ) ) ) ) ).
% mult_of.divides_cong_r
thf(fact_1213_mult__of_Odivides__cong__l,axiom,
! [X3: a,X4: a,Y3: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ X3 @ X4 )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X4 @ Y3 )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X3 @ Y3 ) ) ) ) ).
% mult_of.divides_cong_l
thf(fact_1214_irreducible__imp__irreducible__mult,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A ) ) ) ).
% irreducible_imp_irreducible_mult
thf(fact_1215_ring__irreducibleE_I3_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ R3 ) ) ) ).
% ring_irreducibleE(3)
thf(fact_1216_ring__primeE_I2_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ) ).
% ring_primeE(2)
thf(fact_1217_prime__eq__prime__mult,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P )
= ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ) ).
% prime_eq_prime_mult
thf(fact_1218_mult__of_Oee__factorsD,axiom,
! [As: list_a,Bs: list_a,A: a,B: a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ) ) ).
% mult_of.ee_factorsD
thf(fact_1219_mult__of_Oee__trans,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ Cs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Cs ) ) ) ) ) ) ).
% mult_of.ee_trans
thf(fact_1220_mult__of_Oee__sym,axiom,
! [As: list_a,Bs: list_a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ As ) ) ) ) ).
% mult_of.ee_sym
thf(fact_1221_mult__of_Ofactors__closed,axiom,
! [Fs: list_a,A: a] :
( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.factors_closed
thf(fact_1222_divides__mult__zero,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( zero_a_b @ r ) )
=> ( A
= ( zero_a_b @ r ) ) ) ) ).
% divides_mult_zero
thf(fact_1223_mult__of_Ofactors__mult__single,axiom,
! [A: a,Fb: list_a,B: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fb @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ ( cons_a @ A @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% mult_of.factors_mult_single
thf(fact_1224_mult__of_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X )
= X ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.one_unique
thf(fact_1225_mult__of_Oinv__unique,axiom,
! [Y3: a,X3: a,Y4: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y3 @ X3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y4 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( Y3 = Y4 ) ) ) ) ) ) ).
% mult_of.inv_unique
thf(fact_1226_mult__of_OUnits__r__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.Units_r_inv_ex
thf(fact_1227_mult__of_OUnits__l__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.Units_l_inv_ex
thf(fact_1228_mult__of_Oprime__divides,axiom,
! [A: a,B: a,P: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ A )
| ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ B ) ) ) ) ) ) ).
% mult_of.prime_divides
thf(fact_1229_mult__of_Odivides__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ).
% mult_of.divides_prod_r
thf(fact_1230_mult__of_Odivides__prod__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ) ).
% mult_of.divides_prod_l
thf(fact_1231_mult__of_OUnit__eq__dividesone,axiom,
! [U: a] :
( ( member_a @ U @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ U @ ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.Unit_eq_dividesone
thf(fact_1232_mult__of_Opow__mult__distrib,axiom,
! [X3: a,Y3: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 )
= ( mult_a_ring_ext_a_b @ r @ Y3 @ X3 ) )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X3 @ N ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ Y3 @ N ) ) ) ) ) ) ).
% mult_of.pow_mult_distrib
thf(fact_1233_mult__of_Onat__pow__distrib,axiom,
! [X3: a,Y3: a,N: nat] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X3 @ N ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ Y3 @ N ) ) ) ) ) ).
% mult_of.nat_pow_distrib
thf(fact_1234_mult__of_Onat__pow__comm,axiom,
! [X3: a,N: nat,M: nat] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X3 @ N ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X3 @ M ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X3 @ M ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X3 @ N ) ) ) ) ).
% mult_of.nat_pow_comm
thf(fact_1235_mult__of_Ogroup__commutes__pow,axiom,
! [X3: a,Y3: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 )
= ( mult_a_ring_ext_a_b @ r @ Y3 @ X3 ) )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X3 @ N ) @ Y3 )
= ( mult_a_ring_ext_a_b @ r @ Y3 @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X3 @ N ) ) ) ) ) ) ).
% mult_of.group_commutes_pow
thf(fact_1236_mult__of_Operm__closed,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.perm_closed
thf(fact_1237_mult__of_Ofactors__dividesI,axiom,
! [Fs: list_a,A: a,F: a] :
( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A )
=> ( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ F @ A ) ) ) ) ).
% mult_of.factors_dividesI
thf(fact_1238_mult__of_OfactorsI,axiom,
! [Fs: list_a,A: a] :
( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Fs ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ X ) )
=> ( ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) )
= A )
=> ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A ) ) ) ).
% mult_of.factorsI
thf(fact_1239_mult__of_Oprime__pow__divides__iff,axiom,
! [P: a,A: a,B: a,N: nat] :
( ( member_a @ P @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ~ ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ A )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ P @ N ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ P @ N ) @ B ) ) ) ) ) ) ) ).
% mult_of.prime_pow_divides_iff
thf(fact_1240_mult__of_Ofactors__mult,axiom,
! [Fa: list_a,A: a,Fb: list_a,B: a] :
( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fa @ A )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fb @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% mult_of.factors_mult
thf(fact_1241_mult__of_Omultlist__ee__cong,axiom,
! [Fs: list_a,Fs2: list_a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ Fs2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs2 @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% mult_of.multlist_ee_cong
thf(fact_1242_mult__of_Oassociated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A ) ) ).
% mult_of.associated_refl
thf(fact_1243_mult__of_OUnits__m__closed,axiom,
! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.Units_m_closed
thf(fact_1244_mult__of_OUnits__one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ).
% mult_of.Units_one_closed
thf(fact_1245_Units__mult__eq__Units,axiom,
( ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) )
= ( units_a_ring_ext_a_b @ r ) ) ).
% Units_mult_eq_Units
thf(fact_1246_mult__of_Om__closed,axiom,
! [X3: a,Y3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.m_closed
thf(fact_1247_mult__of_OUnits__l__cancel,axiom,
! [X3: a,Y3: a,Z2: a] :
( ( member_a @ X3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 )
= ( mult_a_ring_ext_a_b @ r @ X3 @ Z2 ) )
= ( Y3 = Z2 ) ) ) ) ) ).
% mult_of.Units_l_cancel
thf(fact_1248_mult__of_Oone__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ).
% mult_of.one_closed
thf(fact_1249_mult__of_Odivides__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A ) ) ).
% mult_of.divides_refl
thf(fact_1250_mult__of_Onat__pow__closed,axiom,
! [X3: a,N: nat] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X3 @ N ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.nat_pow_closed
thf(fact_1251_mult__of_Onat__pow__one,axiom,
! [N: nat] :
( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( one_a_ring_ext_a_b @ r ) ) ).
% mult_of.nat_pow_one
thf(fact_1252_mult__of_Oee__refl,axiom,
! [As: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ As ) ) ).
% mult_of.ee_refl
thf(fact_1253_mult__of_Or__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( one_a_ring_ext_a_b @ r ) )
= X3 ) ) ).
% mult_of.r_one
thf(fact_1254_mult__of_Ol__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X3 )
= X3 ) ) ).
% mult_of.l_one
thf(fact_1255_mult__of_Odivides__mult__rI,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% mult_of.divides_mult_rI
thf(fact_1256_mult__of_Odivides__mult__r,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ) ).
% mult_of.divides_mult_r
thf(fact_1257_mult__of_Odivides__mult__lI,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% mult_of.divides_mult_lI
thf(fact_1258_mult__of_Odivides__mult__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ) ).
% mult_of.divides_mult_l
thf(fact_1259_mult__of_Omultlist__closed,axiom,
! [Fs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.multlist_closed
thf(fact_1260_mult__of_Omultlist__listassoc__cong,axiom,
! [Fs: list_a,Fs2: list_a] :
( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ Fs @ Fs2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs2 @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% mult_of.multlist_listassoc_cong
thf(fact_1261_Span__append__eq__set__add,axiom,
! [K2: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Us2 @ Vs ) )
= ( set_add_a_b @ r @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs ) ) ) ) ) ) ).
% Span_append_eq_set_add
thf(fact_1262_perm__assoc__switch,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Bs )
=> ( ( ( mset_a @ Bs )
= ( mset_a @ Cs ) )
=> ? [Bs2: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs2 ) )
& ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Bs2 @ Cs ) ) ) ) ).
% perm_assoc_switch
thf(fact_1263_perm__assoc__switch__r,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Bs @ Cs )
=> ? [Bs2: list_a] :
( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Bs2 )
& ( ( mset_a @ Bs2 )
= ( mset_a @ Cs ) ) ) ) ) ).
% perm_assoc_switch_r
thf(fact_1264_Span__in__carrier,axiom,
! [K2: set_a,Us2: list_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% Span_in_carrier
thf(fact_1265_mult__of_Operm__assoc__switch__r,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ Bs @ Cs )
=> ? [Bs2: list_a] :
( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ As @ Bs2 )
& ( ( mset_a @ Bs2 )
= ( mset_a @ Cs ) ) ) ) ) ).
% mult_of.perm_assoc_switch_r
thf(fact_1266_mult__of_Operm__assoc__switch,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ As @ Bs )
=> ( ( ( mset_a @ Bs )
= ( mset_a @ Cs ) )
=> ? [Bs2: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs2 ) )
& ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ Bs2 @ Cs ) ) ) ) ).
% mult_of.perm_assoc_switch
thf(fact_1267_essentially__equalI,axiom,
! [Fs1: list_a,Fs12: list_a,Fs22: list_a] :
( ( ( mset_a @ Fs1 )
= ( mset_a @ Fs12 ) )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Fs12 @ Fs22 )
=> ( essent8953798148185448568xt_a_b @ r @ Fs1 @ Fs22 ) ) ) ).
% essentially_equalI
thf(fact_1268_essentially__equalE,axiom,
! [Fs1: list_a,Fs22: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ Fs1 @ Fs22 )
=> ~ ! [Fs13: list_a] :
( ( ( mset_a @ Fs1 )
= ( mset_a @ Fs13 ) )
=> ~ ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Fs13 @ Fs22 ) ) ) ).
% essentially_equalE
thf(fact_1269_Span__subgroup__props_I1_J,axiom,
! [K2: set_a,Us2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% Span_subgroup_props(1)
thf(fact_1270_Span__base__incl,axiom,
! [K2: set_a,Us2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) ) ) ) ).
% Span_base_incl
thf(fact_1271_Span__same__set,axiom,
! [K2: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( set_a2 @ Us2 )
= ( set_a2 @ Vs ) )
=> ( ( embedded_Span_a_b @ r @ K2 @ Us2 )
= ( embedded_Span_a_b @ r @ K2 @ Vs ) ) ) ) ) ).
% Span_same_set
thf(fact_1272_mono__Span__sublist,axiom,
! [K2: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( set_a2 @ Vs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs ) ) ) ) ) ).
% mono_Span_sublist
thf(fact_1273_mono__Span__subset,axiom,
! [K2: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs ) ) ) ) ) ).
% mono_Span_subset
thf(fact_1274_listassoc__sym,axiom,
! [As: list_a,Bs: list_a] :
( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Bs @ As ) ) ) ) ).
% listassoc_sym
thf(fact_1275_listassoc__trans,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Bs )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Bs @ Cs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Cs ) ) ) ) ) ) ).
% listassoc_trans
thf(fact_1276_mult__of_Oessentially__equalI,axiom,
! [Fs1: list_a,Fs12: list_a,Fs22: list_a] :
( ( ( mset_a @ Fs1 )
= ( mset_a @ Fs12 ) )
=> ( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ Fs12 @ Fs22 )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Fs1 @ Fs22 ) ) ) ).
% mult_of.essentially_equalI
% Helper facts (7)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X3: a,Y3: a] :
( ( if_a @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X3: a,Y3: a] :
( ( if_a @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X3: list_a,Y3: list_a] :
( ( if_list_a @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X3: list_a,Y3: list_a] :
( ( if_list_a @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
! [X3: set_list_a,Y3: set_list_a] :
( ( if_set_list_a @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
! [X3: set_list_a,Y3: set_list_a] :
( ( if_set_list_a @ $true @ X3 @ Y3 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ q )
= ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
= ( polyno5814909790663948098es_a_b @ r @ p @ q ) ) ).
%------------------------------------------------------------------------------