TPTP Problem File: SLH0307^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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/0025_Finite_Fields_Isomorphic/prob_00144_005556__18447510_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1713 ( 262 unt; 430 typ; 0 def)
% Number of atoms : 4940 (1260 equ; 0 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 20278 ( 390 ~; 68 |; 164 &;16674 @)
% ( 0 <=>;2982 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 9 avg)
% Number of types : 66 ( 65 usr)
% Number of type conns : 1105 (1105 >; 0 *; 0 +; 0 <<)
% Number of symbols : 368 ( 365 usr; 28 con; 0-4 aty)
% Number of variables : 3600 ( 251 ^;3246 !; 103 ?;3600 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:23:59.449
%------------------------------------------------------------------------------
% Could-be-implicit typings (65)
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% Explicit typings (365)
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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_001_062_It__Complex__Ocomplex_Mt__Nat__Onat_J,type,
collect_complex_nat: ( ( complex > nat ) > $o ) > set_complex_nat ).
thf(sy_c_Set_OCollect_001_062_It__List__Olist_Itf__a_J_Mt__Nat__Onat_J,type,
collect_list_a_nat: ( ( list_a > nat ) > $o ) > set_list_a_nat ).
thf(sy_c_Set_OCollect_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
collect_list_a_a: ( ( list_a > a ) > $o ) > set_list_a_a ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
collect_a_list_a: ( ( a > list_a ) > $o ) > set_a_list_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mt__Nat__Onat_J,type,
collect_a_nat: ( ( a > nat ) > $o ) > set_a_nat ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).
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__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
collect_set_complex: ( set_complex > $o ) > set_set_complex ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
collect_set_list_a: ( set_list_a > $o ) > set_set_list_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
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_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Subrings_Osubcring_001tf__a_001tf__b,type,
subcring_a_b: set_a > partia2175431115845679010xt_a_b > $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_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subrin3541368690557094692t_unit: set_list_list_a > partia2956882679547061052t_unit > $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_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_UnivPoly_Oup_001tf__a_001tf__b,type,
up_a_b: partia2175431115845679010xt_a_b > set_nat_a ).
thf(sy_c_member_001_062_It__Complex__Ocomplex_Mt__Nat__Onat_J,type,
member_complex_nat: ( complex > nat ) > set_complex_nat > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
member8456684049276653052list_a: ( list_list_list_a > list_list_list_a ) > set_li7224484540316392091list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member2200549709344965110list_a: ( list_list_list_a > list_list_a ) > set_li2472003267954637717list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__List__Olist_Itf__a_J_J,type,
member3330872541585647984list_a: ( list_list_list_a > list_a ) > set_li8634773807413628943list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
member8941091444255470082list_a: ( list_list_a > list_list_list_a ) > set_li3897086496743908769list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member8231385768148312316list_a: ( list_list_a > list_list_a ) > set_li5608457238520824219list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member7168557129179038582list_a: ( list_list_a > list_a ) > set_li3422455791611400469list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
member_list_list_a_a: ( list_list_a > a ) > set_list_list_a_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
member7649053982783128200list_a: ( list_a > list_list_list_a ) > set_li8897940509999504679list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member6714375691612171394list_a: ( list_a > list_list_a ) > set_li6773872926390105121list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member_list_a_list_a: ( list_a > list_a ) > set_list_a_list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__Nat__Onat_J,type,
member_list_a_nat: ( list_a > nat ) > set_list_a_nat > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member4263473470251683292list_a: ( list_a > set_list_a ) > set_li1071299071675007611list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
member_list_a_a: ( list_a > a ) > set_list_a_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member5910328476188217884list_a: ( set_list_a > list_a ) > set_se5067313844698916539list_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member5068272912271824380list_a: ( set_list_a > set_list_a ) > set_se1917860372504128155list_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
member_set_list_a_a: ( set_list_a > a ) > set_set_list_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member_a_list_list_a: ( a > list_list_a ) > set_a_list_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
member_a_list_a: ( a > list_a ) > set_a_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Nat__Onat_J,type,
member_a_nat: ( a > nat ) > set_a_nat > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member_a_set_list_a: ( a > set_list_a ) > set_a_set_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
member6842060177613954879list_a: list_l7815035709764258326list_a > set_li3407770045201608054list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member5342144027231129785list_a: list_list_list_a > set_list_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
member352051402189872281list_a: list_list_set_list_a > set_li664707282716828624list_a > $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__Multiset__Omultiset_Itf__a_J,type,
member_multiset_a: multiset_a > set_multiset_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $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_f,type,
f: list_a ).
thf(sy_v_g,type,
g: list_a ).
% Relevant facts (1277)
thf(fact_0_assms_I3_J,axiom,
member_list_a @ g @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).
% assms(3)
thf(fact_1_assms_I2_J,axiom,
member_list_a @ f @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).
% assms(2)
thf(fact_2_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_3__092_060open_062_I_092_060exists_062x_092_060in_062carrier_A_IK_A_091X_093_J_O_Af_A_092_060otimes_062_092_060_094bsub_062K_A_091X_093_092_060_094esub_062_Ax_A_061_Ag_J_A_061_A_I_092_060exists_062x_092_060in_062carrier_A_Ipoly__ring_A_IR_092_060lparr_062carrier_A_058_061_AK_092_060rparr_062_J_J_O_Af_A_092_060otimes_062_092_060_094bsub_062poly__ring_A_IR_092_060lparr_062carrier_A_058_061_AK_092_060rparr_062_J_092_060_094esub_062_Ax_A_061_Ag_J_092_060close_062,axiom,
( ( ? [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ f @ X )
= g ) ) )
= ( ? [X: list_a] :
( ( member_list_a @ X
@ ( partia5361259788508890537t_unit
@ ( univ_poly_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r )
@ ( partia707051561876973205xt_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r ) ) ) ) )
& ( ( mult_l7073676228092353617t_unit
@ ( univ_poly_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r )
@ ( partia707051561876973205xt_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r ) ) )
@ f
@ X )
= g ) ) ) ) ).
% \<open>(\<exists>x\<in>carrier (K [X]). f \<otimes>\<^bsub>K [X]\<^esub> x = g) = (\<exists>x\<in>carrier (poly_ring (R\<lparr>carrier := K\<rparr>)). f \<otimes>\<^bsub>poly_ring (R\<lparr>carrier := K\<rparr>)\<^esub> x = g)\<close>
thf(fact_4_assms_I1_J,axiom,
subfield_a_b @ k @ r ).
% assms(1)
thf(fact_5_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_6_calculation,axiom,
( ( polyno5814909790663948098es_a_b @ r @ f @ g )
= ( ? [X: list_a] :
( ( member_list_a @ X
@ ( partia5361259788508890537t_unit
@ ( univ_poly_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r )
@ ( partia707051561876973205xt_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r ) ) ) ) )
& ( ( mult_l7073676228092353617t_unit
@ ( univ_poly_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r )
@ ( partia707051561876973205xt_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r ) ) )
@ f
@ X )
= g ) ) ) ) ).
% calculation
thf(fact_7_is__root__poly__mult__imp__is__root,axiom,
! [P: list_a,Q: list_a,X2: 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 ) @ X2 )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X2 )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X2 ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_8__092_060open_062f_Adivides_092_060_094bsub_062K_A_091X_093_092_060_094esub_062_Ag_A_061_A_I_092_060exists_062x_092_060in_062carrier_A_IK_A_091X_093_J_O_Af_A_092_060otimes_062_092_060_094bsub_062K_A_091X_093_092_060_094esub_062_Ax_A_061_Ag_J_092_060close_062,axiom,
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ k ) @ f @ g )
= ( ? [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ f @ X )
= g ) ) ) ) ).
% \<open>f divides\<^bsub>K [X]\<^esub> g = (\<exists>x\<in>carrier (K [X]). f \<otimes>\<^bsub>K [X]\<^esub> x = g)\<close>
thf(fact_9_univ__poly__consistent,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ( univ_poly_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) )
= ( univ_poly_a_b @ r ) ) ) ).
% univ_poly_consistent
thf(fact_10_carrier__is__subcring,axiom,
subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subcring
thf(fact_11_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_12_ring__hom__restrict,axiom,
! [F: a > list_a,S: partia2670972154091845814t_unit,G: a > list_a] :
( ( member_a_list_a @ F @ ( ring_h405018892823518980t_unit @ r @ S ) )
=> ( ! [R: a] :
( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( F @ R )
= ( G @ R ) ) )
=> ( member_a_list_a @ G @ ( ring_h405018892823518980t_unit @ r @ S ) ) ) ) ).
% ring_hom_restrict
thf(fact_13_a,axiom,
subring_a_b @ k @ r ).
% a
thf(fact_14_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_15_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_16_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_17_subring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( K != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_18_subcringI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subcring_a_b @ H @ r ) ) ).
% subcringI'
thf(fact_19_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_20_subring__is__domain,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r ) ) ) ).
% subring_is_domain
thf(fact_21_pdivides__iff__shell,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) ) ) ) ) ).
% pdivides_iff_shell
thf(fact_22__092_060open_062f_Apdivides_Ag_A_061_Af_Adivides_092_060_094bsub_062K_A_091X_093_092_060_094esub_062_Ag_092_060close_062,axiom,
( ( polyno5814909790663948098es_a_b @ r @ f @ g )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ k ) @ f @ g ) ) ).
% \<open>f pdivides g = f divides\<^bsub>K [X]\<^esub> g\<close>
thf(fact_23_carrier__polynomial__shell,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% carrier_polynomial_shell
thf(fact_24_univ__poly__is__principal,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_principal
thf(fact_25_pprimeE_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a,R2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R2 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
| ( polyno5814909790663948098es_a_b @ r @ P @ R2 ) ) ) ) ) ) ) ) ).
% pprimeE(3)
thf(fact_26_univ__poly__a__minus__consistent,axiom,
! [K: set_a,Q: list_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) ) ) ) ).
% univ_poly_a_minus_consistent
thf(fact_27_long__division__closed_I1_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_28_dividesI_H,axiom,
! [B: list_a,G2: partia2670972154091845814t_unit,A: list_a,C: list_a] :
( ( B
= ( mult_l7073676228092353617t_unit @ G2 @ A @ C ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( factor1757716651909850160t_unit @ G2 @ A @ B ) ) ) ).
% dividesI'
thf(fact_29_dividesI_H,axiom,
! [B: a,G2: partia2175431115845679010xt_a_b,A: a,C: a] :
( ( B
= ( mult_a_ring_ext_a_b @ G2 @ A @ C ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( factor8216151070175719842xt_a_b @ G2 @ A @ B ) ) ) ).
% dividesI'
thf(fact_30_dividesI_H,axiom,
! [B: list_list_a,G2: partia2956882679547061052t_unit,A: list_list_a,C: list_list_a] :
( ( B
= ( mult_l4853965630390486993t_unit @ G2 @ A @ C ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( factor6954119973539764400t_unit @ G2 @ A @ B ) ) ) ).
% dividesI'
thf(fact_31_subdomain__is__domain,axiom,
! [H: set_a] :
( ( subdomain_a_b @ H @ r )
=> ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r ) ) ) ).
% subdomain_is_domain
thf(fact_32_domain_Opdivides__iff__shell,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R3 @ P @ Q )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P @ Q ) ) ) ) ) ) ).
% domain.pdivides_iff_shell
thf(fact_33_domain_Opdivides__iff__shell,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( polyno8016796738000020810t_unit @ R3 @ P @ Q )
= ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P @ Q ) ) ) ) ) ) ).
% domain.pdivides_iff_shell
thf(fact_34_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a,X2: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( polyno5142720416380192742t_unit @ R3 @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ P @ Q ) @ X2 )
=> ( ( polyno5142720416380192742t_unit @ R3 @ P @ X2 )
| ( polyno5142720416380192742t_unit @ R3 @ Q @ X2 ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_35_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,X2: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ R3 @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ P @ Q ) @ X2 )
=> ( ( polyno6951661231331188332t_unit @ R3 @ P @ X2 )
| ( polyno6951661231331188332t_unit @ R3 @ Q @ X2 ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_36_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,X2: a] :
( ( domain_a_b @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ R3 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ P @ Q ) @ X2 )
=> ( ( polyno4133073214067823460ot_a_b @ R3 @ P @ X2 )
| ( polyno4133073214067823460ot_a_b @ R3 @ Q @ X2 ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_37_pdivides__zero,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P @ nil_a ) ) ) ).
% pdivides_zero
thf(fact_38_finite__number__of__roots,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( finite_finite_a @ ( collect_a @ ( polyno4133073214067823460ot_a_b @ r @ P ) ) ) ) ).
% finite_number_of_roots
thf(fact_39_canonical__embedding__is__hom,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( member_a_list_a @ ( poly_of_const_a_b @ r )
@ ( ring_h405018892823518980t_unit
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r )
@ ( univ_poly_a_b @ r @ K ) ) ) ) ).
% canonical_embedding_is_hom
thf(fact_40_pdivides__def,axiom,
( polyno8016796738000020810t_unit
= ( ^ [R4: partia2670972154091845814t_unit] : ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ R4 @ ( partia5361259788508890537t_unit @ R4 ) ) ) ) ) ).
% pdivides_def
thf(fact_41_pdivides__def,axiom,
( polyno5814909790663948098es_a_b
= ( ^ [R4: partia2175431115845679010xt_a_b] : ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ R4 @ ( partia707051561876973205xt_a_b @ R4 ) ) ) ) ) ).
% pdivides_def
thf(fact_42_pdivides__def,axiom,
( polyno4453881341673752516t_unit
= ( ^ [R4: partia2956882679547061052t_unit] : ( factor652753743487153968t_unit @ ( univ_p2250591967980070728t_unit @ R4 @ ( partia2464479390973590831t_unit @ R4 ) ) ) ) ) ).
% pdivides_def
thf(fact_43_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_44_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_45_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_46_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_47_m__assoc,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X2 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_48_m__comm,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X2 ) ) ) ) ).
% m_comm
thf(fact_49_m__lcomm,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X2 @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_50_subring__props_I6_J,axiom,
! [K: set_a,H1: a,H2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H2 @ K )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H2 ) @ K ) ) ) ) ).
% subring_props(6)
thf(fact_51_mem__Collect__eq,axiom,
! [A: set_a,P2: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
! [A: list_a > a,P2: ( list_a > a ) > $o] :
( ( member_list_a_a @ A @ ( collect_list_a_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A: nat > a,P2: ( nat > a ) > $o] :
( ( member_nat_a @ A @ ( collect_nat_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A: a > list_a,P2: ( a > list_a ) > $o] :
( ( member_a_list_a @ A @ ( collect_a_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_57_mem__Collect__eq,axiom,
! [A: complex,P2: complex > $o] :
( ( member_complex @ A @ ( collect_complex @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_58_mem__Collect__eq,axiom,
! [A: list_a,P2: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( member_set_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_list_a_a] :
( ( collect_list_a_a
@ ^ [X: list_a > a] : ( member_list_a_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A2: set_nat_a] :
( ( collect_nat_a
@ ^ [X: nat > a] : ( member_nat_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A2: set_a_list_a] :
( ( collect_a_list_a
@ ^ [X: a > list_a] : ( member_a_list_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_63_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_65_Collect__mem__eq,axiom,
! [A2: set_complex] :
( ( collect_complex
@ ^ [X: complex] : ( member_complex @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X: list_a] : ( member_list_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_67_Collect__cong,axiom,
! [P2: a > $o,Q2: a > $o] :
( ! [X3: a] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_a @ P2 )
= ( collect_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_68_Collect__cong,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_69_Collect__cong,axiom,
! [P2: complex > $o,Q2: complex > $o] :
( ! [X3: complex] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_complex @ P2 )
= ( collect_complex @ Q2 ) ) ) ).
% Collect_cong
thf(fact_70_Collect__cong,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ! [X3: list_a] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_list_a @ P2 )
= ( collect_list_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_71_univ__poly__is__domain,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_domain
thf(fact_72_zero__pdivides,axiom,
! [P: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P )
= ( P = nil_a ) ) ).
% zero_pdivides
thf(fact_73_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_74_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_75_subdomainI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subdomain_a_b @ H @ r ) ) ).
% subdomainI'
thf(fact_76_poly__of__const__consistent,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ( poly_of_const_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) )
= ( poly_of_const_a_b @ r ) ) ) ).
% poly_of_const_consistent
thf(fact_77_long__division__zero_I1_J,axiom,
! [K: set_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% long_division_zero(1)
thf(fact_78_pprimeE_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( P != nil_a ) ) ) ) ).
% pprimeE(1)
thf(fact_79_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_80_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_81_divides__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ A ) ) ).
% divides_refl
thf(fact_82_m__closed,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_83_ring_Opdiv_Ocong,axiom,
polynomial_pdiv_a_b = polynomial_pdiv_a_b ).
% ring.pdiv.cong
thf(fact_84_domain_Ozero__pdivides__zero,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( polyno8016796738000020810t_unit @ R3 @ nil_list_a @ nil_list_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_85_domain_Ozero__pdivides__zero,axiom,
! [R3: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R3 )
=> ( polyno5814909790663948098es_a_b @ R3 @ nil_a @ nil_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_86_domain_Ozero__pdivides,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( polyno8016796738000020810t_unit @ R3 @ nil_list_a @ P )
= ( P = nil_list_a ) ) ) ).
% domain.zero_pdivides
thf(fact_87_domain_Ozero__pdivides,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( polyno5814909790663948098es_a_b @ R3 @ nil_a @ P )
= ( P = nil_a ) ) ) ).
% domain.zero_pdivides
thf(fact_88_domain_OpprimeE_I1_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P )
=> ( P != nil_a ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_89_domain_OpprimeE_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P )
=> ( P != nil_list_a ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_90_domain_Olong__division__zero_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ R3 @ nil_list_a @ Q )
= nil_list_a ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_91_domain_Olong__division__zero_I1_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ R3 @ nil_a @ Q )
= nil_a ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_92_domain_Opdivides__zero,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( polyno5814909790663948098es_a_b @ R3 @ P @ nil_a ) ) ) ) ).
% domain.pdivides_zero
thf(fact_93_domain_Opdivides__zero,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( polyno8016796738000020810t_unit @ R3 @ P @ nil_list_a ) ) ) ) ).
% domain.pdivides_zero
thf(fact_94_ring_Ois__root_Ocong,axiom,
polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).
% ring.is_root.cong
thf(fact_95_domain_Olong__division__closed_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( member_list_list_a @ ( polyno5893782122288709345t_unit @ R3 @ P @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_96_domain_Olong__division__closed_I1_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ R3 @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_97_domain_Ofinite__number__of__roots,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( finite1660835950917165235list_a @ ( collect_list_list_a @ ( polyno5142720416380192742t_unit @ R3 @ P ) ) ) ) ) ).
% domain.finite_number_of_roots
thf(fact_98_domain_Ofinite__number__of__roots,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( finite_finite_list_a @ ( collect_list_a @ ( polyno6951661231331188332t_unit @ R3 @ P ) ) ) ) ) ).
% domain.finite_number_of_roots
thf(fact_99_domain_Ofinite__number__of__roots,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( finite_finite_a @ ( collect_a @ ( polyno4133073214067823460ot_a_b @ R3 @ P ) ) ) ) ) ).
% domain.finite_number_of_roots
thf(fact_100_domain_OpprimeE_I3_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a,R2: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R3 @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R3 @ K ) @ Q @ R2 ) )
=> ( ( polyno5814909790663948098es_a_b @ R3 @ P @ Q )
| ( polyno5814909790663948098es_a_b @ R3 @ P @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.pprimeE(3)
thf(fact_101_domain_OpprimeE_I3_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a,R2: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( polyno8016796738000020810t_unit @ R3 @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ Q @ R2 ) )
=> ( ( polyno8016796738000020810t_unit @ R3 @ P @ Q )
| ( polyno8016796738000020810t_unit @ R3 @ P @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.pprimeE(3)
thf(fact_102_dividesD,axiom,
! [G2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( factor1757716651909850160t_unit @ G2 @ A @ B )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G2 ) )
& ( B
= ( mult_l7073676228092353617t_unit @ G2 @ A @ X3 ) ) ) ) ).
% dividesD
thf(fact_103_dividesD,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( factor8216151070175719842xt_a_b @ G2 @ A @ B )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G2 ) )
& ( B
= ( mult_a_ring_ext_a_b @ G2 @ A @ X3 ) ) ) ) ).
% dividesD
thf(fact_104_dividesD,axiom,
! [G2: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( factor6954119973539764400t_unit @ G2 @ A @ B )
=> ? [X3: list_list_a] :
( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G2 ) )
& ( B
= ( mult_l4853965630390486993t_unit @ G2 @ A @ X3 ) ) ) ) ).
% dividesD
thf(fact_105_dividesE,axiom,
! [G2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( factor1757716651909850160t_unit @ G2 @ A @ B )
=> ~ ! [C2: list_a] :
( ( B
= ( mult_l7073676228092353617t_unit @ G2 @ A @ C2 ) )
=> ~ ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ G2 ) ) ) ) ).
% dividesE
thf(fact_106_dividesE,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( factor8216151070175719842xt_a_b @ G2 @ A @ B )
=> ~ ! [C2: a] :
( ( B
= ( mult_a_ring_ext_a_b @ G2 @ A @ C2 ) )
=> ~ ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ).
% dividesE
thf(fact_107_dividesE,axiom,
! [G2: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( factor6954119973539764400t_unit @ G2 @ A @ B )
=> ~ ! [C2: list_list_a] :
( ( B
= ( mult_l4853965630390486993t_unit @ G2 @ A @ C2 ) )
=> ~ ( member_list_list_a @ C2 @ ( partia2464479390973590831t_unit @ G2 ) ) ) ) ).
% dividesE
thf(fact_108_dividesI,axiom,
! [C: list_a,G2: partia2670972154091845814t_unit,B: list_a,A: list_a] :
( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( B
= ( mult_l7073676228092353617t_unit @ G2 @ A @ C ) )
=> ( factor1757716651909850160t_unit @ G2 @ A @ B ) ) ) ).
% dividesI
thf(fact_109_dividesI,axiom,
! [C: a,G2: partia2175431115845679010xt_a_b,B: a,A: a] :
( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( B
= ( mult_a_ring_ext_a_b @ G2 @ A @ C ) )
=> ( factor8216151070175719842xt_a_b @ G2 @ A @ B ) ) ) ).
% dividesI
thf(fact_110_dividesI,axiom,
! [C: list_list_a,G2: partia2956882679547061052t_unit,B: list_list_a,A: list_list_a] :
( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( B
= ( mult_l4853965630390486993t_unit @ G2 @ A @ C ) )
=> ( factor6954119973539764400t_unit @ G2 @ A @ B ) ) ) ).
% dividesI
thf(fact_111_factor__def,axiom,
( factor1757716651909850160t_unit
= ( ^ [G3: partia2670972154091845814t_unit,A3: list_a,B2: list_a] :
? [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G3 ) )
& ( B2
= ( mult_l7073676228092353617t_unit @ G3 @ A3 @ X ) ) ) ) ) ).
% factor_def
thf(fact_112_factor__def,axiom,
( factor8216151070175719842xt_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,A3: a,B2: a] :
? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G3 ) )
& ( B2
= ( mult_a_ring_ext_a_b @ G3 @ A3 @ X ) ) ) ) ) ).
% factor_def
thf(fact_113_factor__def,axiom,
( factor6954119973539764400t_unit
= ( ^ [G3: partia2956882679547061052t_unit,A3: list_list_a,B2: list_list_a] :
? [X: list_list_a] :
( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G3 ) )
& ( B2
= ( mult_l4853965630390486993t_unit @ G3 @ A3 @ X ) ) ) ) ) ).
% factor_def
thf(fact_114_exists__unique__long__division,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ? [X3: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ X3 )
& ! [Y2: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ Y2 )
=> ( Y2 = X3 ) ) ) ) ) ) ) ).
% exists_unique_long_division
thf(fact_115_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_116_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_117_domain_Ocanonical__embedding__is__hom,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( member6714375691612171394list_a @ ( poly_o8716471131768098070t_unit @ R3 )
@ ( ring_h8002040739877300486t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ R3 )
@ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) ) ) ) ).
% domain.canonical_embedding_is_hom
thf(fact_118_domain_Ocanonical__embedding__is__hom,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( member_a_list_a @ ( poly_of_const_a_b @ R3 )
@ ( ring_h405018892823518980t_unit
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ R3 )
@ ( univ_poly_a_b @ R3 @ K ) ) ) ) ) ).
% domain.canonical_embedding_is_hom
thf(fact_119_univ__poly__zero__closed,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) ) ).
% univ_poly_zero_closed
thf(fact_120_univ__poly__zero__closed,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a] : ( member_list_list_a @ nil_list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) ) ).
% univ_poly_zero_closed
thf(fact_121_same__pmod__iff__pdivides,axiom,
! [K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( 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 @ K ) @ A @ B ) ) ) ) ) ) ) ).
% same_pmod_iff_pdivides
thf(fact_122_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R3: partia2956882679547061052t_unit,K: set_list_list_a,Q: list_list_list_a,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( subrin3541368690557094692t_unit @ K @ R3 )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) ) )
=> ( ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) @ P @ Q )
= ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ P @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_123_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,Q: list_a,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P @ Q )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ P @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_124_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P @ Q )
= ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ P @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_125_subdomain__iff,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( subdomain_a_b @ H @ r )
= ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r ) ) ) ) ).
% subdomain_iff
thf(fact_126_pprimeI,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q3: list_a,R: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q3 @ R ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q3 )
| ( polyno5814909790663948098es_a_b @ r @ P @ R ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ) ) ).
% pprimeI
thf(fact_127_pmod__zero__iff__pdivides,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ P @ Q )
= nil_a )
= ( polyno5814909790663948098es_a_b @ r @ Q @ P ) ) ) ) ) ).
% pmod_zero_iff_pdivides
thf(fact_128_pprime__iff__pirreducible,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_129_subring__props_I1_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_130_long__division__closed_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(2)
thf(fact_131_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_132_long__division__zero_I2_J,axiom,
! [K: set_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% long_division_zero(2)
thf(fact_133_pirreducibleE_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( P != nil_a ) ) ) ) ).
% pirreducibleE(1)
thf(fact_134_pirreducibleE_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pirreducibleE(2)
thf(fact_135_pprimeE_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pprimeE(2)
thf(fact_136_pirreducibleE_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a,R2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R2 ) )
=> ( ( member_list_a @ Q @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ) ) ) ).
% pirreducibleE(3)
thf(fact_137_pirreducibleI,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q3: list_a,R: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q3 @ R ) )
=> ( ( member_list_a @ Q3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( member_list_a @ R @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ) ) ).
% pirreducibleI
thf(fact_138_ring_Opmod_Ocong,axiom,
polynomial_pmod_a_b = polynomial_pmod_a_b ).
% ring.pmod.cong
thf(fact_139_ring_Olong__divides_Ocong,axiom,
polyno2806191415236617128es_a_b = polyno2806191415236617128es_a_b ).
% ring.long_divides.cong
thf(fact_140_domain_OpirreducibleE_I2_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P )
=> ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) ) ) ) ) ) ).
% domain.pirreducibleE(2)
thf(fact_141_domain_OpirreducibleE_I2_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R3 @ K ) ) ) ) ) ) ) ).
% domain.pirreducibleE(2)
thf(fact_142_domain_OpirreducibleE_I3_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a,R2: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( P
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ Q @ R2 ) )
=> ( ( member_list_list_a @ Q @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
| ( member_list_list_a @ R2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) ) ) ) ) ) ) ) ) ) ).
% domain.pirreducibleE(3)
thf(fact_143_domain_OpirreducibleE_I3_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a,R2: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R3 @ K ) @ Q @ R2 ) )
=> ( ( member_list_a @ Q @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
| ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R3 @ K ) ) ) ) ) ) ) ) ) ) ) ).
% domain.pirreducibleE(3)
thf(fact_144_domain_OpirreducibleI,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( P != nil_list_a )
=> ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ! [Q3: list_list_a,R: list_list_a] :
( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ R @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( P
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ Q3 @ R ) )
=> ( ( member_list_list_a @ Q3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
| ( member_list_list_a @ R @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) ) ) ) ) )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P ) ) ) ) ) ) ) ).
% domain.pirreducibleI
thf(fact_145_domain_OpirreducibleI,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ! [Q3: list_a,R: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R3 @ K ) @ Q3 @ R ) )
=> ( ( member_list_a @ Q3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
| ( member_list_a @ R @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R3 @ K ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P ) ) ) ) ) ) ) ).
% domain.pirreducibleI
thf(fact_146_isgcd__def,axiom,
( isgcd_1118609098697428327t_unit
= ( ^ [G3: partia2670972154091845814t_unit,X: list_a,A3: list_a,B2: list_a] :
( ( factor1757716651909850160t_unit @ G3 @ X @ A3 )
& ( factor1757716651909850160t_unit @ G3 @ X @ B2 )
& ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G3 ) )
=> ( ( ( factor1757716651909850160t_unit @ G3 @ Y3 @ A3 )
& ( factor1757716651909850160t_unit @ G3 @ Y3 @ B2 ) )
=> ( factor1757716651909850160t_unit @ G3 @ Y3 @ X ) ) ) ) ) ) ).
% isgcd_def
thf(fact_147_isgcd__def,axiom,
( isgcd_a_ring_ext_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,X: a,A3: a,B2: a] :
( ( factor8216151070175719842xt_a_b @ G3 @ X @ A3 )
& ( factor8216151070175719842xt_a_b @ G3 @ X @ B2 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( ( ( factor8216151070175719842xt_a_b @ G3 @ Y3 @ A3 )
& ( factor8216151070175719842xt_a_b @ G3 @ Y3 @ B2 ) )
=> ( factor8216151070175719842xt_a_b @ G3 @ Y3 @ X ) ) ) ) ) ) ).
% isgcd_def
thf(fact_148_isgcd__def,axiom,
( isgcd_3804025100609598183t_unit
= ( ^ [G3: partia2956882679547061052t_unit,X: list_list_a,A3: list_list_a,B2: list_list_a] :
( ( factor6954119973539764400t_unit @ G3 @ X @ A3 )
& ( factor6954119973539764400t_unit @ G3 @ X @ B2 )
& ! [Y3: list_list_a] :
( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ G3 ) )
=> ( ( ( factor6954119973539764400t_unit @ G3 @ Y3 @ A3 )
& ( factor6954119973539764400t_unit @ G3 @ Y3 @ B2 ) )
=> ( factor6954119973539764400t_unit @ G3 @ Y3 @ X ) ) ) ) ) ) ).
% isgcd_def
thf(fact_149_ring_Opoly__of__const_Ocong,axiom,
poly_of_const_a_b = poly_of_const_a_b ).
% ring.poly_of_const.cong
thf(fact_150_domain_OpprimeE_I2_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R3 @ K ) ) ) ) ) ) ) ).
% domain.pprimeE(2)
thf(fact_151_domain_OpprimeE_I2_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P )
=> ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) ) ) ) ) ) ).
% domain.pprimeE(2)
thf(fact_152_domain_Olong__division__closed_I2_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( member_list_list_a @ ( polyno1727750685288865234t_unit @ R3 @ P @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(2)
thf(fact_153_domain_Olong__division__closed_I2_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ R3 @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(2)
thf(fact_154_domain_OpirreducibleE_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P )
=> ( P != nil_list_a ) ) ) ) ) ).
% domain.pirreducibleE(1)
thf(fact_155_domain_OpirreducibleE_I1_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P )
=> ( P != nil_a ) ) ) ) ) ).
% domain.pirreducibleE(1)
thf(fact_156_domain_Olong__division__zero_I2_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ R3 @ nil_list_a @ Q )
= nil_list_a ) ) ) ) ).
% domain.long_division_zero(2)
thf(fact_157_domain_Olong__division__zero_I2_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( polynomial_pmod_a_b @ R3 @ nil_a @ Q )
= nil_a ) ) ) ) ).
% domain.long_division_zero(2)
thf(fact_158_domain_Opprime__iff__pirreducible,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_159_domain_Opprime__iff__pirreducible,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_160_domain_Opmod__zero__iff__pdivides,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R3 @ P @ Q )
= nil_list_a )
= ( polyno8016796738000020810t_unit @ R3 @ Q @ P ) ) ) ) ) ) ).
% domain.pmod_zero_iff_pdivides
thf(fact_161_domain_Opmod__zero__iff__pdivides,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ R3 @ P @ Q )
= nil_a )
= ( polyno5814909790663948098es_a_b @ R3 @ Q @ P ) ) ) ) ) ) ).
% domain.pmod_zero_iff_pdivides
thf(fact_162_domain_OpprimeI,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ! [Q3: list_a,R: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R3 @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R3 @ K ) @ Q3 @ R ) )
=> ( ( polyno5814909790663948098es_a_b @ R3 @ P @ Q3 )
| ( polyno5814909790663948098es_a_b @ R3 @ P @ R ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P ) ) ) ) ) ) ) ).
% domain.pprimeI
thf(fact_163_domain_OpprimeI,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( P != nil_list_a )
=> ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ! [Q3: list_list_a,R: list_list_a] :
( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ R @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( polyno8016796738000020810t_unit @ R3 @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ Q3 @ R ) )
=> ( ( polyno8016796738000020810t_unit @ R3 @ P @ Q3 )
| ( polyno8016796738000020810t_unit @ R3 @ P @ R ) ) ) ) )
=> ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P ) ) ) ) ) ) ) ).
% domain.pprimeI
thf(fact_164_domain_Osame__pmod__iff__pdivides,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R3 @ A @ Q )
= ( polyno1727750685288865234t_unit @ R3 @ B @ Q ) )
= ( polyno8016796738000020810t_unit @ R3 @ Q @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ A @ B ) ) ) ) ) ) ) ) ).
% domain.same_pmod_iff_pdivides
thf(fact_165_domain_Osame__pmod__iff__pdivides,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ R3 @ A @ Q )
= ( polynomial_pmod_a_b @ R3 @ B @ Q ) )
= ( polyno5814909790663948098es_a_b @ R3 @ Q @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R3 @ K ) @ A @ B ) ) ) ) ) ) ) ) ).
% domain.same_pmod_iff_pdivides
thf(fact_166_domain_Oexists__unique__long__division,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( Q != nil_list_a )
=> ? [X3: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ R3 @ P @ Q @ X3 )
& ! [Y2: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ R3 @ P @ Q @ Y2 )
=> ( Y2 = X3 ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_167_domain_Oexists__unique__long__division,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( Q != nil_a )
=> ? [X3: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ R3 @ P @ Q @ X3 )
& ! [Y2: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ R3 @ P @ Q @ Y2 )
=> ( Y2 = X3 ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_168_domain_Ouniv__poly__is__domain,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ R3 @ K ) ) ) ) ).
% domain.univ_poly_is_domain
thf(fact_169_domain_Ouniv__poly__is__domain,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( domain7810152921033798211t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) ) ) ).
% domain.univ_poly_is_domain
thf(fact_170_domain_Ouniv__poly__is__principal,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ring_p715737262848045090t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_171_domain_Ouniv__poly__is__principal,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ R3 @ K ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_172_long__divisionI,axiom,
! [K: set_a,P: list_a,Q: list_a,B: list_a,R2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B @ R2 ) )
=> ( ( produc6837034575241423639list_a @ B @ R2 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ) ).
% long_divisionI
thf(fact_173_long__divisionE,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( 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_174_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B3: set_nat] : ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_175_finite__Collect__subsets,axiom,
! [A2: set_complex] :
( ( finite3207457112153483333omplex @ A2 )
=> ( finite6551019134538273531omplex
@ ( collect_set_complex
@ ^ [B3: set_complex] : ( ord_le211207098394363844omplex @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_176_finite__Collect__subsets,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite5282473924520328461list_a
@ ( collect_set_list_a
@ ^ [B3: set_list_a] : ( ord_le8861187494160871172list_a @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_177_finite__Collect__subsets,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B3: set_a] : ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_178_const__term__simprules__shell_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ 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_179_pdiv__pmod,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% pdiv_pmod
thf(fact_180_line__extension__smult__closed,axiom,
! [K: set_a,E: set_a,A: a,K2: a,U: a] :
( ( subfield_a_b @ K @ r )
=> ( ! [K3: a,V: a] :
( ( member_a @ K3 @ K )
=> ( ( member_a @ V @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ V ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ U ) @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_181_monoid__cancelI,axiom,
( ! [A4: a,B4: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A4 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B4 ) )
=> ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A4 = B4 ) ) ) ) )
=> ( ! [A4: a,B4: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A4 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B4 @ C2 ) )
=> ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A4 = B4 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_182_exists__long__division,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ~ ! [B4: list_a] :
( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ! [R: list_a] :
( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B4 @ R ) ) ) ) ) ) ) ) ).
% exists_long_division
thf(fact_183_pirreducible__pow__pdivides__iff,axiom,
! [K: set_a,P: list_a,Q: list_a,R2: list_a,N: nat] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ~ ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R2 ) )
= ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) @ R2 ) ) ) ) ) ) ) ) ).
% pirreducible_pow_pdivides_iff
thf(fact_184_domain_Oring__irreducibleE_I5_J,axiom,
! [R3: partia2956882679547061052t_unit,R2: list_list_a,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( ring_r360171070648044744t_unit @ R3 @ R2 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( R2
= ( mult_l4853965630390486993t_unit @ R3 @ A @ B ) )
=> ( ( member_list_list_a @ A @ ( units_4903515905731149798t_unit @ R3 ) )
| ( member_list_list_a @ B @ ( units_4903515905731149798t_unit @ R3 ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_185_domain_Oring__irreducibleE_I5_J,axiom,
! [R3: partia2670972154091845814t_unit,R2: list_a,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( ring_r932985474545269838t_unit @ R3 @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( R2
= ( mult_l7073676228092353617t_unit @ R3 @ A @ B ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ R3 ) )
| ( member_list_a @ B @ ( units_2932844235741507942t_unit @ R3 ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_186_domain_Oring__irreducibleE_I5_J,axiom,
! [R3: partia2175431115845679010xt_a_b,R2: a,A: a,B: a] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( ring_r999134135267193926le_a_b @ R3 @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ R3 @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ R3 ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ R3 ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_187_poly__add_Ocases,axiom,
! [X2: produc9164743771328383783list_a] :
~ ! [P1: list_a,P22: list_a] :
( X2
!= ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).
% poly_add.cases
thf(fact_188_ring__irreducibleE_I4_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_189_Units__closed,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_190_ring__irreducibleE_I5_J,axiom,
! [R2: a,A: a,B: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R2
= ( 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_191_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_192_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_193_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_194_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_195_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_196_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_197_line__extension__in__carrier,axiom,
! [K: set_a,A: a,E: set_a] :
( ( ord_less_eq_set_a @ K @ ( 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 @ K @ A @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_198_line__extension__consistent,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ( embedd971793762689825387on_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) )
= ( embedd971793762689825387on_a_b @ r ) ) ) ).
% line_extension_consistent
thf(fact_199_const__term__simprules__shell_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_a @ ( const_term_a_b @ r @ P ) @ K ) ) ) ).
% const_term_simprules_shell(1)
thf(fact_200_finite__Collect__disjI,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P2 @ X )
| ( Q2 @ X ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P2 ) )
& ( finite_finite_a @ ( collect_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_201_finite__Collect__disjI,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P2 @ X )
| ( Q2 @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
& ( finite_finite_nat @ ( collect_nat @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_202_finite__Collect__disjI,axiom,
! [P2: complex > $o,Q2: complex > $o] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X: complex] :
( ( P2 @ X )
| ( Q2 @ X ) ) ) )
= ( ( finite3207457112153483333omplex @ ( collect_complex @ P2 ) )
& ( finite3207457112153483333omplex @ ( collect_complex @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_203_finite__Collect__disjI,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X: list_a] :
( ( P2 @ X )
| ( Q2 @ X ) ) ) )
= ( ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
& ( finite_finite_list_a @ ( collect_list_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_204_finite__Collect__conjI,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P2 ) )
| ( finite_finite_a @ ( collect_a @ Q2 ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_205_finite__Collect__conjI,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
| ( finite_finite_nat @ ( collect_nat @ Q2 ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_206_finite__Collect__conjI,axiom,
! [P2: complex > $o,Q2: complex > $o] :
( ( ( finite3207457112153483333omplex @ ( collect_complex @ P2 ) )
| ( finite3207457112153483333omplex @ ( collect_complex @ Q2 ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X: complex] :
( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_207_finite__Collect__conjI,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
| ( finite_finite_list_a @ ( collect_list_a @ Q2 ) ) )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X: list_a] :
( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_208_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_209_subring__polynomial__pow__not__zero,axiom,
! [K: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N )
!= nil_a ) ) ) ) ).
% subring_polynomial_pow_not_zero
thf(fact_210_long__division__add_I2_J,axiom,
! [K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ A @ Q ) @ ( polynomial_pmod_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(2)
thf(fact_211_long__division__add__iff,axiom,
! [K: set_a,A: list_a,B: list_a,C: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( 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 @ K ) @ A @ C ) @ Q )
= ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ).
% long_division_add_iff
thf(fact_212_long__division__add_I1_J,axiom,
! [K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ A @ Q ) @ ( polynomial_pdiv_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(1)
thf(fact_213_Units__m__closed,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_214_Units__l__cancel,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y )
= ( mult_a_ring_ext_a_b @ r @ X2 @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% Units_l_cancel
thf(fact_215_finite__ring__finite__units,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( finite_finite_a @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% finite_ring_finite_units
thf(fact_216_ring_Oconst__term_Ocong,axiom,
const_term_a_b = const_term_a_b ).
% ring.const_term.cong
thf(fact_217_monoid__cancel_Ois__monoid__cancel,axiom,
! [G2: partia2175431115845679010xt_a_b] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( monoid5798828371819920185xt_a_b @ G2 ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_218_domain_Oconst__term__simprules__shell_I3_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( const_term_a_b @ R3 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P @ Q ) )
= ( add_a_b @ R3 @ ( const_term_a_b @ R3 @ P ) @ ( const_term_a_b @ R3 @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(3)
thf(fact_219_domain_Oconst__term__simprules__shell_I3_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ R3 @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P @ Q ) )
= ( add_li7652885771158616974t_unit @ R3 @ ( const_6738166269504826821t_unit @ R3 @ P ) @ ( const_6738166269504826821t_unit @ R3 @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(3)
thf(fact_220_monoid__cancel_Ol__cancel,axiom,
! [G2: partia2670972154091845814t_unit,C: list_a,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G2 )
=> ( ( ( mult_l7073676228092353617t_unit @ G2 @ C @ A )
= ( mult_l7073676228092353617t_unit @ G2 @ C @ B ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_221_monoid__cancel_Ol__cancel,axiom,
! [G2: partia2175431115845679010xt_a_b,C: a,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( ( mult_a_ring_ext_a_b @ G2 @ C @ A )
= ( mult_a_ring_ext_a_b @ G2 @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_222_monoid__cancel_Ol__cancel,axiom,
! [G2: partia2956882679547061052t_unit,C: list_list_a,A: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G2 )
=> ( ( ( mult_l4853965630390486993t_unit @ G2 @ C @ A )
= ( mult_l4853965630390486993t_unit @ G2 @ C @ B ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_223_monoid__cancel_Or__cancel,axiom,
! [G2: partia2670972154091845814t_unit,A: list_a,C: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G2 )
=> ( ( ( mult_l7073676228092353617t_unit @ G2 @ A @ C )
= ( mult_l7073676228092353617t_unit @ G2 @ B @ C ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_224_monoid__cancel_Or__cancel,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,C: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( ( mult_a_ring_ext_a_b @ G2 @ A @ C )
= ( mult_a_ring_ext_a_b @ G2 @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_225_monoid__cancel_Or__cancel,axiom,
! [G2: partia2956882679547061052t_unit,A: list_list_a,C: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G2 )
=> ( ( ( mult_l4853965630390486993t_unit @ G2 @ A @ C )
= ( mult_l4853965630390486993t_unit @ G2 @ B @ C ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_226_monoid__cancel_Odivides__mult__l,axiom,
! [G2: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( monoid4303264861975686087t_unit @ G2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( factor1757716651909850160t_unit @ G2 @ ( mult_l7073676228092353617t_unit @ G2 @ C @ A ) @ ( mult_l7073676228092353617t_unit @ G2 @ C @ B ) )
= ( factor1757716651909850160t_unit @ G2 @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_227_monoid__cancel_Odivides__mult__l,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ ( mult_a_ring_ext_a_b @ G2 @ C @ A ) @ ( mult_a_ring_ext_a_b @ G2 @ C @ B ) )
= ( factor8216151070175719842xt_a_b @ G2 @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_228_monoid__cancel_Odivides__mult__l,axiom,
! [G2: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,C: list_list_a] :
( ( monoid576229335242748231t_unit @ G2 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( factor6954119973539764400t_unit @ G2 @ ( mult_l4853965630390486993t_unit @ G2 @ C @ A ) @ ( mult_l4853965630390486993t_unit @ G2 @ C @ B ) )
= ( factor6954119973539764400t_unit @ G2 @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_229_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B5: set_a,R3: a > a > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B5 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B5 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B5 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A2 )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_230_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B5: set_nat,R3: a > nat > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ B5 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B5 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B5 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A2 )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_231_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B5: set_complex,R3: a > complex > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B5 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: complex] :
( ( member_complex @ X3 @ B5 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A2 )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_232_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B5: set_a,R3: nat > a > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a @ B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B5 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B5 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_233_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B5: set_nat,R3: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B5 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B5 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_234_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B5: set_complex,R3: nat > complex > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B5 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: complex] :
( ( member_complex @ X3 @ B5 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_235_pigeonhole__infinite__rel,axiom,
! [A2: set_complex,B5: set_a,R3: complex > a > $o] :
( ~ ( finite3207457112153483333omplex @ A2 )
=> ( ( finite_finite_a @ B5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B5 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B5 )
& ~ ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [A3: complex] :
( ( member_complex @ A3 @ A2 )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_236_pigeonhole__infinite__rel,axiom,
! [A2: set_complex,B5: set_nat,R3: complex > nat > $o] :
( ~ ( finite3207457112153483333omplex @ A2 )
=> ( ( finite_finite_nat @ B5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B5 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B5 )
& ~ ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [A3: complex] :
( ( member_complex @ A3 @ A2 )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_237_pigeonhole__infinite__rel,axiom,
! [A2: set_complex,B5: set_complex,R3: complex > complex > $o] :
( ~ ( finite3207457112153483333omplex @ A2 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B5 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: complex] :
( ( member_complex @ X3 @ B5 )
& ~ ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [A3: complex] :
( ( member_complex @ A3 @ A2 )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_238_pigeonhole__infinite__rel,axiom,
! [A2: set_set_a,B5: set_a,R3: set_a > a > $o] :
( ~ ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_a @ B5 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B5 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B5 )
& ~ ( finite_finite_set_a
@ ( collect_set_a
@ ^ [A3: set_a] :
( ( member_set_a @ A3 @ A2 )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_239_not__finite__existsD,axiom,
! [P2: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P2 ) )
=> ? [X_1: a] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_240_not__finite__existsD,axiom,
! [P2: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ? [X_1: nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_241_not__finite__existsD,axiom,
! [P2: complex > $o] :
( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P2 ) )
=> ? [X_1: complex] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_242_not__finite__existsD,axiom,
! [P2: list_a > $o] :
( ~ ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
=> ? [X_1: list_a] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_243_domain_Opolynomial__pow__not__zero,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( P != nil_list_list_a )
=> ( ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ P @ N )
!= nil_list_list_a ) ) ) ) ).
% domain.polynomial_pow_not_zero
thf(fact_244_domain_Opolynomial__pow__not__zero,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a,N: nat] :
( ( domain_a_b @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ P @ N )
!= nil_a ) ) ) ) ).
% domain.polynomial_pow_not_zero
thf(fact_245_domain_Opolynomial__pow__not__zero,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ P @ N )
!= nil_list_a ) ) ) ) ).
% domain.polynomial_pow_not_zero
thf(fact_246_domain_Osubring__polynomial__pow__not__zero,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,N: nat] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R3 @ K ) @ P @ N )
!= nil_a ) ) ) ) ) ).
% domain.subring_polynomial_pow_not_zero
thf(fact_247_domain_Osubring__polynomial__pow__not__zero,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( P != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P @ N )
!= nil_list_a ) ) ) ) ) ).
% domain.subring_polynomial_pow_not_zero
thf(fact_248_domain_Olong__division__add__iff,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,C: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R3 @ A @ Q )
= ( polyno1727750685288865234t_unit @ R3 @ B @ Q ) )
= ( ( polyno1727750685288865234t_unit @ R3 @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ A @ C ) @ Q )
= ( polyno1727750685288865234t_unit @ R3 @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ) ).
% domain.long_division_add_iff
thf(fact_249_domain_Olong__division__add__iff,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,C: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ R3 @ A @ Q )
= ( polynomial_pmod_a_b @ R3 @ B @ Q ) )
= ( ( polynomial_pmod_a_b @ R3 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R3 @ K ) @ A @ C ) @ Q )
= ( polynomial_pmod_a_b @ R3 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R3 @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ) ).
% domain.long_division_add_iff
thf(fact_250_domain_Olong__division__add_I2_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ R3 @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ A @ B ) @ Q )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ ( polyno1727750685288865234t_unit @ R3 @ A @ Q ) @ ( polyno1727750685288865234t_unit @ R3 @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(2)
thf(fact_251_domain_Olong__division__add_I2_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( polynomial_pmod_a_b @ R3 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R3 @ K ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R3 @ K ) @ ( polynomial_pmod_a_b @ R3 @ A @ Q ) @ ( polynomial_pmod_a_b @ R3 @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(2)
thf(fact_252_domain_Olong__division__add_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ R3 @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ A @ B ) @ Q )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ ( polyno5893782122288709345t_unit @ R3 @ A @ Q ) @ ( polyno5893782122288709345t_unit @ R3 @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_253_domain_Olong__division__add_I1_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ R3 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R3 @ K ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R3 @ K ) @ ( polynomial_pdiv_a_b @ R3 @ A @ Q ) @ ( polynomial_pdiv_a_b @ R3 @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_254_domain_Oconst__term__simprules__shell_I1_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( member_a @ ( const_term_a_b @ R3 @ P ) @ K ) ) ) ) ).
% domain.const_term_simprules_shell(1)
thf(fact_255_domain_Oconst__term__simprules__shell_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ R3 @ P ) @ K ) ) ) ) ).
% domain.const_term_simprules_shell(1)
thf(fact_256_domain_Oexists__long__division,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( Q != nil_list_a )
=> ~ ! [B4: list_list_a] :
( ( member_list_list_a @ B4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ! [R: list_list_a] :
( ( member_list_list_a @ R @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ~ ( polyno6947042923167803568t_unit @ R3 @ P @ Q @ ( produc8696003437204565271list_a @ B4 @ R ) ) ) ) ) ) ) ) ) ).
% domain.exists_long_division
thf(fact_257_domain_Oexists__long__division,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( Q != nil_a )
=> ~ ! [B4: list_a] :
( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ! [R: list_a] :
( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ R3 @ P @ Q @ ( produc6837034575241423639list_a @ B4 @ R ) ) ) ) ) ) ) ) ) ).
% domain.exists_long_division
thf(fact_258_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_259_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_260_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_261_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_262_rev__finite__subset,axiom,
! [B5: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_263_rev__finite__subset,axiom,
! [B5: set_complex,A2: set_complex] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B5 )
=> ( finite3207457112153483333omplex @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_264_rev__finite__subset,axiom,
! [B5: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B5 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B5 )
=> ( finite_finite_list_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_265_rev__finite__subset,axiom,
! [B5: set_a,A2: set_a] :
( ( finite_finite_a @ B5 )
=> ( ( ord_less_eq_set_a @ A2 @ B5 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_266_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_267_infinite__super,axiom,
! [S: set_complex,T: set_complex] :
( ( ord_le211207098394363844omplex @ S @ T )
=> ( ~ ( finite3207457112153483333omplex @ S )
=> ~ ( finite3207457112153483333omplex @ T ) ) ) ).
% infinite_super
thf(fact_268_infinite__super,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_super
thf(fact_269_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_270_finite__subset,axiom,
! [A2: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( finite_finite_nat @ B5 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_271_finite__subset,axiom,
! [A2: set_complex,B5: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ B5 )
=> ( ( finite3207457112153483333omplex @ B5 )
=> ( finite3207457112153483333omplex @ A2 ) ) ) ).
% finite_subset
thf(fact_272_finite__subset,axiom,
! [A2: set_list_a,B5: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B5 )
=> ( ( finite_finite_list_a @ B5 )
=> ( finite_finite_list_a @ A2 ) ) ) ).
% finite_subset
thf(fact_273_finite__subset,axiom,
! [A2: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A2 @ B5 )
=> ( ( finite_finite_a @ B5 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_274_infinite__imp__nonempty,axiom,
! [S: set_complex] :
( ~ ( finite3207457112153483333omplex @ S )
=> ( S != bot_bot_set_complex ) ) ).
% infinite_imp_nonempty
thf(fact_275_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_276_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_277_infinite__imp__nonempty,axiom,
! [S: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ( S != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_278_finite_OemptyI,axiom,
finite3207457112153483333omplex @ bot_bot_set_complex ).
% finite.emptyI
thf(fact_279_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_280_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_281_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_282_domain_Opirreducible__pow__pdivides__iff,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a,R2: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P )
=> ( ~ ( polyno8016796738000020810t_unit @ R3 @ P @ Q )
=> ( ( polyno8016796738000020810t_unit @ R3 @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P @ N ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ Q @ R2 ) )
= ( polyno8016796738000020810t_unit @ R3 @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P @ N ) @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.pirreducible_pow_pdivides_iff
thf(fact_283_domain_Opirreducible__pow__pdivides__iff,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a,R2: list_a,N: nat] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P )
=> ( ~ ( polyno5814909790663948098es_a_b @ R3 @ P @ Q )
=> ( ( polyno5814909790663948098es_a_b @ R3 @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R3 @ K ) @ P @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R3 @ K ) @ Q @ R2 ) )
= ( polyno5814909790663948098es_a_b @ R3 @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R3 @ K ) @ P @ N ) @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.pirreducible_pow_pdivides_iff
thf(fact_284_domain_Opdiv__pmod,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( P
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ Q @ ( polyno5893782122288709345t_unit @ R3 @ P @ Q ) ) @ ( polyno1727750685288865234t_unit @ R3 @ P @ Q ) ) ) ) ) ) ) ).
% domain.pdiv_pmod
thf(fact_285_domain_Opdiv__pmod,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R3 @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R3 @ K ) @ Q @ ( polynomial_pdiv_a_b @ R3 @ P @ Q ) ) @ ( polynomial_pmod_a_b @ R3 @ P @ Q ) ) ) ) ) ) ) ).
% domain.pdiv_pmod
thf(fact_286_domain_Oconst__term__simprules__shell_I2_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( const_term_a_b @ R3 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ R3 @ ( const_term_a_b @ R3 @ P ) @ ( const_term_a_b @ R3 @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(2)
thf(fact_287_domain_Oconst__term__simprules__shell_I2_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ R3 @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P @ Q ) )
= ( mult_l7073676228092353617t_unit @ R3 @ ( const_6738166269504826821t_unit @ R3 @ P ) @ ( const_6738166269504826821t_unit @ R3 @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(2)
thf(fact_288_principal__domain_Oaxioms_I1_J,axiom,
! [R3: partia2175431115845679010xt_a_b] :
( ( ring_p8803135361686045600in_a_b @ R3 )
=> ( domain_a_b @ R3 ) ) ).
% principal_domain.axioms(1)
thf(fact_289_principal__domain_Oaxioms_I1_J,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( ring_p8098905331641078952t_unit @ R3 )
=> ( domain6553523120543210313t_unit @ R3 ) ) ).
% principal_domain.axioms(1)
thf(fact_290_domain_Olong__divisionI,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a,B: list_list_a,R2: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno6947042923167803568t_unit @ R3 @ P @ Q @ ( produc8696003437204565271list_a @ B @ R2 ) )
=> ( ( produc8696003437204565271list_a @ B @ R2 )
= ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ R3 @ P @ Q ) @ ( polyno1727750685288865234t_unit @ R3 @ P @ Q ) ) ) ) ) ) ) ) ) ).
% domain.long_divisionI
thf(fact_291_domain_Olong__divisionI,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a,B: list_a,R2: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ R3 @ P @ Q @ ( produc6837034575241423639list_a @ B @ R2 ) )
=> ( ( produc6837034575241423639list_a @ B @ R2 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ R3 @ P @ Q ) @ ( polynomial_pmod_a_b @ R3 @ P @ Q ) ) ) ) ) ) ) ) ) ).
% domain.long_divisionI
thf(fact_292_domain_Olong__divisionE,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( Q != nil_list_a )
=> ( polyno6947042923167803568t_unit @ R3 @ P @ Q @ ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ R3 @ P @ Q ) @ ( polyno1727750685288865234t_unit @ R3 @ P @ Q ) ) ) ) ) ) ) ) ).
% domain.long_divisionE
thf(fact_293_domain_Olong__divisionE,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( Q != nil_a )
=> ( polyno2806191415236617128es_a_b @ R3 @ P @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ R3 @ P @ Q ) @ ( polynomial_pmod_a_b @ R3 @ P @ Q ) ) ) ) ) ) ) ) ).
% domain.long_divisionE
thf(fact_294_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_295_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_296_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_297_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_298_domain_Oring__irreducibleE_I4_J,axiom,
! [R3: partia2956882679547061052t_unit,R2: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( ring_r360171070648044744t_unit @ R3 @ R2 )
=> ~ ( member_list_list_a @ R2 @ ( units_4903515905731149798t_unit @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_299_domain_Oring__irreducibleE_I4_J,axiom,
! [R3: partia2670972154091845814t_unit,R2: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( ring_r932985474545269838t_unit @ R3 @ R2 )
=> ~ ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_300_domain_Oring__irreducibleE_I4_J,axiom,
! [R3: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( ring_r999134135267193926le_a_b @ R3 @ R2 )
=> ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_301_principal__domain_Oprimeness__condition,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_a] :
( ( ring_p715737262848045090t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( ring_r360171070648044744t_unit @ R3 @ P )
= ( ring_r5437400583859147359t_unit @ R3 @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_302_principal__domain_Oprimeness__condition,axiom,
! [R3: partia2670972154091845814t_unit,P: list_a] :
( ( ring_p8098905331641078952t_unit @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( ring_r932985474545269838t_unit @ R3 @ P )
= ( ring_r6430282645014804837t_unit @ R3 @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_303_principal__domain_Oprimeness__condition,axiom,
! [R3: partia2175431115845679010xt_a_b,P: a] :
( ( ring_p8803135361686045600in_a_b @ R3 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( ring_r999134135267193926le_a_b @ R3 @ P )
= ( ring_ring_prime_a_b @ R3 @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_304_local_Opderiv__mult,axiom,
! [K: set_a,F: list_a,G: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( formal4452980811800949548iv_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ F @ G ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ ( formal4452980811800949548iv_a_b @ r @ F ) @ G ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ F @ ( formal4452980811800949548iv_a_b @ r @ G ) ) ) ) ) ) ) ).
% local.pderiv_mult
thf(fact_305_subring__polynomial__pow__division,axiom,
! [K: set_a,P: list_a,N: nat,M: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ M ) ) ) ) ) ).
% subring_polynomial_pow_division
thf(fact_306_subring__degree__one__imp__pirreducible,axiom,
! [K: set_a,A: a,B: a] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A
@ ( units_a_ring_ext_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) ) )
=> ( ( member_a @ B @ K )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) ) ) ) ) ).
% subring_degree_one_imp_pirreducible
thf(fact_307_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_308_local_Opderiv__add,axiom,
! [K: set_a,F: list_a,G: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( formal4452980811800949548iv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ F @ G ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( formal4452980811800949548iv_a_b @ r @ F ) @ ( formal4452980811800949548iv_a_b @ r @ G ) ) ) ) ) ) ).
% local.pderiv_add
thf(fact_309_const__term__simprules__shell_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ 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_310_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_311_empty__subsetI,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% empty_subsetI
thf(fact_312_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_313_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_314_subset__empty,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_315_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_316_divides__pirreducible__condition,axiom,
! [K: set_a,Q: list_a,P: list_a] :
( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ Q )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
=> ( ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) ) ) ) ) ).
% divides_pirreducible_condition
thf(fact_317_normalize_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ~ ! [V: a,Va: list_a] :
( X2
!= ( cons_a @ V @ Va ) ) ) ).
% normalize.cases
thf(fact_318_a__lcomm,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X2 @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X2 @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_319_a__comm,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X2 @ Y )
= ( add_a_b @ r @ Y @ X2 ) ) ) ) ).
% a_comm
thf(fact_320_a__assoc,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X2 @ Y ) @ Z )
= ( add_a_b @ r @ X2 @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_321_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_322_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_323_combine_Ocases,axiom,
! [X2: produc9164743771328383783list_a] :
( ! [K3: a,Ks: list_a,U2: a,Us: list_a] :
( X2
!= ( produc6837034575241423639list_a @ ( cons_a @ K3 @ Ks ) @ ( cons_a @ U2 @ Us ) ) )
=> ( ! [Us: list_a] :
( X2
!= ( produc6837034575241423639list_a @ nil_a @ Us ) )
=> ~ ! [Ks: list_a] :
( X2
!= ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).
% combine.cases
thf(fact_324_poly__mult_Ocases,axiom,
! [X2: produc9164743771328383783list_a] :
( ! [P22: list_a] :
( X2
!= ( produc6837034575241423639list_a @ nil_a @ P22 ) )
=> ~ ! [V: a,Va: list_a,P22: list_a] :
( X2
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ Va ) @ P22 ) ) ) ).
% poly_mult.cases
thf(fact_325_subring__props_I7_J,axiom,
! [K: set_a,H1: a,H2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H2 @ K )
=> ( member_a @ ( add_a_b @ r @ H1 @ H2 ) @ K ) ) ) ) ).
% subring_props(7)
thf(fact_326_r__distr,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X2 @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X2 ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_327_l__distr,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X2 @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_328_div__sum__iff,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 @ ( add_a_b @ r @ B @ C ) )
= ( factor8216151070175719842xt_a_b @ r @ A @ C ) ) ) ) ) ) ).
% div_sum_iff
thf(fact_329_div__sum,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 @ C )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( add_a_b @ r @ B @ C ) ) ) ) ) ) ) ).
% div_sum
thf(fact_330_subsetI,axiom,
! [A2: set_set_a,B5: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B5 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B5 ) ) ).
% subsetI
thf(fact_331_subsetI,axiom,
! [A2: set_list_a_a,B5: set_list_a_a] :
( ! [X3: list_a > a] :
( ( member_list_a_a @ X3 @ A2 )
=> ( member_list_a_a @ X3 @ B5 ) )
=> ( ord_le6942402695062981877st_a_a @ A2 @ B5 ) ) ).
% subsetI
thf(fact_332_subsetI,axiom,
! [A2: set_nat_a,B5: set_nat_a] :
( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ A2 )
=> ( member_nat_a @ X3 @ B5 ) )
=> ( ord_le871467723717165285_nat_a @ A2 @ B5 ) ) ).
% subsetI
thf(fact_333_subsetI,axiom,
! [A2: set_a_list_a,B5: set_a_list_a] :
( ! [X3: a > list_a] :
( ( member_a_list_a @ X3 @ A2 )
=> ( member_a_list_a @ X3 @ B5 ) )
=> ( ord_le50412136050534657list_a @ A2 @ B5 ) ) ).
% subsetI
thf(fact_334_subsetI,axiom,
! [A2: set_a,B5: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B5 ) )
=> ( ord_less_eq_set_a @ A2 @ B5 ) ) ).
% subsetI
thf(fact_335_subset__antisym,axiom,
! [A2: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A2 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ A2 )
=> ( A2 = B5 ) ) ) ).
% subset_antisym
thf(fact_336_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_337_empty__iff,axiom,
! [C: list_a > a] :
~ ( member_list_a_a @ C @ bot_bot_set_list_a_a ) ).
% empty_iff
thf(fact_338_empty__iff,axiom,
! [C: nat > a] :
~ ( member_nat_a @ C @ bot_bot_set_nat_a ) ).
% empty_iff
thf(fact_339_empty__iff,axiom,
! [C: a > list_a] :
~ ( member_a_list_a @ C @ bot_bot_set_a_list_a ) ).
% empty_iff
thf(fact_340_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_341_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_342_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_343_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X: set_a] :
~ ( member_set_a @ X @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_344_all__not__in__conv,axiom,
! [A2: set_list_a_a] :
( ( ! [X: list_a > a] :
~ ( member_list_a_a @ X @ A2 ) )
= ( A2 = bot_bot_set_list_a_a ) ) ).
% all_not_in_conv
thf(fact_345_all__not__in__conv,axiom,
! [A2: set_nat_a] :
( ( ! [X: nat > a] :
~ ( member_nat_a @ X @ A2 ) )
= ( A2 = bot_bot_set_nat_a ) ) ).
% all_not_in_conv
thf(fact_346_all__not__in__conv,axiom,
! [A2: set_a_list_a] :
( ( ! [X: a > list_a] :
~ ( member_a_list_a @ X @ A2 ) )
= ( A2 = bot_bot_set_a_list_a ) ) ).
% all_not_in_conv
thf(fact_347_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_348_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_349_all__not__in__conv,axiom,
! [A2: set_list_a] :
( ( ! [X: list_a] :
~ ( member_list_a @ X @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_350_Collect__empty__eq,axiom,
! [P2: complex > $o] :
( ( ( collect_complex @ P2 )
= bot_bot_set_complex )
= ( ! [X: complex] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_351_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_352_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_353_Collect__empty__eq,axiom,
! [P2: list_a > $o] :
( ( ( collect_list_a @ P2 )
= bot_bot_set_list_a )
= ( ! [X: list_a] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_354_empty__Collect__eq,axiom,
! [P2: complex > $o] :
( ( bot_bot_set_complex
= ( collect_complex @ P2 ) )
= ( ! [X: complex] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_355_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X: a] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_356_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X: nat] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_357_empty__Collect__eq,axiom,
! [P2: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P2 ) )
= ( ! [X: list_a] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_358_line__extension__mem__iff,axiom,
! [U: a,K: set_a,A: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) )
= ( ? [X: a] :
( ( member_a @ X @ K )
& ? [Y3: a] :
( ( member_a @ Y3 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ A ) @ Y3 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_359_pderiv__carr,axiom,
! [K: set_a,F: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( formal4452980811800949548iv_a_b @ r @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ).
% pderiv_carr
thf(fact_360_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_361_associated__polynomials__imp__same__is__root,axiom,
! [P: list_a,Q: list_a,X2: 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 @ X2 )
= ( polyno4133073214067823460ot_a_b @ r @ Q @ X2 ) ) ) ) ) ).
% associated_polynomials_imp_same_is_root
thf(fact_362_a__closed,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X2 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_363_local_Oadd_Oright__cancel,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X2 )
= ( add_a_b @ r @ Z @ X2 ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_364_less__eq__set__def,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B3: set_set_a] :
( ord_less_eq_set_a_o
@ ^ [X: set_a] : ( member_set_a @ X @ A5 )
@ ^ [X: set_a] : ( member_set_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_365_less__eq__set__def,axiom,
( ord_le6942402695062981877st_a_a
= ( ^ [A5: set_list_a_a,B3: set_list_a_a] :
( ord_le5538412863658560464_a_a_o
@ ^ [X: list_a > a] : ( member_list_a_a @ X @ A5 )
@ ^ [X: list_a > a] : ( member_list_a_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_366_less__eq__set__def,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A5: set_nat_a,B3: set_nat_a] :
( ord_less_eq_nat_a_o
@ ^ [X: nat > a] : ( member_nat_a @ X @ A5 )
@ ^ [X: nat > a] : ( member_nat_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_367_less__eq__set__def,axiom,
( ord_le50412136050534657list_a
= ( ^ [A5: set_a_list_a,B3: set_a_list_a] :
( ord_le3190646433912094404st_a_o
@ ^ [X: a > list_a] : ( member_a_list_a @ X @ A5 )
@ ^ [X: a > list_a] : ( member_a_list_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_368_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B3: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ A5 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_369_associatedD,axiom,
! [G2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( associ8407585678920448409t_unit @ G2 @ A @ B )
=> ( factor1757716651909850160t_unit @ G2 @ A @ B ) ) ).
% associatedD
thf(fact_370_associatedD,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ G2 @ A @ B )
=> ( factor8216151070175719842xt_a_b @ G2 @ A @ B ) ) ).
% associatedD
thf(fact_371_associatedE,axiom,
! [G2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( associ8407585678920448409t_unit @ G2 @ A @ B )
=> ~ ( ( factor1757716651909850160t_unit @ G2 @ A @ B )
=> ~ ( factor1757716651909850160t_unit @ G2 @ B @ A ) ) ) ).
% associatedE
thf(fact_372_associatedE,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ G2 @ A @ B )
=> ~ ( ( factor8216151070175719842xt_a_b @ G2 @ A @ B )
=> ~ ( factor8216151070175719842xt_a_b @ G2 @ B @ A ) ) ) ).
% associatedE
thf(fact_373_associated__def,axiom,
( associ8407585678920448409t_unit
= ( ^ [G3: partia2670972154091845814t_unit,A3: list_a,B2: list_a] :
( ( factor1757716651909850160t_unit @ G3 @ A3 @ B2 )
& ( factor1757716651909850160t_unit @ G3 @ B2 @ A3 ) ) ) ) ).
% associated_def
thf(fact_374_associated__def,axiom,
( associ5860276527279195403xt_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( factor8216151070175719842xt_a_b @ G3 @ A3 @ B2 )
& ( factor8216151070175719842xt_a_b @ G3 @ B2 @ A3 ) ) ) ) ).
% associated_def
thf(fact_375_divides__antisym,axiom,
! [G2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( factor1757716651909850160t_unit @ G2 @ A @ B )
=> ( ( factor1757716651909850160t_unit @ G2 @ B @ A )
=> ( associ8407585678920448409t_unit @ G2 @ A @ B ) ) ) ).
% divides_antisym
thf(fact_376_divides__antisym,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( factor8216151070175719842xt_a_b @ G2 @ A @ B )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ B @ A )
=> ( associ5860276527279195403xt_a_b @ G2 @ A @ B ) ) ) ).
% divides_antisym
thf(fact_377_monoid__cancel_Oassoc__l__cancel,axiom,
! [G2: partia2670972154091845814t_unit,A: list_a,B: list_a,B6: list_a] :
( ( monoid4303264861975686087t_unit @ G2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ B6 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( associ8407585678920448409t_unit @ G2 @ ( mult_l7073676228092353617t_unit @ G2 @ A @ B ) @ ( mult_l7073676228092353617t_unit @ G2 @ A @ B6 ) )
=> ( associ8407585678920448409t_unit @ G2 @ B @ B6 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_378_monoid__cancel_Oassoc__l__cancel,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a,B6: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ ( mult_a_ring_ext_a_b @ G2 @ A @ B ) @ ( mult_a_ring_ext_a_b @ G2 @ A @ B6 ) )
=> ( associ5860276527279195403xt_a_b @ G2 @ B @ B6 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_379_monoid__cancel_Oassoc__l__cancel,axiom,
! [G2: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,B6: list_list_a] :
( ( monoid576229335242748231t_unit @ G2 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ B6 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( associ5603075271488036121t_unit @ G2 @ ( mult_l4853965630390486993t_unit @ G2 @ A @ B ) @ ( mult_l4853965630390486993t_unit @ G2 @ A @ B6 ) )
=> ( associ5603075271488036121t_unit @ G2 @ B @ B6 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_380_monoid__cancel_Oassoc__unit__r,axiom,
! [G2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G2 )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G2 ) )
=> ( ( associ8407585678920448409t_unit @ G2 @ A @ B )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_381_monoid__cancel_Oassoc__unit__r,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G2 ) )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ A @ B )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ G2 ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_382_monoid__cancel_Oassoc__unit__r,axiom,
! [G2: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G2 )
=> ( ( member_list_list_a @ A @ ( units_4903515905731149798t_unit @ G2 ) )
=> ( ( associ5603075271488036121t_unit @ G2 @ A @ B )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( member_list_list_a @ B @ ( units_4903515905731149798t_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_383_monoid__cancel_Oassoc__unit__l,axiom,
! [G2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G2 )
=> ( ( associ8407585678920448409t_unit @ G2 @ A @ B )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G2 ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_384_monoid__cancel_Oassoc__unit__l,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ A @ B )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ G2 ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_385_monoid__cancel_Oassoc__unit__l,axiom,
! [G2: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G2 )
=> ( ( associ5603075271488036121t_unit @ G2 @ A @ B )
=> ( ( member_list_list_a @ B @ ( units_4903515905731149798t_unit @ G2 ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( member_list_list_a @ A @ ( units_4903515905731149798t_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_386_in__mono,axiom,
! [A2: set_set_a,B5: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B5 )
=> ( ( member_set_a @ X2 @ A2 )
=> ( member_set_a @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_387_in__mono,axiom,
! [A2: set_list_a_a,B5: set_list_a_a,X2: list_a > a] :
( ( ord_le6942402695062981877st_a_a @ A2 @ B5 )
=> ( ( member_list_a_a @ X2 @ A2 )
=> ( member_list_a_a @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_388_in__mono,axiom,
! [A2: set_nat_a,B5: set_nat_a,X2: nat > a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B5 )
=> ( ( member_nat_a @ X2 @ A2 )
=> ( member_nat_a @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_389_in__mono,axiom,
! [A2: set_a_list_a,B5: set_a_list_a,X2: a > list_a] :
( ( ord_le50412136050534657list_a @ A2 @ B5 )
=> ( ( member_a_list_a @ X2 @ A2 )
=> ( member_a_list_a @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_390_in__mono,axiom,
! [A2: set_a,B5: set_a,X2: a] :
( ( ord_less_eq_set_a @ A2 @ B5 )
=> ( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B5 ) ) ) ).
% in_mono
thf(fact_391_subsetD,axiom,
! [A2: set_set_a,B5: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B5 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_392_subsetD,axiom,
! [A2: set_list_a_a,B5: set_list_a_a,C: list_a > a] :
( ( ord_le6942402695062981877st_a_a @ A2 @ B5 )
=> ( ( member_list_a_a @ C @ A2 )
=> ( member_list_a_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_393_subsetD,axiom,
! [A2: set_nat_a,B5: set_nat_a,C: nat > a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B5 )
=> ( ( member_nat_a @ C @ A2 )
=> ( member_nat_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_394_subsetD,axiom,
! [A2: set_a_list_a,B5: set_a_list_a,C: a > list_a] :
( ( ord_le50412136050534657list_a @ A2 @ B5 )
=> ( ( member_a_list_a @ C @ A2 )
=> ( member_a_list_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_395_subsetD,axiom,
! [A2: set_a,B5: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B5 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_396_equalityE,axiom,
! [A2: set_a,B5: set_a] :
( ( A2 = B5 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B5 )
=> ~ ( ord_less_eq_set_a @ B5 @ A2 ) ) ) ).
% equalityE
thf(fact_397_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B3: set_set_a] :
! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( member_set_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_398_subset__eq,axiom,
( ord_le6942402695062981877st_a_a
= ( ^ [A5: set_list_a_a,B3: set_list_a_a] :
! [X: list_a > a] :
( ( member_list_a_a @ X @ A5 )
=> ( member_list_a_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_399_subset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A5: set_nat_a,B3: set_nat_a] :
! [X: nat > a] :
( ( member_nat_a @ X @ A5 )
=> ( member_nat_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_400_subset__eq,axiom,
( ord_le50412136050534657list_a
= ( ^ [A5: set_a_list_a,B3: set_a_list_a] :
! [X: a > list_a] :
( ( member_a_list_a @ X @ A5 )
=> ( member_a_list_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_401_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B3: set_a] :
! [X: a] :
( ( member_a @ X @ A5 )
=> ( member_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_402_equalityD1,axiom,
! [A2: set_a,B5: set_a] :
( ( A2 = B5 )
=> ( ord_less_eq_set_a @ A2 @ B5 ) ) ).
% equalityD1
thf(fact_403_equalityD2,axiom,
! [A2: set_a,B5: set_a] :
( ( A2 = B5 )
=> ( ord_less_eq_set_a @ B5 @ A2 ) ) ).
% equalityD2
thf(fact_404_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B3: set_set_a] :
! [T2: set_a] :
( ( member_set_a @ T2 @ A5 )
=> ( member_set_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_405_subset__iff,axiom,
( ord_le6942402695062981877st_a_a
= ( ^ [A5: set_list_a_a,B3: set_list_a_a] :
! [T2: list_a > a] :
( ( member_list_a_a @ T2 @ A5 )
=> ( member_list_a_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_406_subset__iff,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A5: set_nat_a,B3: set_nat_a] :
! [T2: nat > a] :
( ( member_nat_a @ T2 @ A5 )
=> ( member_nat_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_407_subset__iff,axiom,
( ord_le50412136050534657list_a
= ( ^ [A5: set_a_list_a,B3: set_a_list_a] :
! [T2: a > list_a] :
( ( member_a_list_a @ T2 @ A5 )
=> ( member_a_list_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_408_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B3: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A5 )
=> ( member_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_409_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_410_Collect__mono,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_mono
thf(fact_411_Collect__mono,axiom,
! [P2: complex > $o,Q2: complex > $o] :
( ! [X3: complex] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le211207098394363844omplex @ ( collect_complex @ P2 ) @ ( collect_complex @ Q2 ) ) ) ).
% Collect_mono
thf(fact_412_Collect__mono,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ! [X3: list_a] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_413_Collect__mono,axiom,
! [P2: a > $o,Q2: a > $o] :
( ! [X3: a] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_414_subset__trans,axiom,
! [A2: set_a,B5: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ C3 )
=> ( ord_less_eq_set_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_415_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A5: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A5 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_416_Collect__mono__iff,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) )
= ( ! [X: nat] :
( ( P2 @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_417_Collect__mono__iff,axiom,
! [P2: complex > $o,Q2: complex > $o] :
( ( ord_le211207098394363844omplex @ ( collect_complex @ P2 ) @ ( collect_complex @ Q2 ) )
= ( ! [X: complex] :
( ( P2 @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_418_Collect__mono__iff,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q2 ) )
= ( ! [X: list_a] :
( ( P2 @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_419_Collect__mono__iff,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) )
= ( ! [X: a] :
( ( P2 @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_420_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_421_emptyE,axiom,
! [A: list_a > a] :
~ ( member_list_a_a @ A @ bot_bot_set_list_a_a ) ).
% emptyE
thf(fact_422_emptyE,axiom,
! [A: nat > a] :
~ ( member_nat_a @ A @ bot_bot_set_nat_a ) ).
% emptyE
thf(fact_423_emptyE,axiom,
! [A: a > list_a] :
~ ( member_a_list_a @ A @ bot_bot_set_a_list_a ) ).
% emptyE
thf(fact_424_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_425_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_426_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_427_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_428_equals0D,axiom,
! [A2: set_list_a_a,A: list_a > a] :
( ( A2 = bot_bot_set_list_a_a )
=> ~ ( member_list_a_a @ A @ A2 ) ) ).
% equals0D
thf(fact_429_equals0D,axiom,
! [A2: set_nat_a,A: nat > a] :
( ( A2 = bot_bot_set_nat_a )
=> ~ ( member_nat_a @ A @ A2 ) ) ).
% equals0D
thf(fact_430_equals0D,axiom,
! [A2: set_a_list_a,A: a > list_a] :
( ( A2 = bot_bot_set_a_list_a )
=> ~ ( member_a_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_431_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_432_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_433_equals0D,axiom,
! [A2: set_list_a,A: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_434_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y5: set_a] :
~ ( member_set_a @ Y5 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_435_equals0I,axiom,
! [A2: set_list_a_a] :
( ! [Y5: list_a > a] :
~ ( member_list_a_a @ Y5 @ A2 )
=> ( A2 = bot_bot_set_list_a_a ) ) ).
% equals0I
thf(fact_436_equals0I,axiom,
! [A2: set_nat_a] :
( ! [Y5: nat > a] :
~ ( member_nat_a @ Y5 @ A2 )
=> ( A2 = bot_bot_set_nat_a ) ) ).
% equals0I
thf(fact_437_equals0I,axiom,
! [A2: set_a_list_a] :
( ! [Y5: a > list_a] :
~ ( member_a_list_a @ Y5 @ A2 )
=> ( A2 = bot_bot_set_a_list_a ) ) ).
% equals0I
thf(fact_438_equals0I,axiom,
! [A2: set_a] :
( ! [Y5: a] :
~ ( member_a @ Y5 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_439_equals0I,axiom,
! [A2: set_nat] :
( ! [Y5: nat] :
~ ( member_nat @ Y5 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_440_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y5: list_a] :
~ ( member_list_a @ Y5 @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_441_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X: set_a] : ( member_set_a @ X @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_442_ex__in__conv,axiom,
! [A2: set_list_a_a] :
( ( ? [X: list_a > a] : ( member_list_a_a @ X @ A2 ) )
= ( A2 != bot_bot_set_list_a_a ) ) ).
% ex_in_conv
thf(fact_443_ex__in__conv,axiom,
! [A2: set_nat_a] :
( ( ? [X: nat > a] : ( member_nat_a @ X @ A2 ) )
= ( A2 != bot_bot_set_nat_a ) ) ).
% ex_in_conv
thf(fact_444_ex__in__conv,axiom,
! [A2: set_a_list_a] :
( ( ? [X: a > list_a] : ( member_a_list_a @ X @ A2 ) )
= ( A2 != bot_bot_set_a_list_a ) ) ).
% ex_in_conv
thf(fact_445_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X: a] : ( member_a @ X @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_446_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_447_ex__in__conv,axiom,
! [A2: set_list_a] :
( ( ? [X: list_a] : ( member_list_a @ X @ A2 ) )
= ( A2 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_448_domain_Oring__associated__iff,axiom,
! [R3: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( associ8407585678920448409t_unit @ R3 @ A @ B )
= ( ? [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ R3 ) )
& ( A
= ( mult_l7073676228092353617t_unit @ R3 @ X @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_449_domain_Oring__associated__iff,axiom,
! [R3: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( associ5860276527279195403xt_a_b @ R3 @ A @ B )
= ( ? [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ R3 ) )
& ( A
= ( mult_a_ring_ext_a_b @ R3 @ X @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_450_domain_Oring__associated__iff,axiom,
! [R3: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( associ5603075271488036121t_unit @ R3 @ A @ B )
= ( ? [X: list_list_a] :
( ( member_list_list_a @ X @ ( units_4903515905731149798t_unit @ R3 ) )
& ( A
= ( mult_l4853965630390486993t_unit @ R3 @ X @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_451_monoid__cancel_OassociatedD2,axiom,
! [G2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G2 )
=> ( ( associ8407585678920448409t_unit @ G2 @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G2 ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ G2 ) )
& ( A
= ( mult_l7073676228092353617t_unit @ G2 @ B @ X3 ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_452_monoid__cancel_OassociatedD2,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ G2 ) )
& ( A
= ( mult_a_ring_ext_a_b @ G2 @ B @ X3 ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_453_monoid__cancel_OassociatedD2,axiom,
! [G2: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G2 )
=> ( ( associ5603075271488036121t_unit @ G2 @ A @ B )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G2 ) )
=> ? [X3: list_list_a] :
( ( member_list_list_a @ X3 @ ( units_4903515905731149798t_unit @ G2 ) )
& ( A
= ( mult_l4853965630390486993t_unit @ G2 @ B @ X3 ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_454_monoid__cancel_OassociatedE2,axiom,
! [G2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G2 )
=> ( ( associ8407585678920448409t_unit @ G2 @ A @ B )
=> ( ! [U2: list_a] :
( ( A
= ( mult_l7073676228092353617t_unit @ G2 @ B @ U2 ) )
=> ~ ( member_list_a @ U2 @ ( units_2932844235741507942t_unit @ G2 ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G2 ) )
=> ~ ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_455_monoid__cancel_OassociatedE2,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ A @ B )
=> ( ! [U2: a] :
( ( A
= ( mult_a_ring_ext_a_b @ G2 @ B @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_ring_ext_a_b @ G2 ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ~ ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_456_monoid__cancel_OassociatedE2,axiom,
! [G2: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G2 )
=> ( ( associ5603075271488036121t_unit @ G2 @ A @ B )
=> ( ! [U2: list_list_a] :
( ( A
= ( mult_l4853965630390486993t_unit @ G2 @ B @ U2 ) )
=> ~ ( member_list_list_a @ U2 @ ( units_4903515905731149798t_unit @ G2 ) ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G2 ) )
=> ~ ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_457_monoid__cancel_Oassociated__iff,axiom,
! [G2: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( associ8407585678920448409t_unit @ G2 @ A @ B )
= ( ? [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G2 ) )
& ( A
= ( mult_l7073676228092353617t_unit @ G2 @ B @ X ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_458_monoid__cancel_Oassociated__iff,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ A @ B )
= ( ? [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ G2 ) )
& ( A
= ( mult_a_ring_ext_a_b @ G2 @ B @ X ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_459_monoid__cancel_Oassociated__iff,axiom,
! [G2: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G2 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( associ5603075271488036121t_unit @ G2 @ A @ B )
= ( ? [X: list_list_a] :
( ( member_list_list_a @ X @ ( units_4903515905731149798t_unit @ G2 ) )
& ( A
= ( mult_l4853965630390486993t_unit @ G2 @ B @ X ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_460_Collect__subset,axiom,
! [A2: set_set_a,P2: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ A2 )
& ( P2 @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_461_Collect__subset,axiom,
! [A2: set_list_a_a,P2: ( list_a > a ) > $o] :
( ord_le6942402695062981877st_a_a
@ ( collect_list_a_a
@ ^ [X: list_a > a] :
( ( member_list_a_a @ X @ A2 )
& ( P2 @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_462_Collect__subset,axiom,
! [A2: set_nat_a,P2: ( nat > a ) > $o] :
( ord_le871467723717165285_nat_a
@ ( collect_nat_a
@ ^ [X: nat > a] :
( ( member_nat_a @ X @ A2 )
& ( P2 @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_463_Collect__subset,axiom,
! [A2: set_a_list_a,P2: ( a > list_a ) > $o] :
( ord_le50412136050534657list_a
@ ( collect_a_list_a
@ ^ [X: a > list_a] :
( ( member_a_list_a @ X @ A2 )
& ( P2 @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_464_Collect__subset,axiom,
! [A2: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P2 @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_465_Collect__subset,axiom,
! [A2: set_complex,P2: complex > $o] :
( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [X: complex] :
( ( member_complex @ X @ A2 )
& ( P2 @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_466_Collect__subset,axiom,
! [A2: set_list_a,P2: list_a > $o] :
( ord_le8861187494160871172list_a
@ ( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ A2 )
& ( P2 @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_467_Collect__subset,axiom,
! [A2: set_a,P2: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( P2 @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_468_empty__def,axiom,
( bot_bot_set_complex
= ( collect_complex
@ ^ [X: complex] : $false ) ) ).
% empty_def
thf(fact_469_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X: a] : $false ) ) ).
% empty_def
thf(fact_470_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% empty_def
thf(fact_471_empty__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a
@ ^ [X: list_a] : $false ) ) ).
% empty_def
thf(fact_472_domain_Oassociated__polynomials__imp__same__is__root,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a,X2: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( associ9038253669175192217t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ P @ Q )
=> ( ( polyno5142720416380192742t_unit @ R3 @ P @ X2 )
= ( polyno5142720416380192742t_unit @ R3 @ Q @ X2 ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_is_root
thf(fact_473_domain_Oassociated__polynomials__imp__same__is__root,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,X2: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ P @ Q )
=> ( ( polyno6951661231331188332t_unit @ R3 @ P @ X2 )
= ( polyno6951661231331188332t_unit @ R3 @ Q @ X2 ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_is_root
thf(fact_474_domain_Oassociated__polynomials__imp__same__is__root,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,X2: a] :
( ( domain_a_b @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ P @ Q )
=> ( ( polyno4133073214067823460ot_a_b @ R3 @ P @ X2 )
= ( polyno4133073214067823460ot_a_b @ R3 @ Q @ X2 ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_is_root
thf(fact_475_domain_Opolynomial__pow__division,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a,N: nat,M: nat] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno4453881341673752516t_unit @ R3 @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ P @ N ) @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ P @ M ) ) ) ) ) ).
% domain.polynomial_pow_division
thf(fact_476_domain_Opolynomial__pow__division,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a,N: nat,M: nat] :
( ( domain_a_b @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno5814909790663948098es_a_b @ R3 @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ P @ M ) ) ) ) ) ).
% domain.polynomial_pow_division
thf(fact_477_domain_Opolynomial__pow__division,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a,N: nat,M: nat] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno8016796738000020810t_unit @ R3 @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ P @ N ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ P @ M ) ) ) ) ) ).
% domain.polynomial_pow_division
thf(fact_478_domain_Osubring__polynomial__pow__division,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,N: nat,M: nat] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ R3 @ K ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R3 @ K ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R3 @ K ) @ P @ M ) ) ) ) ) ) ).
% domain.subring_polynomial_pow_division
thf(fact_479_domain_Osubring__polynomial__pow__division,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,N: nat,M: nat] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P @ N ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P @ M ) ) ) ) ) ) ).
% domain.subring_polynomial_pow_division
thf(fact_480_domain_Osubring__degree__one__imp__pirreducible,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_a @ A
@ ( units_2932844235741507942t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ R3 ) ) )
=> ( ( member_list_a @ B @ K )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ ( cons_list_a @ A @ ( cons_list_a @ B @ nil_list_a ) ) ) ) ) ) ) ).
% domain.subring_degree_one_imp_pirreducible
thf(fact_481_domain_Osubring__degree__one__imp__pirreducible,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,A: a,B: a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_a @ A
@ ( units_a_ring_ext_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ R3 ) ) )
=> ( ( member_a @ B @ K )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R3 @ K ) @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) ) ) ) ) ) ).
% domain.subring_degree_one_imp_pirreducible
thf(fact_482_domain_Opderiv__mult,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,F: list_list_a,G: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ G @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( formal6075833236969493044t_unit @ R3 @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ F @ G ) )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ ( formal6075833236969493044t_unit @ R3 @ F ) @ G ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ F @ ( formal6075833236969493044t_unit @ R3 @ G ) ) ) ) ) ) ) ) ).
% domain.pderiv_mult
thf(fact_483_domain_Opderiv__mult,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,F: list_a,G: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( formal4452980811800949548iv_a_b @ R3 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R3 @ K ) @ F @ G ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R3 @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R3 @ K ) @ ( formal4452980811800949548iv_a_b @ R3 @ F ) @ G ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R3 @ K ) @ F @ ( formal4452980811800949548iv_a_b @ R3 @ G ) ) ) ) ) ) ) ) ).
% domain.pderiv_mult
thf(fact_484_domain_Opderiv__add,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,F: list_list_a,G: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ G @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( formal6075833236969493044t_unit @ R3 @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ F @ G ) )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ ( formal6075833236969493044t_unit @ R3 @ F ) @ ( formal6075833236969493044t_unit @ R3 @ G ) ) ) ) ) ) ) ).
% domain.pderiv_add
thf(fact_485_domain_Opderiv__add,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,F: list_a,G: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( formal4452980811800949548iv_a_b @ R3 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R3 @ K ) @ F @ G ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R3 @ K ) @ ( formal4452980811800949548iv_a_b @ R3 @ F ) @ ( formal4452980811800949548iv_a_b @ R3 @ G ) ) ) ) ) ) ) ).
% domain.pderiv_add
thf(fact_486_subring__degree__one__associatedI,axiom,
! [K: set_a,A: a,A6: a,B: a] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A @ K )
=> ( ( member_a @ A6 @ K )
=> ( ( member_a @ B @ K )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ A6 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ ( 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 @ A6 @ B ) @ nil_a ) ) ) ) ) ) ) ) ).
% subring_degree_one_associatedI
thf(fact_487_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_488_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_489_domain_Opderiv__carr,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,F: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( member_list_list_a @ ( formal6075833236969493044t_unit @ R3 @ F ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) ) ) ) ) ).
% domain.pderiv_carr
thf(fact_490_domain_Opderiv__carr,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,F: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( member_list_a @ ( formal4452980811800949548iv_a_b @ R3 @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) ) ) ) ) ).
% domain.pderiv_carr
thf(fact_491_a__lcos__m__assoc,axiom,
! [M2: set_a,G: a,H3: a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H3 @ M2 ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H3 ) @ M2 ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_492_cgenideal__pirreducible,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ Q )
=> ( ( member_list_a @ Q @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) ) ) ) ) ) ).
% cgenideal_pirreducible
thf(fact_493_associated__sym,axiom,
! [A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( associ5860276527279195403xt_a_b @ r @ B @ A ) ) ).
% associated_sym
thf(fact_494_assoc__subst,axiom,
! [A: a,B: a,F: a > a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ! [A4: a,B4: a] :
( ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ A4 @ B4 ) )
=> ( ( member_a @ ( F @ A4 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ ( F @ B4 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ ( F @ A4 ) @ ( F @ B4 ) ) ) )
=> ( ( 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_495_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_496_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_497_mult__cong__l,axiom,
! [A: a,A6: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ A6 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A6 @ ( 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 @ A6 @ B ) ) ) ) ) ) ).
% mult_cong_l
thf(fact_498_mult__cong__r,axiom,
! [B: a,B6: a,A: a] :
( ( associ5860276527279195403xt_a_b @ r @ B @ B6 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B6 @ ( 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 @ B6 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_499_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_500_divides__cong__l,axiom,
! [X2: a,X4: a,Y: a] :
( ( associ5860276527279195403xt_a_b @ r @ X2 @ X4 )
=> ( ( factor8216151070175719842xt_a_b @ r @ X4 @ Y )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ X2 @ Y ) ) ) ) ).
% divides_cong_l
thf(fact_501_divides__cong__r,axiom,
! [X2: a,Y: a,Y6: a] :
( ( factor8216151070175719842xt_a_b @ r @ X2 @ Y )
=> ( ( associ5860276527279195403xt_a_b @ r @ Y @ Y6 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ X2 @ Y6 ) ) ) ) ).
% divides_cong_r
thf(fact_502_subring__props_I3_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K ) ) ).
% subring_props(3)
thf(fact_503_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_504_inv__unique,axiom,
! [Y: a,X2: a,Y6: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y @ X2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y6 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y6 ) ) ) ) ) ) ).
% inv_unique
thf(fact_505_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X3 )
= X3 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_506_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_507_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_508_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 )
= ( ? [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
& ( A
= ( mult_a_ring_ext_a_b @ r @ X @ B ) ) ) ) ) ) ) ).
% ring_associated_iff
thf(fact_509_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_510_Units__inv__comm,axiom,
! [X2: a,Y: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_inv_comm
thf(fact_511_a__l__coset__subset__G,axiom,
! [H: set_a,X2: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X2 @ H ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_512_Units__l__inv__ex,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_513_Units__r__inv__ex,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X2 @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_514_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_515_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_516_associated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ A ) ) ).
% associated_refl
thf(fact_517_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_518_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_519_l__one,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X2 )
= X2 ) ) ).
% l_one
thf(fact_520_r__one,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( one_a_ring_ext_a_b @ r ) )
= X2 ) ) ).
% r_one
thf(fact_521_ring_Oroots_Ocong,axiom,
polynomial_roots_a_b = polynomial_roots_a_b ).
% ring.roots.cong
thf(fact_522_univ__poly__one,axiom,
! [R3: partia7496981018696276118t_unit,K: set_set_list_a] :
( ( one_li1622763072977731901t_unit @ ( univ_p863672496597069550t_unit @ R3 @ K ) )
= ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R3 ) @ nil_set_list_a ) ) ).
% univ_poly_one
thf(fact_523_univ__poly__one,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a] :
( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R3 @ K ) )
= ( cons_a @ ( one_a_ring_ext_a_b @ R3 ) @ nil_a ) ) ).
% univ_poly_one
thf(fact_524_univ__poly__one,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a] :
( ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) )
= ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ nil_list_a ) ) ).
% univ_poly_one
thf(fact_525_domain_Oring__primeE_I3_J,axiom,
! [R3: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( ring_r6430282645014804837t_unit @ R3 @ P )
=> ( prime_2011924034616061926t_unit @ R3 @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_526_domain_Oring__primeE_I3_J,axiom,
! [R3: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( ring_ring_prime_a_b @ R3 @ P )
=> ( prime_a_ring_ext_a_b @ R3 @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_527_domain_Oring__primeE_I3_J,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( ring_r5437400583859147359t_unit @ R3 @ P )
=> ( prime_1232919612140715622t_unit @ R3 @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_528_monoid__cancel_Oprime__cong,axiom,
! [G2: partia2670972154091845814t_unit,P: list_a,P3: list_a] :
( ( monoid4303264861975686087t_unit @ G2 )
=> ( ( prime_2011924034616061926t_unit @ G2 @ P )
=> ( ( associ8407585678920448409t_unit @ G2 @ P @ P3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( prime_2011924034616061926t_unit @ G2 @ P3 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_529_monoid__cancel_Oprime__cong,axiom,
! [G2: partia2175431115845679010xt_a_b,P: a,P3: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( prime_a_ring_ext_a_b @ G2 @ P )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ P @ P3 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ P3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( prime_a_ring_ext_a_b @ G2 @ P3 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_530_monoid__cancel_Oprime__cong,axiom,
! [G2: partia2956882679547061052t_unit,P: list_list_a,P3: list_list_a] :
( ( monoid576229335242748231t_unit @ G2 )
=> ( ( prime_1232919612140715622t_unit @ G2 @ P )
=> ( ( associ5603075271488036121t_unit @ G2 @ P @ P3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( prime_1232919612140715622t_unit @ G2 @ P3 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_531_domain_Oassociated__polynomials__imp__same__roots,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( associ9038253669175192217t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ P @ Q )
=> ( ( polyno3707469075594375645t_unit @ R3 @ P )
= ( polyno3707469075594375645t_unit @ R3 @ Q ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_roots
thf(fact_532_domain_Oassociated__polynomials__imp__same__roots,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ P @ Q )
=> ( ( polyno7858422826990252003t_unit @ R3 @ P )
= ( polyno7858422826990252003t_unit @ R3 @ Q ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_roots
thf(fact_533_domain_Oassociated__polynomials__imp__same__roots,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ P @ Q )
=> ( ( polynomial_roots_a_b @ R3 @ P )
= ( polynomial_roots_a_b @ R3 @ Q ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_roots
thf(fact_534_primeE,axiom,
! [G2: partia2670972154091845814t_unit,P: list_a] :
( ( prime_2011924034616061926t_unit @ G2 @ P )
=> ~ ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ G2 ) )
=> ~ ! [X5: list_a] :
( ( member_list_a @ X5 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( factor1757716651909850160t_unit @ G2 @ P @ ( mult_l7073676228092353617t_unit @ G2 @ X5 @ Xa ) )
=> ( ( factor1757716651909850160t_unit @ G2 @ P @ X5 )
| ( factor1757716651909850160t_unit @ G2 @ P @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_535_primeE,axiom,
! [G2: partia2175431115845679010xt_a_b,P: a] :
( ( prime_a_ring_ext_a_b @ G2 @ P )
=> ~ ( ~ ( member_a @ P @ ( units_a_ring_ext_a_b @ G2 ) )
=> ~ ! [X5: a] :
( ( member_a @ X5 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ P @ ( mult_a_ring_ext_a_b @ G2 @ X5 @ Xa ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ P @ X5 )
| ( factor8216151070175719842xt_a_b @ G2 @ P @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_536_primeE,axiom,
! [G2: partia2956882679547061052t_unit,P: list_list_a] :
( ( prime_1232919612140715622t_unit @ G2 @ P )
=> ~ ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ G2 ) )
=> ~ ! [X5: list_list_a] :
( ( member_list_list_a @ X5 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( factor6954119973539764400t_unit @ G2 @ P @ ( mult_l4853965630390486993t_unit @ G2 @ X5 @ Xa ) )
=> ( ( factor6954119973539764400t_unit @ G2 @ P @ X5 )
| ( factor6954119973539764400t_unit @ G2 @ P @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_537_Divisibility_OprimeI,axiom,
! [P: list_a,G2: partia2670972154091845814t_unit] :
( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ G2 ) )
=> ( ! [A4: list_a,B4: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( factor1757716651909850160t_unit @ G2 @ P @ ( mult_l7073676228092353617t_unit @ G2 @ A4 @ B4 ) )
=> ( ( factor1757716651909850160t_unit @ G2 @ P @ A4 )
| ( factor1757716651909850160t_unit @ G2 @ P @ B4 ) ) ) ) )
=> ( prime_2011924034616061926t_unit @ G2 @ P ) ) ) ).
% Divisibility.primeI
thf(fact_538_Divisibility_OprimeI,axiom,
! [P: a,G2: partia2175431115845679010xt_a_b] :
( ~ ( member_a @ P @ ( units_a_ring_ext_a_b @ G2 ) )
=> ( ! [A4: a,B4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ P @ ( mult_a_ring_ext_a_b @ G2 @ A4 @ B4 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ P @ A4 )
| ( factor8216151070175719842xt_a_b @ G2 @ P @ B4 ) ) ) ) )
=> ( prime_a_ring_ext_a_b @ G2 @ P ) ) ) ).
% Divisibility.primeI
thf(fact_539_Divisibility_OprimeI,axiom,
! [P: list_list_a,G2: partia2956882679547061052t_unit] :
( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ G2 ) )
=> ( ! [A4: list_list_a,B4: list_list_a] :
( ( member_list_list_a @ A4 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ B4 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( factor6954119973539764400t_unit @ G2 @ P @ ( mult_l4853965630390486993t_unit @ G2 @ A4 @ B4 ) )
=> ( ( factor6954119973539764400t_unit @ G2 @ P @ A4 )
| ( factor6954119973539764400t_unit @ G2 @ P @ B4 ) ) ) ) )
=> ( prime_1232919612140715622t_unit @ G2 @ P ) ) ) ).
% Divisibility.primeI
thf(fact_540_Divisibility_Oprime__def,axiom,
( prime_2011924034616061926t_unit
= ( ^ [G3: partia2670972154091845814t_unit,P4: list_a] :
( ~ ( member_list_a @ P4 @ ( units_2932844235741507942t_unit @ G3 ) )
& ! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G3 ) )
=> ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G3 ) )
=> ( ( factor1757716651909850160t_unit @ G3 @ P4 @ ( mult_l7073676228092353617t_unit @ G3 @ X @ Y3 ) )
=> ( ( factor1757716651909850160t_unit @ G3 @ P4 @ X )
| ( factor1757716651909850160t_unit @ G3 @ P4 @ Y3 ) ) ) ) ) ) ) ) ).
% Divisibility.prime_def
thf(fact_541_Divisibility_Oprime__def,axiom,
( prime_a_ring_ext_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,P4: a] :
( ~ ( member_a @ P4 @ ( units_a_ring_ext_a_b @ G3 ) )
& ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( ( factor8216151070175719842xt_a_b @ G3 @ P4 @ ( mult_a_ring_ext_a_b @ G3 @ X @ Y3 ) )
=> ( ( factor8216151070175719842xt_a_b @ G3 @ P4 @ X )
| ( factor8216151070175719842xt_a_b @ G3 @ P4 @ Y3 ) ) ) ) ) ) ) ) ).
% Divisibility.prime_def
thf(fact_542_Divisibility_Oprime__def,axiom,
( prime_1232919612140715622t_unit
= ( ^ [G3: partia2956882679547061052t_unit,P4: list_list_a] :
( ~ ( member_list_list_a @ P4 @ ( units_4903515905731149798t_unit @ G3 ) )
& ! [X: list_list_a] :
( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G3 ) )
=> ! [Y3: list_list_a] :
( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ G3 ) )
=> ( ( factor6954119973539764400t_unit @ G3 @ P4 @ ( mult_l4853965630390486993t_unit @ G3 @ X @ Y3 ) )
=> ( ( factor6954119973539764400t_unit @ G3 @ P4 @ X )
| ( factor6954119973539764400t_unit @ G3 @ P4 @ Y3 ) ) ) ) ) ) ) ) ).
% Divisibility.prime_def
thf(fact_543_domain_Ocgenideal__pirreducible,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ Q )
=> ( ( member_list_list_a @ Q @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P ) )
=> ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P @ Q ) ) ) ) ) ) ) ).
% domain.cgenideal_pirreducible
thf(fact_544_domain_Ocgenideal__pirreducible,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R3 @ K ) @ Q )
=> ( ( member_list_a @ Q @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P @ Q ) ) ) ) ) ) ) ).
% domain.cgenideal_pirreducible
thf(fact_545_domain_Osubring__degree__one__associatedI,axiom,
! [R3: partia7496981018696276118t_unit,K: set_set_list_a,A: set_list_a,A6: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( subrin5643252653130547402t_unit @ K @ R3 )
=> ( ( member_set_list_a @ A @ K )
=> ( ( member_set_list_a @ A6 @ K )
=> ( ( member_set_list_a @ B @ K )
=> ( ( ( mult_s7802724872828879953t_unit @ R3 @ A @ A6 )
= ( one_se1127990129394575805t_unit @ R3 ) )
=> ( associ8249012953061539097t_unit @ ( univ_p863672496597069550t_unit @ R3 @ K ) @ ( cons_set_list_a @ A @ ( cons_set_list_a @ B @ nil_set_list_a ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R3 ) @ ( cons_set_list_a @ ( mult_s7802724872828879953t_unit @ R3 @ A6 @ B ) @ nil_set_list_a ) ) ) ) ) ) ) ) ) ).
% domain.subring_degree_one_associatedI
thf(fact_546_domain_Osubring__degree__one__associatedI,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,A: list_a,A6: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_a @ A @ K )
=> ( ( member_list_a @ A6 @ K )
=> ( ( member_list_a @ B @ K )
=> ( ( ( mult_l7073676228092353617t_unit @ R3 @ A @ A6 )
= ( one_li8328186300101108157t_unit @ R3 ) )
=> ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ ( cons_list_a @ A @ ( cons_list_a @ B @ nil_list_a ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( cons_list_a @ ( mult_l7073676228092353617t_unit @ R3 @ A6 @ B ) @ nil_list_a ) ) ) ) ) ) ) ) ) ).
% domain.subring_degree_one_associatedI
thf(fact_547_domain_Osubring__degree__one__associatedI,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,A: a,A6: a,B: a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_a @ A @ K )
=> ( ( member_a @ A6 @ K )
=> ( ( member_a @ B @ K )
=> ( ( ( mult_a_ring_ext_a_b @ R3 @ A @ A6 )
= ( one_a_ring_ext_a_b @ R3 ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R3 @ K ) @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( cons_a @ ( mult_a_ring_ext_a_b @ R3 @ A6 @ B ) @ nil_a ) ) ) ) ) ) ) ) ) ).
% domain.subring_degree_one_associatedI
thf(fact_548_is__root__imp__pdivides,axiom,
! [P: list_a,X2: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X2 )
=> ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X2 ) @ nil_a ) ) @ P ) ) ) ).
% is_root_imp_pdivides
thf(fact_549_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_550_pdivides__imp__is__root,axiom,
! [P: list_a,X2: a] :
( ( P != nil_a )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X2 ) @ nil_a ) ) @ P )
=> ( polyno4133073214067823460ot_a_b @ r @ P @ X2 ) ) ) ) ).
% pdivides_imp_is_root
thf(fact_551_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_552_ring__hom__memI,axiom,
! [R3: partia2175431115845679010xt_a_b,H3: a > a,S: partia2175431115845679010xt_a_b] :
( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_a @ ( H3 @ X3 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X3: a,Y5: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R3 @ X3 @ Y5 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X3: a,Y5: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H3 @ ( add_a_b @ R3 @ X3 @ Y5 ) )
= ( add_a_b @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_a_ring_ext_a_b @ R3 ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_a_a @ H3 @ ( ring_hom_a_b_a_b @ R3 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_553_ring__hom__memI,axiom,
! [R3: partia2670972154091845814t_unit,H3: list_a > a,S: partia2175431115845679010xt_a_b] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_a @ ( H3 @ X3 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X3: list_a,Y5: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R3 @ X3 @ Y5 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X3: list_a,Y5: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R3 @ X3 @ Y5 ) )
= ( add_a_b @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_list_a_a @ H3 @ ( ring_h2895973938487309444it_a_b @ R3 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_554_ring__hom__memI,axiom,
! [R3: partia2175431115845679010xt_a_b,H3: a > list_a,S: partia2670972154091845814t_unit] :
( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_list_a @ ( H3 @ X3 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X3: a,Y5: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R3 @ X3 @ Y5 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X3: a,Y5: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H3 @ ( add_a_b @ R3 @ X3 @ Y5 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_a_ring_ext_a_b @ R3 ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( member_a_list_a @ H3 @ ( ring_h405018892823518980t_unit @ R3 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_555_ring__hom__memI,axiom,
! [R3: partia7496981018696276118t_unit,H3: set_list_a > a,S: partia2175431115845679010xt_a_b] :
( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( member_a @ ( H3 @ X3 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X3: set_list_a,Y5: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ Y5 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( H3 @ ( mult_s7802724872828879953t_unit @ R3 @ X3 @ Y5 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X3: set_list_a,Y5: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ Y5 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( H3 @ ( add_se2486902527185523630t_unit @ R3 @ X3 @ Y5 ) )
= ( add_a_b @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_se1127990129394575805t_unit @ R3 ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_set_list_a_a @ H3 @ ( ring_h8906680420194085028it_a_b @ R3 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_556_ring__hom__memI,axiom,
! [R3: partia2670972154091845814t_unit,H3: list_a > list_a,S: partia2670972154091845814t_unit] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_list_a @ ( H3 @ X3 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X3: list_a,Y5: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R3 @ X3 @ Y5 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X3: list_a,Y5: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R3 @ X3 @ Y5 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( member_list_a_list_a @ H3 @ ( ring_h7399960747407462284t_unit @ R3 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_557_ring__hom__memI,axiom,
! [R3: partia2175431115845679010xt_a_b,H3: a > set_list_a,S: partia7496981018696276118t_unit] :
( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_set_list_a @ ( H3 @ X3 ) @ ( partia141011252114345353t_unit @ S ) ) )
=> ( ! [X3: a,Y5: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R3 @ X3 @ Y5 ) )
= ( mult_s7802724872828879953t_unit @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X3: a,Y5: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H3 @ ( add_a_b @ R3 @ X3 @ Y5 ) )
= ( add_se2486902527185523630t_unit @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_a_ring_ext_a_b @ R3 ) )
= ( one_se1127990129394575805t_unit @ S ) )
=> ( member_a_set_list_a @ H3 @ ( ring_h6109298854714515236t_unit @ R3 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_558_ring__hom__memI,axiom,
! [R3: partia2175431115845679010xt_a_b,H3: a > list_list_a,S: partia2956882679547061052t_unit] :
( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_list_list_a @ ( H3 @ X3 ) @ ( partia2464479390973590831t_unit @ S ) ) )
=> ( ! [X3: a,Y5: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R3 @ X3 @ Y5 ) )
= ( mult_l4853965630390486993t_unit @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X3: a,Y5: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H3 @ ( add_a_b @ R3 @ X3 @ Y5 ) )
= ( add_li174743652000525320t_unit @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_a_ring_ext_a_b @ R3 ) )
= ( one_li8234411390022467901t_unit @ S ) )
=> ( member_a_list_list_a @ H3 @ ( ring_h6858658657455840382t_unit @ R3 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_559_ring__hom__memI,axiom,
! [R3: partia2956882679547061052t_unit,H3: list_list_a > a,S: partia2175431115845679010xt_a_b] :
( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( member_a @ ( H3 @ X3 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X3: list_list_a,Y5: list_list_a] :
( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( H3 @ ( mult_l4853965630390486993t_unit @ R3 @ X3 @ Y5 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X3: list_list_a,Y5: list_list_a] :
( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( H3 @ ( add_li174743652000525320t_unit @ R3 @ X3 @ Y5 ) )
= ( add_a_b @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_li8234411390022467901t_unit @ R3 ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_list_list_a_a @ H3 @ ( ring_h8078271382950527358it_a_b @ R3 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_560_ring__hom__memI,axiom,
! [R3: partia7496981018696276118t_unit,H3: set_list_a > list_a,S: partia2670972154091845814t_unit] :
( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( member_list_a @ ( H3 @ X3 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X3: set_list_a,Y5: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ Y5 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( H3 @ ( mult_s7802724872828879953t_unit @ R3 @ X3 @ Y5 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X3: set_list_a,Y5: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ Y5 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( H3 @ ( add_se2486902527185523630t_unit @ R3 @ X3 @ Y5 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_se1127990129394575805t_unit @ R3 ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( member5910328476188217884list_a @ H3 @ ( ring_h8038483918290310060t_unit @ R3 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_561_ring__hom__memI,axiom,
! [R3: partia2670972154091845814t_unit,H3: list_a > set_list_a,S: partia7496981018696276118t_unit] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_set_list_a @ ( H3 @ X3 ) @ ( partia141011252114345353t_unit @ S ) ) )
=> ( ! [X3: list_a,Y5: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R3 @ X3 @ Y5 ) )
= ( mult_s7802724872828879953t_unit @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X3: list_a,Y5: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R3 @ X3 @ Y5 ) )
= ( add_se2486902527185523630t_unit @ S @ ( H3 @ X3 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_se1127990129394575805t_unit @ S ) )
=> ( member4263473470251683292list_a @ H3 @ ( ring_h6188449271506562988t_unit @ R3 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_562_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_563_roots__mem__iff__is__root,axiom,
! [P: list_a,X2: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a @ X2 @ ( set_mset_a @ ( polynomial_roots_a_b @ r @ P ) ) )
= ( polyno4133073214067823460ot_a_b @ r @ P @ X2 ) ) ) ).
% roots_mem_iff_is_root
thf(fact_564_subring__props_I2_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K ) ) ).
% subring_props(2)
thf(fact_565_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_566_zero__divides,axiom,
! [A: a] :
( ( factor8216151070175719842xt_a_b @ r @ ( zero_a_b @ r ) @ A )
= ( A
= ( zero_a_b @ r ) ) ) ).
% zero_divides
thf(fact_567_subring__props_I5_J,axiom,
! [K: set_a,H3: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H3 @ K )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ K ) ) ) ).
% subring_props(5)
thf(fact_568_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_569_local_Ominus__unique,axiom,
! [Y: a,X2: a,Y6: a] :
( ( ( add_a_b @ r @ Y @ X2 )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X2 @ Y6 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y6 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_570_add_Or__inv__ex,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X2 @ X3 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_571_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X3 )
= X3 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_572_add_Ol__inv__ex,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X3 @ X2 )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_573_add_Oinv__comm,axiom,
! [X2: a,Y: a] :
( ( ( add_a_b @ r @ X2 @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_574_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_575_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_576_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_577_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_578_r__neg2,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X2 @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_579_r__neg1,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ ( add_a_b @ r @ X2 @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_580_local_Ominus__add,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X2 @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_581_a__transpose__inv,axiom,
! [X2: a,Y: a,Z: a] :
( ( ( add_a_b @ r @ X2 @ Y )
= Z )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ Z )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_582_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_583_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_584_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_585_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_586_add_Oinv__mult__group,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X2 @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X2 ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_587_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_588_r__minus,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( a_inv_a_b @ r @ Y ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) ) ) ) ) ).
% r_minus
thf(fact_589_l__minus,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ Y )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) ) ) ) ) ).
% l_minus
thf(fact_590_div__neg,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( a_inv_a_b @ r @ B ) ) ) ) ) ).
% div_neg
thf(fact_591_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_592_ring__irreducibleE_I1_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( R2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_593_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_594_sum__zero__eq__neg,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X2 @ Y )
= ( zero_a_b @ r ) )
=> ( X2
= ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% sum_zero_eq_neg
thf(fact_595_r__neg,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X2 @ ( a_inv_a_b @ r @ X2 ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_596_minus__equality,axiom,
! [Y: a,X2: a] :
( ( ( add_a_b @ r @ Y @ X2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X2 )
= Y ) ) ) ) ).
% minus_equality
thf(fact_597_l__neg,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ X2 )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_598_square__eq__one,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X2 @ X2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( X2
= ( one_a_ring_ext_a_b @ r ) )
| ( X2
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% square_eq_one
thf(fact_599_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_600_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_601_add_Oone__in__subset,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H != bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ H )
=> ( member_a @ ( a_inv_a_b @ r @ X3 ) @ H ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ H )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ H )
=> ( member_a @ ( add_a_b @ r @ X3 @ Xa2 ) @ H ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H ) ) ) ) ) ).
% add.one_in_subset
thf(fact_602_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_603_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_604_local_Ominus__minus,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X2 ) )
= X2 ) ) ).
% local.minus_minus
thf(fact_605_a__inv__closed,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_606_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_607_r__zero,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X2 @ ( zero_a_b @ r ) )
= X2 ) ) ).
% r_zero
thf(fact_608_l__zero,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X2 )
= X2 ) ) ).
% l_zero
thf(fact_609_add_Or__cancel__one_H,axiom,
! [X2: a,A: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X2
= ( add_a_b @ r @ A @ X2 ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_610_add_Or__cancel__one,axiom,
! [X2: a,A: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X2 )
= X2 )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_611_add_Ol__cancel__one_H,axiom,
! [X2: a,A: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X2
= ( add_a_b @ r @ X2 @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_612_add_Ol__cancel__one,axiom,
! [X2: a,A: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X2 @ A )
= X2 )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_613_r__null,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_614_l__null,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X2 )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_615_add_Oinv__eq__1__iff,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X2 )
= ( zero_a_b @ r ) )
= ( X2
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_616_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_617_a__minus__def,axiom,
( a_minu3984020753470702548t_unit
= ( ^ [R4: partia2670972154091845814t_unit,X: list_a,Y3: list_a] : ( add_li7652885771158616974t_unit @ R4 @ X @ ( a_inv_8944721093294617173t_unit @ R4 @ Y3 ) ) ) ) ).
% a_minus_def
thf(fact_618_a__minus__def,axiom,
( a_minus_a_b
= ( ^ [R4: partia2175431115845679010xt_a_b,X: a,Y3: a] : ( add_a_b @ R4 @ X @ ( a_inv_a_b @ R4 @ Y3 ) ) ) ) ).
% a_minus_def
thf(fact_619_domain_Oone__not__zero,axiom,
! [R3: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R3 )
=> ( ( one_a_ring_ext_a_b @ R3 )
!= ( zero_a_b @ R3 ) ) ) ).
% domain.one_not_zero
thf(fact_620_domain_Oone__not__zero,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( one_li8328186300101108157t_unit @ R3 )
!= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ).
% domain.one_not_zero
thf(fact_621_domain_Oone__not__zero,axiom,
! [R3: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( one_se1127990129394575805t_unit @ R3 )
!= ( zero_s2910681146719230829t_unit @ R3 ) ) ) ).
% domain.one_not_zero
thf(fact_622_domain_Ozero__not__one,axiom,
! [R3: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R3 )
=> ( ( zero_a_b @ R3 )
!= ( one_a_ring_ext_a_b @ R3 ) ) ) ).
% domain.zero_not_one
thf(fact_623_domain_Ozero__not__one,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( zero_l4142658623432671053t_unit @ R3 )
!= ( one_li8328186300101108157t_unit @ R3 ) ) ) ).
% domain.zero_not_one
thf(fact_624_domain_Ozero__not__one,axiom,
! [R3: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( zero_s2910681146719230829t_unit @ R3 )
!= ( one_se1127990129394575805t_unit @ R3 ) ) ) ).
% domain.zero_not_one
thf(fact_625_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R3 ) @ ( partia141011252114345353t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_626_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R3 ) @ ( partia5361259788508890537t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_627_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R3 )
=> ( member_a @ ( zero_a_b @ R3 ) @ ( partia707051561876973205xt_a_b @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_628_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R3 ) @ ( partia2464479390973590831t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_629_domain_Ointegral__iff,axiom,
! [R3: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R3 @ A @ B )
= ( zero_s2910681146719230829t_unit @ R3 ) )
= ( ( A
= ( zero_s2910681146719230829t_unit @ R3 ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R3 ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_630_domain_Ointegral__iff,axiom,
! [R3: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R3 @ A @ B )
= ( zero_l4142658623432671053t_unit @ R3 ) )
= ( ( A
= ( zero_l4142658623432671053t_unit @ R3 ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_631_domain_Ointegral__iff,axiom,
! [R3: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( ( mult_a_ring_ext_a_b @ R3 @ A @ B )
= ( zero_a_b @ R3 ) )
= ( ( A
= ( zero_a_b @ R3 ) )
| ( B
= ( zero_a_b @ R3 ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_632_domain_Ointegral__iff,axiom,
! [R3: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R3 @ A @ B )
= ( zero_l347298301471573063t_unit @ R3 ) )
= ( ( A
= ( zero_l347298301471573063t_unit @ R3 ) )
| ( B
= ( zero_l347298301471573063t_unit @ R3 ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_633_domain_Om__rcancel,axiom,
! [R3: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a,C: set_list_a] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( A
!= ( zero_s2910681146719230829t_unit @ R3 ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R3 @ B @ A )
= ( mult_s7802724872828879953t_unit @ R3 @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_634_domain_Om__rcancel,axiom,
! [R3: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ R3 ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R3 @ B @ A )
= ( mult_l7073676228092353617t_unit @ R3 @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_635_domain_Om__rcancel,axiom,
! [R3: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R3 )
=> ( ( A
!= ( zero_a_b @ R3 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( ( mult_a_ring_ext_a_b @ R3 @ B @ A )
= ( mult_a_ring_ext_a_b @ R3 @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_636_domain_Om__rcancel,axiom,
! [R3: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,C: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( A
!= ( zero_l347298301471573063t_unit @ R3 ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R3 @ B @ A )
= ( mult_l4853965630390486993t_unit @ R3 @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_637_domain_Om__lcancel,axiom,
! [R3: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a,C: set_list_a] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( A
!= ( zero_s2910681146719230829t_unit @ R3 ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R3 @ A @ B )
= ( mult_s7802724872828879953t_unit @ R3 @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_638_domain_Om__lcancel,axiom,
! [R3: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ R3 ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R3 @ A @ B )
= ( mult_l7073676228092353617t_unit @ R3 @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_639_domain_Om__lcancel,axiom,
! [R3: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R3 )
=> ( ( A
!= ( zero_a_b @ R3 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( ( mult_a_ring_ext_a_b @ R3 @ A @ B )
= ( mult_a_ring_ext_a_b @ R3 @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_640_domain_Om__lcancel,axiom,
! [R3: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,C: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( A
!= ( zero_l347298301471573063t_unit @ R3 ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R3 @ A @ B )
= ( mult_l4853965630390486993t_unit @ R3 @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_641_domain_Ointegral,axiom,
! [R3: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( ( mult_s7802724872828879953t_unit @ R3 @ A @ B )
= ( zero_s2910681146719230829t_unit @ R3 ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( A
= ( zero_s2910681146719230829t_unit @ R3 ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R3 ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_642_domain_Ointegral,axiom,
! [R3: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( ( mult_l7073676228092353617t_unit @ R3 @ A @ B )
= ( zero_l4142658623432671053t_unit @ R3 ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( A
= ( zero_l4142658623432671053t_unit @ R3 ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_643_domain_Ointegral,axiom,
! [R3: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R3 )
=> ( ( ( mult_a_ring_ext_a_b @ R3 @ A @ B )
= ( zero_a_b @ R3 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( A
= ( zero_a_b @ R3 ) )
| ( B
= ( zero_a_b @ R3 ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_644_domain_Ointegral,axiom,
! [R3: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( ( mult_l4853965630390486993t_unit @ R3 @ A @ B )
= ( zero_l347298301471573063t_unit @ R3 ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( A
= ( zero_l347298301471573063t_unit @ R3 ) )
| ( B
= ( zero_l347298301471573063t_unit @ R3 ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_645_semiring_Osemiring__simprules_I11_J,axiom,
! [R3: partia7496981018696276118t_unit,X2: set_list_a] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( add_se2486902527185523630t_unit @ R3 @ X2 @ ( zero_s2910681146719230829t_unit @ R3 ) )
= X2 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_646_semiring_Osemiring__simprules_I11_J,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( add_li7652885771158616974t_unit @ R3 @ X2 @ ( zero_l4142658623432671053t_unit @ R3 ) )
= X2 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_647_semiring_Osemiring__simprules_I11_J,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( add_a_b @ R3 @ X2 @ ( zero_a_b @ R3 ) )
= X2 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_648_semiring_Osemiring__simprules_I11_J,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( add_li174743652000525320t_unit @ R3 @ X2 @ ( zero_l347298301471573063t_unit @ R3 ) )
= X2 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_649_semiring_Osemiring__simprules_I6_J,axiom,
! [R3: partia7496981018696276118t_unit,X2: set_list_a] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( add_se2486902527185523630t_unit @ R3 @ ( zero_s2910681146719230829t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_650_semiring_Osemiring__simprules_I6_J,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( add_li7652885771158616974t_unit @ R3 @ ( zero_l4142658623432671053t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_651_semiring_Osemiring__simprules_I6_J,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( add_a_b @ R3 @ ( zero_a_b @ R3 ) @ X2 )
= X2 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_652_semiring_Osemiring__simprules_I6_J,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( add_li174743652000525320t_unit @ R3 @ ( zero_l347298301471573063t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_653_semiring_Or__null,axiom,
! [R3: partia7496981018696276118t_unit,X2: set_list_a] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( mult_s7802724872828879953t_unit @ R3 @ X2 @ ( zero_s2910681146719230829t_unit @ R3 ) )
= ( zero_s2910681146719230829t_unit @ R3 ) ) ) ) ).
% semiring.r_null
thf(fact_654_semiring_Or__null,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( mult_l7073676228092353617t_unit @ R3 @ X2 @ ( zero_l4142658623432671053t_unit @ R3 ) )
= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ) ).
% semiring.r_null
thf(fact_655_semiring_Or__null,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ X2 @ ( zero_a_b @ R3 ) )
= ( zero_a_b @ R3 ) ) ) ) ).
% semiring.r_null
thf(fact_656_semiring_Or__null,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( mult_l4853965630390486993t_unit @ R3 @ X2 @ ( zero_l347298301471573063t_unit @ R3 ) )
= ( zero_l347298301471573063t_unit @ R3 ) ) ) ) ).
% semiring.r_null
thf(fact_657_semiring_Ol__null,axiom,
! [R3: partia7496981018696276118t_unit,X2: set_list_a] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( mult_s7802724872828879953t_unit @ R3 @ ( zero_s2910681146719230829t_unit @ R3 ) @ X2 )
= ( zero_s2910681146719230829t_unit @ R3 ) ) ) ) ).
% semiring.l_null
thf(fact_658_semiring_Ol__null,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( mult_l7073676228092353617t_unit @ R3 @ ( zero_l4142658623432671053t_unit @ R3 ) @ X2 )
= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ) ).
% semiring.l_null
thf(fact_659_semiring_Ol__null,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ ( zero_a_b @ R3 ) @ X2 )
= ( zero_a_b @ R3 ) ) ) ) ).
% semiring.l_null
thf(fact_660_semiring_Ol__null,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( mult_l4853965630390486993t_unit @ R3 @ ( zero_l347298301471573063t_unit @ R3 ) @ X2 )
= ( zero_l347298301471573063t_unit @ R3 ) ) ) ) ).
% semiring.l_null
thf(fact_661_domain_Oconst__term__simprules__shell_I4_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( const_term_a_b @ R3 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P ) )
= ( a_inv_a_b @ R3 @ ( const_term_a_b @ R3 @ P ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(4)
thf(fact_662_domain_Oconst__term__simprules__shell_I4_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ R3 @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P ) )
= ( a_inv_8944721093294617173t_unit @ R3 @ ( const_6738166269504826821t_unit @ R3 @ P ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(4)
thf(fact_663_domain_Ozero__is__prime_I1_J,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( prime_2011924034616061926t_unit @ R3 @ ( zero_l4142658623432671053t_unit @ R3 ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_664_domain_Ozero__is__prime_I1_J,axiom,
! [R3: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( prime_5738381090551951334t_unit @ R3 @ ( zero_s2910681146719230829t_unit @ R3 ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_665_domain_Ozero__is__prime_I1_J,axiom,
! [R3: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R3 )
=> ( prime_a_ring_ext_a_b @ R3 @ ( zero_a_b @ R3 ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_666_domain_Osquare__eq__one,axiom,
! [R3: partia7496981018696276118t_unit,X2: set_list_a] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R3 @ X2 @ X2 )
= ( one_se1127990129394575805t_unit @ R3 ) )
=> ( ( X2
= ( one_se1127990129394575805t_unit @ R3 ) )
| ( X2
= ( a_inv_5715216516650856053t_unit @ R3 @ ( one_se1127990129394575805t_unit @ R3 ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_667_domain_Osquare__eq__one,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R3 @ X2 @ X2 )
= ( one_li8328186300101108157t_unit @ R3 ) )
=> ( ( X2
= ( one_li8328186300101108157t_unit @ R3 ) )
| ( X2
= ( a_inv_8944721093294617173t_unit @ R3 @ ( one_li8328186300101108157t_unit @ R3 ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_668_domain_Osquare__eq__one,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( ( mult_a_ring_ext_a_b @ R3 @ X2 @ X2 )
= ( one_a_ring_ext_a_b @ R3 ) )
=> ( ( X2
= ( one_a_ring_ext_a_b @ R3 ) )
| ( X2
= ( a_inv_a_b @ R3 @ ( one_a_ring_ext_a_b @ R3 ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_669_domain_Osquare__eq__one,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R3 @ X2 @ X2 )
= ( one_li8234411390022467901t_unit @ R3 ) )
=> ( ( X2
= ( one_li8234411390022467901t_unit @ R3 ) )
| ( X2
= ( a_inv_7033018035630854991t_unit @ R3 @ ( one_li8234411390022467901t_unit @ R3 ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_670_ring__prime__def,axiom,
( ring_r1091214237498979717t_unit
= ( ^ [R4: partia7496981018696276118t_unit,A3: set_list_a] :
( ( A3
!= ( zero_s2910681146719230829t_unit @ R4 ) )
& ( prime_5738381090551951334t_unit @ R4 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_671_ring__prime__def,axiom,
( ring_r6430282645014804837t_unit
= ( ^ [R4: partia2670972154091845814t_unit,A3: list_a] :
( ( A3
!= ( zero_l4142658623432671053t_unit @ R4 ) )
& ( prime_2011924034616061926t_unit @ R4 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_672_ring__prime__def,axiom,
( ring_ring_prime_a_b
= ( ^ [R4: partia2175431115845679010xt_a_b,A3: a] :
( ( A3
!= ( zero_a_b @ R4 ) )
& ( prime_a_ring_ext_a_b @ R4 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_673_domain_Oring__irreducibleE_I1_J,axiom,
! [R3: partia7496981018696276118t_unit,R2: set_list_a] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( member_set_list_a @ R2 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( ring_r5115406448772830318t_unit @ R3 @ R2 )
=> ( R2
!= ( zero_s2910681146719230829t_unit @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_674_domain_Oring__irreducibleE_I1_J,axiom,
! [R3: partia2956882679547061052t_unit,R2: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( ring_r360171070648044744t_unit @ R3 @ R2 )
=> ( R2
!= ( zero_l347298301471573063t_unit @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_675_domain_Oring__irreducibleE_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,R2: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( ring_r932985474545269838t_unit @ R3 @ R2 )
=> ( R2
!= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_676_domain_Oring__irreducibleE_I1_J,axiom,
! [R3: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( ring_r999134135267193926le_a_b @ R3 @ R2 )
=> ( R2
!= ( zero_a_b @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_677_domain_Oring__primeE_I1_J,axiom,
! [R3: partia7496981018696276118t_unit,P: set_list_a] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( member_set_list_a @ P @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( ring_r1091214237498979717t_unit @ R3 @ P )
=> ( P
!= ( zero_s2910681146719230829t_unit @ R3 ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_678_domain_Oring__primeE_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( ring_r6430282645014804837t_unit @ R3 @ P )
=> ( P
!= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_679_domain_Oring__primeE_I1_J,axiom,
! [R3: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( ring_ring_prime_a_b @ R3 @ P )
=> ( P
!= ( zero_a_b @ R3 ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_680_domain_Oring__primeE_I1_J,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( ring_r5437400583859147359t_unit @ R3 @ P )
=> ( P
!= ( zero_l347298301471573063t_unit @ R3 ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_681_domain_Oroots__mem__iff__is__root,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a,X2: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( member_list_list_a @ X2 @ ( set_mset_list_list_a @ ( polyno3707469075594375645t_unit @ R3 @ P ) ) )
= ( polyno5142720416380192742t_unit @ R3 @ P @ X2 ) ) ) ) ).
% domain.roots_mem_iff_is_root
thf(fact_682_domain_Oroots__mem__iff__is__root,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a,X2: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( member_list_a @ X2 @ ( set_mset_list_a @ ( polyno7858422826990252003t_unit @ R3 @ P ) ) )
= ( polyno6951661231331188332t_unit @ R3 @ P @ X2 ) ) ) ) ).
% domain.roots_mem_iff_is_root
thf(fact_683_domain_Oroots__mem__iff__is__root,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a,X2: a] :
( ( domain_a_b @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( member_a @ X2 @ ( set_mset_a @ ( polynomial_roots_a_b @ R3 @ P ) ) )
= ( polyno4133073214067823460ot_a_b @ R3 @ P @ X2 ) ) ) ) ).
% domain.roots_mem_iff_is_root
thf(fact_684_domain_Opdivides__imp__is__root,axiom,
! [R3: partia7496981018696276118t_unit,P: list_set_list_a,X2: set_list_a] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( P != nil_set_list_a )
=> ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( polyno9075941895896075626t_unit @ R3 @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R3 ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R3 @ X2 ) @ nil_set_list_a ) ) @ P )
=> ( polyno4320237611291262604t_unit @ R3 @ P @ X2 ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_685_domain_Opdivides__imp__is__root,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a,X2: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( P != nil_list_a )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( polyno8016796738000020810t_unit @ R3 @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R3 @ X2 ) @ nil_list_a ) ) @ P )
=> ( polyno6951661231331188332t_unit @ R3 @ P @ X2 ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_686_domain_Opdivides__imp__is__root,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a,X2: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( P != nil_list_list_a )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( polyno4453881341673752516t_unit @ R3 @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R3 ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R3 @ X2 ) @ nil_list_list_a ) ) @ P )
=> ( polyno5142720416380192742t_unit @ R3 @ P @ X2 ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_687_domain_Opdivides__imp__is__root,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a,X2: a] :
( ( domain_a_b @ R3 )
=> ( ( P != nil_a )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( polyno5814909790663948098es_a_b @ R3 @ ( cons_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( cons_a @ ( a_inv_a_b @ R3 @ X2 ) @ nil_a ) ) @ P )
=> ( polyno4133073214067823460ot_a_b @ R3 @ P @ X2 ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_688_ring__hom__closed,axiom,
! [H3: list_a > list_a,R3: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X2: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_h7399960747407462284t_unit @ R3 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_list_a @ ( H3 @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_689_ring__hom__closed,axiom,
! [H3: list_a > a,R3: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X2: list_a] :
( ( member_list_a_a @ H3 @ ( ring_h2895973938487309444it_a_b @ R3 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_a @ ( H3 @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_690_ring__hom__closed,axiom,
! [H3: list_a > list_list_a,R3: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,X2: list_a] :
( ( member6714375691612171394list_a @ H3 @ ( ring_h8002040739877300486t_unit @ R3 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_list_list_a @ ( H3 @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_691_ring__hom__closed,axiom,
! [H3: a > list_a,R3: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X2: a] :
( ( member_a_list_a @ H3 @ ( ring_h405018892823518980t_unit @ R3 @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_list_a @ ( H3 @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_692_ring__hom__closed,axiom,
! [H3: a > a,R3: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X2: a] :
( ( member_a_a @ H3 @ ( ring_hom_a_b_a_b @ R3 @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_a @ ( H3 @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_693_ring__hom__closed,axiom,
! [H3: a > list_list_a,R3: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,X2: a] :
( ( member_a_list_list_a @ H3 @ ( ring_h6858658657455840382t_unit @ R3 @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_list_list_a @ ( H3 @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_694_ring__hom__closed,axiom,
! [H3: list_list_a > list_a,R3: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X2: list_list_a] :
( ( member7168557129179038582list_a @ H3 @ ( ring_h5031276006722532742t_unit @ R3 @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( member_list_a @ ( H3 @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_695_ring__hom__closed,axiom,
! [H3: list_list_a > a,R3: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X2: list_list_a] :
( ( member_list_list_a_a @ H3 @ ( ring_h8078271382950527358it_a_b @ R3 @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( member_a @ ( H3 @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_696_ring__hom__closed,axiom,
! [H3: list_list_a > list_list_a,R3: partia2956882679547061052t_unit,S: partia2956882679547061052t_unit,X2: list_list_a] :
( ( member8231385768148312316list_a @ H3 @ ( ring_h8129544334414776832t_unit @ R3 @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( member_list_list_a @ ( H3 @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_697_domain_Opdivides__imp__roots__incl,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( Q != nil_list_list_a )
=> ( ( polyno4453881341673752516t_unit @ R3 @ P @ Q )
=> ( subset8447756916971205105list_a @ ( polyno3707469075594375645t_unit @ R3 @ P ) @ ( polyno3707469075594375645t_unit @ R3 @ Q ) ) ) ) ) ) ) ).
% domain.pdivides_imp_roots_incl
thf(fact_698_domain_Opdivides__imp__roots__incl,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ R3 @ P @ Q )
=> ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ R3 @ P ) @ ( polyno7858422826990252003t_unit @ R3 @ Q ) ) ) ) ) ) ) ).
% domain.pdivides_imp_roots_incl
thf(fact_699_domain_Opdivides__imp__roots__incl,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ R3 @ P @ Q )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ R3 @ P ) @ ( polynomial_roots_a_b @ R3 @ Q ) ) ) ) ) ) ) ).
% domain.pdivides_imp_roots_incl
thf(fact_700_ring__hom__one,axiom,
! [H3: a > a,R3: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H3 @ ( ring_hom_a_b_a_b @ R3 @ S ) )
=> ( ( H3 @ ( one_a_ring_ext_a_b @ R3 ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_hom_one
thf(fact_701_ring__hom__one,axiom,
! [H3: a > list_a,R3: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H3 @ ( ring_h405018892823518980t_unit @ R3 @ S ) )
=> ( ( H3 @ ( one_a_ring_ext_a_b @ R3 ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_702_ring__hom__one,axiom,
! [H3: a > set_list_a,R3: partia2175431115845679010xt_a_b,S: partia7496981018696276118t_unit] :
( ( member_a_set_list_a @ H3 @ ( ring_h6109298854714515236t_unit @ R3 @ S ) )
=> ( ( H3 @ ( one_a_ring_ext_a_b @ R3 ) )
= ( one_se1127990129394575805t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_703_ring__hom__one,axiom,
! [H3: list_a > a,R3: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_list_a_a @ H3 @ ( ring_h2895973938487309444it_a_b @ R3 @ S ) )
=> ( ( H3 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_hom_one
thf(fact_704_ring__hom__one,axiom,
! [H3: list_a > list_a,R3: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit] :
( ( member_list_a_list_a @ H3 @ ( ring_h7399960747407462284t_unit @ R3 @ S ) )
=> ( ( H3 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_705_ring__hom__one,axiom,
! [H3: list_a > set_list_a,R3: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit] :
( ( member4263473470251683292list_a @ H3 @ ( ring_h6188449271506562988t_unit @ R3 @ S ) )
=> ( ( H3 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_se1127990129394575805t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_706_ring__hom__one,axiom,
! [H3: set_list_a > a,R3: partia7496981018696276118t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_set_list_a_a @ H3 @ ( ring_h8906680420194085028it_a_b @ R3 @ S ) )
=> ( ( H3 @ ( one_se1127990129394575805t_unit @ R3 ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_hom_one
thf(fact_707_ring__hom__one,axiom,
! [H3: set_list_a > list_a,R3: partia7496981018696276118t_unit,S: partia2670972154091845814t_unit] :
( ( member5910328476188217884list_a @ H3 @ ( ring_h8038483918290310060t_unit @ R3 @ S ) )
=> ( ( H3 @ ( one_se1127990129394575805t_unit @ R3 ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_708_ring__hom__one,axiom,
! [H3: set_list_a > set_list_a,R3: partia7496981018696276118t_unit,S: partia7496981018696276118t_unit] :
( ( member5068272912271824380list_a @ H3 @ ( ring_h6076331213207892940t_unit @ R3 @ S ) )
=> ( ( H3 @ ( one_se1127990129394575805t_unit @ R3 ) )
= ( one_se1127990129394575805t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_709_domain_Ois__root__imp__pdivides,axiom,
! [R3: partia7496981018696276118t_unit,P: list_set_list_a,X2: set_list_a] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) ) )
=> ( ( polyno4320237611291262604t_unit @ R3 @ P @ X2 )
=> ( polyno9075941895896075626t_unit @ R3 @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R3 ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R3 @ X2 ) @ nil_set_list_a ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_710_domain_Ois__root__imp__pdivides,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a,X2: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( polyno5142720416380192742t_unit @ R3 @ P @ X2 )
=> ( polyno4453881341673752516t_unit @ R3 @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R3 ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R3 @ X2 ) @ nil_list_list_a ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_711_domain_Ois__root__imp__pdivides,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a,X2: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ R3 @ P @ X2 )
=> ( polyno8016796738000020810t_unit @ R3 @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R3 @ X2 ) @ nil_list_a ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_712_domain_Ois__root__imp__pdivides,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a,X2: a] :
( ( domain_a_b @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ R3 @ P @ X2 )
=> ( polyno5814909790663948098es_a_b @ R3 @ ( cons_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( cons_a @ ( a_inv_a_b @ R3 @ X2 ) @ nil_a ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_713_ring__hom__add,axiom,
! [H3: list_a > list_a,R3: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_h7399960747407462284t_unit @ R3 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R3 @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_714_ring__hom__add,axiom,
! [H3: list_a > a,R3: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X2: list_a,Y: list_a] :
( ( member_list_a_a @ H3 @ ( ring_h2895973938487309444it_a_b @ R3 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R3 @ X2 @ Y ) )
= ( add_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_715_ring__hom__add,axiom,
! [H3: a > a,R3: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( member_a_a @ H3 @ ( ring_hom_a_b_a_b @ R3 @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H3 @ ( add_a_b @ R3 @ X2 @ Y ) )
= ( add_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_716_ring__hom__add,axiom,
! [H3: a > list_a,R3: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X2: a,Y: a] :
( ( member_a_list_a @ H3 @ ( ring_h405018892823518980t_unit @ R3 @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H3 @ ( add_a_b @ R3 @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_717_ring__hom__add,axiom,
! [H3: list_list_a > list_a,R3: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X2: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H3 @ ( ring_h5031276006722532742t_unit @ R3 @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( H3 @ ( add_li174743652000525320t_unit @ R3 @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_718_ring__hom__add,axiom,
! [H3: list_list_a > a,R3: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X2: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H3 @ ( ring_h8078271382950527358it_a_b @ R3 @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( H3 @ ( add_li174743652000525320t_unit @ R3 @ X2 @ Y ) )
= ( add_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_719_ring__hom__mult,axiom,
! [H3: list_a > list_a,R3: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_h7399960747407462284t_unit @ R3 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R3 @ X2 @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_720_ring__hom__mult,axiom,
! [H3: list_a > a,R3: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X2: list_a,Y: list_a] :
( ( member_list_a_a @ H3 @ ( ring_h2895973938487309444it_a_b @ R3 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R3 @ X2 @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_721_ring__hom__mult,axiom,
! [H3: a > list_a,R3: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X2: a,Y: a] :
( ( member_a_list_a @ H3 @ ( ring_h405018892823518980t_unit @ R3 @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R3 @ X2 @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_722_ring__hom__mult,axiom,
! [H3: a > a,R3: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( member_a_a @ H3 @ ( ring_hom_a_b_a_b @ R3 @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R3 @ X2 @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_723_ring__hom__mult,axiom,
! [H3: list_list_a > list_a,R3: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X2: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H3 @ ( ring_h5031276006722532742t_unit @ R3 @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( H3 @ ( mult_l4853965630390486993t_unit @ R3 @ X2 @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_724_ring__hom__mult,axiom,
! [H3: list_list_a > a,R3: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X2: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H3 @ ( ring_h8078271382950527358it_a_b @ R3 @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( H3 @ ( mult_l4853965630390486993t_unit @ R3 @ X2 @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_725_semiring_Osemiring__simprules_I12_J,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( add_li7652885771158616974t_unit @ R3 @ X2 @ ( add_li7652885771158616974t_unit @ R3 @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ R3 @ Y @ ( add_li7652885771158616974t_unit @ R3 @ X2 @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_726_semiring_Osemiring__simprules_I12_J,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a,Y: a,Z: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( add_a_b @ R3 @ X2 @ ( add_a_b @ R3 @ Y @ Z ) )
= ( add_a_b @ R3 @ Y @ ( add_a_b @ R3 @ X2 @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_727_semiring_Osemiring__simprules_I12_J,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( add_li174743652000525320t_unit @ R3 @ X2 @ ( add_li174743652000525320t_unit @ R3 @ Y @ Z ) )
= ( add_li174743652000525320t_unit @ R3 @ Y @ ( add_li174743652000525320t_unit @ R3 @ X2 @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_728_semiring_Osemiring__simprules_I7_J,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( add_li7652885771158616974t_unit @ R3 @ X2 @ Y )
= ( add_li7652885771158616974t_unit @ R3 @ Y @ X2 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_729_semiring_Osemiring__simprules_I7_J,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( add_a_b @ R3 @ X2 @ Y )
= ( add_a_b @ R3 @ Y @ X2 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_730_semiring_Osemiring__simprules_I7_J,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( add_li174743652000525320t_unit @ R3 @ X2 @ Y )
= ( add_li174743652000525320t_unit @ R3 @ Y @ X2 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_731_semiring_Osemiring__simprules_I5_J,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( add_li7652885771158616974t_unit @ R3 @ ( add_li7652885771158616974t_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R3 @ X2 @ ( add_li7652885771158616974t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_732_semiring_Osemiring__simprules_I5_J,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a,Y: a,Z: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( add_a_b @ R3 @ ( add_a_b @ R3 @ X2 @ Y ) @ Z )
= ( add_a_b @ R3 @ X2 @ ( add_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_733_semiring_Osemiring__simprules_I5_J,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( add_li174743652000525320t_unit @ R3 @ ( add_li174743652000525320t_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ R3 @ X2 @ ( add_li174743652000525320t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_734_semiring_Osemiring__simprules_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R3 @ X2 @ Y ) @ ( partia5361259788508890537t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_735_semiring_Osemiring__simprules_I1_J,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_a @ ( add_a_b @ R3 @ X2 @ Y ) @ ( partia707051561876973205xt_a_b @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_736_semiring_Osemiring__simprules_I1_J,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R3 @ X2 @ Y ) @ ( partia2464479390973590831t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_737_semiring_Osemiring__simprules_I3_J,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R3 @ X2 @ Y ) @ ( partia5361259788508890537t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_738_semiring_Osemiring__simprules_I3_J,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R3 @ X2 @ Y ) @ ( partia707051561876973205xt_a_b @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_739_semiring_Osemiring__simprules_I3_J,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ R3 @ X2 @ Y ) @ ( partia2464479390973590831t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_740_semiring_Osemiring__simprules_I8_J,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( mult_l7073676228092353617t_unit @ R3 @ ( mult_l7073676228092353617t_unit @ R3 @ X2 @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ R3 @ X2 @ ( mult_l7073676228092353617t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_741_semiring_Osemiring__simprules_I8_J,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a,Y: a,Z: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ X2 @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R3 @ X2 @ ( mult_a_ring_ext_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_742_semiring_Osemiring__simprules_I8_J,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( mult_l4853965630390486993t_unit @ R3 @ ( mult_l4853965630390486993t_unit @ R3 @ X2 @ Y ) @ Z )
= ( mult_l4853965630390486993t_unit @ R3 @ X2 @ ( mult_l4853965630390486993t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_743_semiring_Osemiring__simprules_I4_J,axiom,
! [R3: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R3 ) @ ( partia141011252114345353t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_744_semiring_Osemiring__simprules_I4_J,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( partia5361259788508890537t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_745_semiring_Osemiring__simprules_I4_J,axiom,
! [R3: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R3 )
=> ( member_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( partia707051561876973205xt_a_b @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_746_semiring_Osemiring__simprules_I4_J,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( member_list_list_a @ ( one_li8234411390022467901t_unit @ R3 ) @ ( partia2464479390973590831t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_747_semiring_Or__distr,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( mult_l7073676228092353617t_unit @ R3 @ Z @ ( add_li7652885771158616974t_unit @ R3 @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ R3 @ ( mult_l7073676228092353617t_unit @ R3 @ Z @ X2 ) @ ( mult_l7073676228092353617t_unit @ R3 @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_748_semiring_Or__distr,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a,Y: a,Z: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ Z @ ( add_a_b @ R3 @ X2 @ Y ) )
= ( add_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ Z @ X2 ) @ ( mult_a_ring_ext_a_b @ R3 @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_749_semiring_Or__distr,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( mult_l4853965630390486993t_unit @ R3 @ Z @ ( add_li174743652000525320t_unit @ R3 @ X2 @ Y ) )
= ( add_li174743652000525320t_unit @ R3 @ ( mult_l4853965630390486993t_unit @ R3 @ Z @ X2 ) @ ( mult_l4853965630390486993t_unit @ R3 @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_750_semiring_Ol__distr,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( mult_l7073676228092353617t_unit @ R3 @ ( add_li7652885771158616974t_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R3 @ ( mult_l7073676228092353617t_unit @ R3 @ X2 @ Z ) @ ( mult_l7073676228092353617t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_751_semiring_Ol__distr,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a,Y: a,Z: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ ( add_a_b @ R3 @ X2 @ Y ) @ Z )
= ( add_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ X2 @ Z ) @ ( mult_a_ring_ext_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_752_semiring_Ol__distr,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( mult_l4853965630390486993t_unit @ R3 @ ( add_li174743652000525320t_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ R3 @ ( mult_l4853965630390486993t_unit @ R3 @ X2 @ Z ) @ ( mult_l4853965630390486993t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_753_semiring_Osemiring__simprules_I9_J,axiom,
! [R3: partia7496981018696276118t_unit,X2: set_list_a] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( mult_s7802724872828879953t_unit @ R3 @ ( one_se1127990129394575805t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_754_semiring_Osemiring__simprules_I9_J,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( mult_l7073676228092353617t_unit @ R3 @ ( one_li8328186300101108157t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_755_semiring_Osemiring__simprules_I9_J,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ ( one_a_ring_ext_a_b @ R3 ) @ X2 )
= X2 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_756_semiring_Osemiring__simprules_I9_J,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( mult_l4853965630390486993t_unit @ R3 @ ( one_li8234411390022467901t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_757_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 )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( 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 @ A4 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P @ A4 ) ) @ Q ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).
% roots_inclI
thf(fact_758_le__alg__mult__imp__pdivides,axiom,
! [X2: a,P: list_a,N: nat] :
( ( member_a @ X2 @ ( 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 @ X2 ) )
=> ( 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 @ X2 ) @ nil_a ) ) @ N ) @ P ) ) ) ) ).
% le_alg_mult_imp_pdivides
thf(fact_759_alg__multE_I2_J,axiom,
! [X2: a,P: list_a,N: nat] :
( ( member_a @ X2 @ ( 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 @ X2 ) @ nil_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P @ X2 ) ) ) ) ) ) ).
% alg_multE(2)
thf(fact_760_alg__multE_I1_J,axiom,
! [X2: a,P: list_a] :
( ( member_a @ X2 @ ( 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 @ X2 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P @ X2 ) ) @ P ) ) ) ) ).
% alg_multE(1)
thf(fact_761_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 )
=> ~ ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A4 @ ( 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 @ A4 ) @ 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 @ A4 ) @ nil_a ) ) @ P ) ) ) ) ) ) ).
% not_empty_rootsE
thf(fact_762_ring__iso__imp__img__domain,axiom,
! [H3: a > a,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ r @ S ) )
=> ( domain_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H3 @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_domain
thf(fact_763_ring__iso__imp__img__domain,axiom,
! [H3: a > list_a,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ r @ S ) )
=> ( domain6553523120543210313t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H3 @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_domain
thf(fact_764_minus__eq,axiom,
! [X2: a,Y: a] :
( ( a_minus_a_b @ r @ X2 @ Y )
= ( add_a_b @ r @ X2 @ ( a_inv_a_b @ r @ Y ) ) ) ).
% minus_eq
thf(fact_765_pderiv__zero,axiom,
! [K: set_a] :
( ( formal4452980811800949548iv_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% pderiv_zero
thf(fact_766_univ__poly__a__inv__consistent,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ) ).
% univ_poly_a_inv_consistent
thf(fact_767_long__division__a__inv_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% long_division_a_inv(2)
thf(fact_768_pderiv__inv,axiom,
! [K: set_a,F: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( formal4452980811800949548iv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ F ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( formal4452980811800949548iv_a_b @ r @ F ) ) ) ) ) ).
% pderiv_inv
thf(fact_769_long__division__a__inv_I1_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% long_division_a_inv(1)
thf(fact_770_const__term__simprules__shell_I4_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
= ( a_inv_a_b @ r @ ( const_term_a_b @ r @ P ) ) ) ) ) ).
% const_term_simprules_shell(4)
thf(fact_771_minus__closed,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X2 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_772_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_773_ring_Oalg__mult_Ocong,axiom,
polyno4422430861927485590lt_a_b = polyno4422430861927485590lt_a_b ).
% ring.alg_mult.cong
thf(fact_774_univ__poly__zero,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a] :
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R3 @ K ) )
= nil_a ) ).
% univ_poly_zero
thf(fact_775_univ__poly__zero,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a] :
( ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) )
= nil_list_a ) ).
% univ_poly_zero
thf(fact_776_domain_Opderiv__zero,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( formal6075833236969493044t_unit @ R3 @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) ) ) ).
% domain.pderiv_zero
thf(fact_777_domain_Opderiv__zero,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R3 )
=> ( ( formal4452980811800949548iv_a_b @ R3 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R3 @ K ) ) ) ) ).
% domain.pderiv_zero
thf(fact_778_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R3: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( subrin3541368690557094692t_unit @ K @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) ) )
=> ( ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) @ P )
= ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_779_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_780_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_781_domain_Olong__division__a__inv_I2_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ R3 @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P ) @ Q )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ ( polyno1727750685288865234t_unit @ R3 @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(2)
thf(fact_782_domain_Olong__division__a__inv_I2_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( polynomial_pmod_a_b @ R3 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R3 @ K ) @ ( polynomial_pmod_a_b @ R3 @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(2)
thf(fact_783_domain_Opderiv__inv,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,F: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( formal6075833236969493044t_unit @ R3 @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ F ) )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ ( formal6075833236969493044t_unit @ R3 @ F ) ) ) ) ) ) ).
% domain.pderiv_inv
thf(fact_784_domain_Opderiv__inv,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,F: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( formal4452980811800949548iv_a_b @ R3 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R3 @ K ) @ F ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R3 @ K ) @ ( formal4452980811800949548iv_a_b @ R3 @ F ) ) ) ) ) ) ).
% domain.pderiv_inv
thf(fact_785_domain_Olong__division__a__inv_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ R3 @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P ) @ Q )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ ( polyno5893782122288709345t_unit @ R3 @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_786_domain_Olong__division__a__inv_I1_J,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subfield_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ R3 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R3 @ K ) @ ( polynomial_pdiv_a_b @ R3 @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_787_domain_Onot__empty__rootsE,axiom,
! [R3: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) ) )
=> ( ( ( polyno4169377219242390531t_unit @ R3 @ P )
!= zero_z7061913751530476641list_a )
=> ~ ! [A4: set_list_a] :
( ( member_set_list_a @ A4 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ A4 @ ( set_mset_set_list_a @ ( polyno4169377219242390531t_unit @ R3 @ P ) ) )
=> ( ( member5524387281408368019list_a @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R3 ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R3 @ A4 ) @ nil_set_list_a ) ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) ) )
=> ~ ( polyno9075941895896075626t_unit @ R3 @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R3 ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R3 @ A4 ) @ nil_set_list_a ) ) @ P ) ) ) ) ) ) ) ).
% domain.not_empty_rootsE
thf(fact_788_domain_Onot__empty__rootsE,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( ( polyno3707469075594375645t_unit @ R3 @ P )
!= zero_z1542645121299710087list_a )
=> ~ ! [A4: list_list_a] :
( ( member_list_list_a @ A4 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ A4 @ ( set_mset_list_list_a @ ( polyno3707469075594375645t_unit @ R3 @ P ) ) )
=> ( ( member5342144027231129785list_a @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R3 ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R3 @ A4 ) @ nil_list_list_a ) ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ~ ( polyno4453881341673752516t_unit @ R3 @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R3 ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R3 @ A4 ) @ nil_list_list_a ) ) @ P ) ) ) ) ) ) ) ).
% domain.not_empty_rootsE
thf(fact_789_domain_Onot__empty__rootsE,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( ( polyno7858422826990252003t_unit @ R3 @ P )
!= zero_z4454100511807792257list_a )
=> ~ ! [A4: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ A4 @ ( set_mset_list_a @ ( polyno7858422826990252003t_unit @ R3 @ P ) ) )
=> ( ( member_list_list_a @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R3 @ A4 ) @ nil_list_a ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ~ ( polyno8016796738000020810t_unit @ R3 @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R3 @ A4 ) @ nil_list_a ) ) @ P ) ) ) ) ) ) ) ).
% domain.not_empty_rootsE
thf(fact_790_domain_Onot__empty__rootsE,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( ( polynomial_roots_a_b @ R3 @ P )
!= zero_zero_multiset_a )
=> ~ ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ A4 @ ( set_mset_a @ ( polynomial_roots_a_b @ R3 @ P ) ) )
=> ( ( member_list_a @ ( cons_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( cons_a @ ( a_inv_a_b @ R3 @ A4 ) @ nil_a ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ~ ( polyno5814909790663948098es_a_b @ R3 @ ( cons_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( cons_a @ ( a_inv_a_b @ R3 @ A4 ) @ nil_a ) ) @ P ) ) ) ) ) ) ) ).
% domain.not_empty_rootsE
thf(fact_791_domain_Oalg__multE_I1_J,axiom,
! [R3: partia7496981018696276118t_unit,X2: set_list_a,P: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) ) )
=> ( ( P != nil_set_list_a )
=> ( polyno9075941895896075626t_unit @ R3 @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R3 ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R3 @ X2 ) @ nil_set_list_a ) ) @ ( polyno1088517687229135038t_unit @ R3 @ P @ X2 ) ) @ P ) ) ) ) ) ).
% domain.alg_multE(1)
thf(fact_792_domain_Oalg__multE_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( P != nil_list_a )
=> ( polyno8016796738000020810t_unit @ R3 @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R3 @ X2 ) @ nil_list_a ) ) @ ( polyno4259638811958763678t_unit @ R3 @ P @ X2 ) ) @ P ) ) ) ) ) ).
% domain.alg_multE(1)
thf(fact_793_domain_Oalg__multE_I1_J,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( P != nil_list_list_a )
=> ( polyno4453881341673752516t_unit @ R3 @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R3 ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R3 @ X2 ) @ nil_list_list_a ) ) @ ( polyno1672195411705137432t_unit @ R3 @ P @ X2 ) ) @ P ) ) ) ) ) ).
% domain.alg_multE(1)
thf(fact_794_domain_Oalg__multE_I1_J,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( P != nil_a )
=> ( polyno5814909790663948098es_a_b @ R3 @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( cons_a @ ( a_inv_a_b @ R3 @ X2 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ R3 @ P @ X2 ) ) @ P ) ) ) ) ) ).
% domain.alg_multE(1)
thf(fact_795_domain_Ole__alg__mult__imp__pdivides,axiom,
! [R3: partia7496981018696276118t_unit,X2: set_list_a,P: list_set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno1088517687229135038t_unit @ R3 @ P @ X2 ) )
=> ( polyno9075941895896075626t_unit @ R3 @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R3 ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R3 @ X2 ) @ nil_set_list_a ) ) @ N ) @ P ) ) ) ) ) ).
% domain.le_alg_mult_imp_pdivides
thf(fact_796_domain_Ole__alg__mult__imp__pdivides,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a,P: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ R3 @ P @ X2 ) )
=> ( polyno8016796738000020810t_unit @ R3 @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R3 @ X2 ) @ nil_list_a ) ) @ N ) @ P ) ) ) ) ) ).
% domain.le_alg_mult_imp_pdivides
thf(fact_797_domain_Ole__alg__mult__imp__pdivides,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a,P: list_list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno1672195411705137432t_unit @ R3 @ P @ X2 ) )
=> ( polyno4453881341673752516t_unit @ R3 @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R3 ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R3 @ X2 ) @ nil_list_list_a ) ) @ N ) @ P ) ) ) ) ) ).
% domain.le_alg_mult_imp_pdivides
thf(fact_798_domain_Ole__alg__mult__imp__pdivides,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a,P: list_a,N: nat] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ R3 @ P @ X2 ) )
=> ( polyno5814909790663948098es_a_b @ R3 @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( cons_a @ ( a_inv_a_b @ R3 @ X2 ) @ nil_a ) ) @ N ) @ P ) ) ) ) ) ).
% domain.le_alg_mult_imp_pdivides
thf(fact_799_domain_Oalg__multE_I2_J,axiom,
! [R3: partia7496981018696276118t_unit,X2: set_list_a,P: list_set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) ) )
=> ( ( P != nil_set_list_a )
=> ( ( polyno9075941895896075626t_unit @ R3 @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R3 ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R3 @ X2 ) @ nil_set_list_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno1088517687229135038t_unit @ R3 @ P @ X2 ) ) ) ) ) ) ) ).
% domain.alg_multE(2)
thf(fact_800_domain_Oalg__multE_I2_J,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a,P: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ R3 @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R3 @ X2 ) @ nil_list_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ R3 @ P @ X2 ) ) ) ) ) ) ) ).
% domain.alg_multE(2)
thf(fact_801_domain_Oalg__multE_I2_J,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a,P: list_list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( P != nil_list_list_a )
=> ( ( polyno4453881341673752516t_unit @ R3 @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R3 ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R3 @ X2 ) @ nil_list_list_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno1672195411705137432t_unit @ R3 @ P @ X2 ) ) ) ) ) ) ) ).
% domain.alg_multE(2)
thf(fact_802_domain_Oalg__multE_I2_J,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a,P: list_a,N: nat] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( P != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ R3 @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( cons_a @ ( a_inv_a_b @ R3 @ X2 ) @ nil_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ R3 @ P @ X2 ) ) ) ) ) ) ) ).
% domain.alg_multE(2)
thf(fact_803_domain_Oroots__inclI,axiom,
! [R3: partia7496981018696276118t_unit,P: list_set_list_a,Q: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) ) )
=> ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) ) )
=> ( ( Q != nil_set_list_a )
=> ( ! [A4: set_list_a] :
( ( member_set_list_a @ A4 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( P != nil_set_list_a )
=> ( polyno9075941895896075626t_unit @ R3 @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R3 ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R3 @ A4 ) @ nil_set_list_a ) ) @ ( polyno1088517687229135038t_unit @ R3 @ P @ A4 ) ) @ Q ) ) )
=> ( subset4236506274861796683list_a @ ( polyno4169377219242390531t_unit @ R3 @ P ) @ ( polyno4169377219242390531t_unit @ R3 @ Q ) ) ) ) ) ) ) ).
% domain.roots_inclI
thf(fact_804_domain_Oroots__inclI,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( Q != nil_list_list_a )
=> ( ! [A4: list_list_a] :
( ( member_list_list_a @ A4 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( P != nil_list_list_a )
=> ( polyno4453881341673752516t_unit @ R3 @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R3 ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R3 @ A4 ) @ nil_list_list_a ) ) @ ( polyno1672195411705137432t_unit @ R3 @ P @ A4 ) ) @ Q ) ) )
=> ( subset8447756916971205105list_a @ ( polyno3707469075594375645t_unit @ R3 @ P ) @ ( polyno3707469075594375645t_unit @ R3 @ Q ) ) ) ) ) ) ) ).
% domain.roots_inclI
thf(fact_805_domain_Oroots__inclI,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( Q != nil_list_a )
=> ( ! [A4: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( P != nil_list_a )
=> ( polyno8016796738000020810t_unit @ R3 @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R3 @ A4 ) @ nil_list_a ) ) @ ( polyno4259638811958763678t_unit @ R3 @ P @ A4 ) ) @ Q ) ) )
=> ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ R3 @ P ) @ ( polyno7858422826990252003t_unit @ R3 @ Q ) ) ) ) ) ) ) ).
% domain.roots_inclI
thf(fact_806_domain_Oroots__inclI,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( Q != nil_a )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( P != nil_a )
=> ( polyno5814909790663948098es_a_b @ R3 @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( cons_a @ ( a_inv_a_b @ R3 @ A4 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ R3 @ P @ A4 ) ) @ Q ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ R3 @ P ) @ ( polynomial_roots_a_b @ R3 @ Q ) ) ) ) ) ) ) ).
% domain.roots_inclI
thf(fact_807_set__mset__eq__empty__iff,axiom,
! [M2: multiset_a] :
( ( ( set_mset_a @ M2 )
= bot_bot_set_a )
= ( M2 = zero_zero_multiset_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_808_set__mset__eq__empty__iff,axiom,
! [M2: multiset_nat] :
( ( ( set_mset_nat @ M2 )
= bot_bot_set_nat )
= ( M2 = zero_z7348594199698428585et_nat ) ) ).
% set_mset_eq_empty_iff
thf(fact_809_set__mset__eq__empty__iff,axiom,
! [M2: multiset_list_a] :
( ( ( set_mset_list_a @ M2 )
= bot_bot_set_list_a )
= ( M2 = zero_z4454100511807792257list_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_810_set__mset__empty,axiom,
( ( set_mset_a @ zero_zero_multiset_a )
= bot_bot_set_a ) ).
% set_mset_empty
thf(fact_811_set__mset__empty,axiom,
( ( set_mset_nat @ zero_z7348594199698428585et_nat )
= bot_bot_set_nat ) ).
% set_mset_empty
thf(fact_812_set__mset__empty,axiom,
( ( set_mset_list_a @ zero_z4454100511807792257list_a )
= bot_bot_set_list_a ) ).
% set_mset_empty
thf(fact_813_degree__one__roots,axiom,
! [A: a,A6: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ A6 )
= ( 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 @ A6 @ B ) ) @ zero_zero_multiset_a ) ) ) ) ) ) ).
% degree_one_roots
thf(fact_814_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 ) ) ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( ord_less_eq_nat @ ( polyno4422430861927485590lt_a_b @ r @ P @ A4 ) @ ( count_a @ M @ A4 ) ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ M ) ) ) ).
% roots_inclI'
thf(fact_815_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_816_finite__set__mset,axiom,
! [M2: multiset_nat] : ( finite_finite_nat @ ( set_mset_nat @ M2 ) ) ).
% finite_set_mset
thf(fact_817_finite__set__mset,axiom,
! [M2: multiset_complex] : ( finite3207457112153483333omplex @ ( set_mset_complex @ M2 ) ) ).
% finite_set_mset
thf(fact_818_finite__set__mset,axiom,
! [M2: multiset_list_a] : ( finite_finite_list_a @ ( set_mset_list_a @ M2 ) ) ).
% finite_set_mset
thf(fact_819_finite__set__mset,axiom,
! [M2: multiset_a] : ( finite_finite_a @ ( set_mset_a @ M2 ) ) ).
% finite_set_mset
thf(fact_820_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_821_mset__subset__eqI,axiom,
! [A2: multiset_a,B5: multiset_a] :
( ! [A4: a] : ( ord_less_eq_nat @ ( count_a @ A2 @ A4 ) @ ( count_a @ B5 @ A4 ) )
=> ( subseteq_mset_a @ A2 @ B5 ) ) ).
% mset_subset_eqI
thf(fact_822_subseteq__mset__def,axiom,
( subseteq_mset_a
= ( ^ [A5: multiset_a,B3: multiset_a] :
! [A3: a] : ( ord_less_eq_nat @ ( count_a @ A5 @ A3 ) @ ( count_a @ B3 @ A3 ) ) ) ) ).
% subseteq_mset_def
thf(fact_823_mset__subset__eq__count,axiom,
! [A2: multiset_a,B5: multiset_a,A: a] :
( ( subseteq_mset_a @ A2 @ B5 )
=> ( ord_less_eq_nat @ ( count_a @ A2 @ A ) @ ( count_a @ B5 @ A ) ) ) ).
% mset_subset_eq_count
thf(fact_824_multiset__induct__min,axiom,
! [P2: multiset_nat > $o,M2: multiset_nat] :
( ( P2 @ zero_z7348594199698428585et_nat )
=> ( ! [X3: nat,M3: multiset_nat] :
( ( P2 @ M3 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M3 ) )
=> ( ord_less_eq_nat @ X3 @ Xa ) )
=> ( P2 @ ( add_mset_nat @ X3 @ M3 ) ) ) )
=> ( P2 @ M2 ) ) ) ).
% multiset_induct_min
thf(fact_825_multiset__induct__max,axiom,
! [P2: multiset_nat > $o,M2: multiset_nat] :
( ( P2 @ zero_z7348594199698428585et_nat )
=> ( ! [X3: nat,M3: multiset_nat] :
( ( P2 @ M3 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M3 ) )
=> ( ord_less_eq_nat @ Xa @ X3 ) )
=> ( P2 @ ( add_mset_nat @ X3 @ M3 ) ) ) )
=> ( P2 @ M2 ) ) ) ).
% multiset_induct_max
thf(fact_826_subset__mset_Ofinite__has__maximal,axiom,
! [A2: set_multiset_a] :
( ( finite2463020702752857069iset_a @ A2 )
=> ( ( A2 != bot_bo6997605411617904272iset_a )
=> ? [X3: multiset_a] :
( ( member_multiset_a @ X3 @ A2 )
& ! [Xa: multiset_a] :
( ( member_multiset_a @ Xa @ A2 )
=> ( ( subseteq_mset_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_maximal
thf(fact_827_subset__mset_Ofinite__has__minimal,axiom,
! [A2: set_multiset_a] :
( ( finite2463020702752857069iset_a @ A2 )
=> ( ( A2 != bot_bo6997605411617904272iset_a )
=> ? [X3: multiset_a] :
( ( member_multiset_a @ X3 @ A2 )
& ! [Xa: multiset_a] :
( ( member_multiset_a @ Xa @ A2 )
=> ( ( subseteq_mset_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_minimal
thf(fact_828_domain_Oalg__mult__eq__count__roots,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( polyno1672195411705137432t_unit @ R3 @ P )
= ( count_list_list_a @ ( polyno3707469075594375645t_unit @ R3 @ P ) ) ) ) ) ).
% domain.alg_mult_eq_count_roots
thf(fact_829_domain_Oalg__mult__eq__count__roots,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( polyno4259638811958763678t_unit @ R3 @ P )
= ( count_list_a @ ( polyno7858422826990252003t_unit @ R3 @ P ) ) ) ) ) ).
% domain.alg_mult_eq_count_roots
thf(fact_830_domain_Oalg__mult__eq__count__roots,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( polyno4422430861927485590lt_a_b @ R3 @ P )
= ( count_a @ ( polynomial_roots_a_b @ R3 @ P ) ) ) ) ) ).
% domain.alg_mult_eq_count_roots
thf(fact_831_domain_Omonic__degree__one__roots,axiom,
! [R3: partia7496981018696276118t_unit,A: set_list_a] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( polyno4169377219242390531t_unit @ R3 @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R3 ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R3 @ A ) @ nil_set_list_a ) ) )
= ( add_mset_set_list_a @ A @ zero_z7061913751530476641list_a ) ) ) ) ).
% domain.monic_degree_one_roots
thf(fact_832_domain_Omonic__degree__one__roots,axiom,
! [R3: partia2670972154091845814t_unit,A: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( polyno7858422826990252003t_unit @ R3 @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R3 @ A ) @ nil_list_a ) ) )
= ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) ) ) ).
% domain.monic_degree_one_roots
thf(fact_833_domain_Omonic__degree__one__roots,axiom,
! [R3: partia2956882679547061052t_unit,A: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( polyno3707469075594375645t_unit @ R3 @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R3 ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R3 @ A ) @ nil_list_list_a ) ) )
= ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) ) ) ) ).
% domain.monic_degree_one_roots
thf(fact_834_domain_Omonic__degree__one__roots,axiom,
! [R3: partia2175431115845679010xt_a_b,A: a] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( polynomial_roots_a_b @ R3 @ ( cons_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( cons_a @ ( a_inv_a_b @ R3 @ A ) @ nil_a ) ) )
= ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ).
% domain.monic_degree_one_roots
thf(fact_835_domain_Oroots__inclI_H,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a,M: multiset_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ! [A4: list_list_a] :
( ( member_list_list_a @ A4 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( P != nil_list_list_a )
=> ( ord_less_eq_nat @ ( polyno1672195411705137432t_unit @ R3 @ P @ A4 ) @ ( count_list_list_a @ M @ A4 ) ) ) )
=> ( subset8447756916971205105list_a @ ( polyno3707469075594375645t_unit @ R3 @ P ) @ M ) ) ) ) ).
% domain.roots_inclI'
thf(fact_836_domain_Oroots__inclI_H,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a,M: multiset_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ! [A4: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( P != nil_list_a )
=> ( ord_less_eq_nat @ ( polyno4259638811958763678t_unit @ R3 @ P @ A4 ) @ ( count_list_a @ M @ A4 ) ) ) )
=> ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ R3 @ P ) @ M ) ) ) ) ).
% domain.roots_inclI'
thf(fact_837_domain_Oroots__inclI_H,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a,M: multiset_a] :
( ( domain_a_b @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( P != nil_a )
=> ( ord_less_eq_nat @ ( polyno4422430861927485590lt_a_b @ R3 @ P @ A4 ) @ ( count_a @ M @ A4 ) ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ R3 @ P ) @ M ) ) ) ) ).
% domain.roots_inclI'
thf(fact_838_domain_Odegree__one__roots,axiom,
! [R3: partia7496981018696276118t_unit,A: set_list_a,A6: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ A6 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R3 @ A @ A6 )
= ( one_se1127990129394575805t_unit @ R3 ) )
=> ( ( polyno4169377219242390531t_unit @ R3 @ ( cons_set_list_a @ A @ ( cons_set_list_a @ B @ nil_set_list_a ) ) )
= ( add_mset_set_list_a @ ( a_inv_5715216516650856053t_unit @ R3 @ ( mult_s7802724872828879953t_unit @ R3 @ A6 @ B ) ) @ zero_z7061913751530476641list_a ) ) ) ) ) ) ) ).
% domain.degree_one_roots
thf(fact_839_domain_Odegree__one__roots,axiom,
! [R3: partia2670972154091845814t_unit,A: list_a,A6: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ A6 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R3 @ A @ A6 )
= ( one_li8328186300101108157t_unit @ R3 ) )
=> ( ( polyno7858422826990252003t_unit @ R3 @ ( cons_list_a @ A @ ( cons_list_a @ B @ nil_list_a ) ) )
= ( add_mset_list_a @ ( a_inv_8944721093294617173t_unit @ R3 @ ( mult_l7073676228092353617t_unit @ R3 @ A6 @ B ) ) @ zero_z4454100511807792257list_a ) ) ) ) ) ) ) ).
% domain.degree_one_roots
thf(fact_840_domain_Odegree__one__roots,axiom,
! [R3: partia2956882679547061052t_unit,A: list_list_a,A6: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ A6 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R3 @ A @ A6 )
= ( one_li8234411390022467901t_unit @ R3 ) )
=> ( ( polyno3707469075594375645t_unit @ R3 @ ( cons_list_list_a @ A @ ( cons_list_list_a @ B @ nil_list_list_a ) ) )
= ( add_mset_list_list_a @ ( a_inv_7033018035630854991t_unit @ R3 @ ( mult_l4853965630390486993t_unit @ R3 @ A6 @ B ) ) @ zero_z1542645121299710087list_a ) ) ) ) ) ) ) ).
% domain.degree_one_roots
thf(fact_841_domain_Odegree__one__roots,axiom,
! [R3: partia2175431115845679010xt_a_b,A: a,A6: a,B: a] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( ( mult_a_ring_ext_a_b @ R3 @ A @ A6 )
= ( one_a_ring_ext_a_b @ R3 ) )
=> ( ( polynomial_roots_a_b @ R3 @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) )
= ( add_mset_a @ ( a_inv_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ A6 @ B ) ) @ zero_zero_multiset_a ) ) ) ) ) ) ) ).
% domain.degree_one_roots
thf(fact_842_set__mset__mono,axiom,
! [A2: multiset_a,B5: multiset_a] :
( ( subseteq_mset_a @ A2 @ B5 )
=> ( ord_less_eq_set_a @ ( set_mset_a @ A2 ) @ ( set_mset_a @ B5 ) ) ) ).
% set_mset_mono
thf(fact_843_subset__eq__mset__impl_Ocases,axiom,
! [X2: produc9164743771328383783list_a] :
( ! [Ys: list_a] :
( X2
!= ( produc6837034575241423639list_a @ nil_a @ Ys ) )
=> ~ ! [X3: a,Xs: list_a,Ys: list_a] :
( X2
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs ) @ Ys ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_844_domain_Oring__iso__imp__img__domain,axiom,
! [R3: partia2175431115845679010xt_a_b,H3: a > a,S: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R3 )
=> ( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R3 @ S ) )
=> ( domain_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H3 @ ( zero_a_b @ R3 ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_845_domain_Oring__iso__imp__img__domain,axiom,
! [R3: partia2175431115845679010xt_a_b,H3: a > list_a,S: partia2670972154091845814t_unit] :
( ( domain_a_b @ R3 )
=> ( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R3 @ S ) )
=> ( domain6553523120543210313t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H3 @ ( zero_a_b @ R3 ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_846_domain_Oring__iso__imp__img__domain,axiom,
! [R3: partia2670972154091845814t_unit,H3: list_a > a,S: partia2175431115845679010xt_a_b] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R3 @ S ) )
=> ( domain_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H3 @ ( zero_l4142658623432671053t_unit @ R3 ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_847_domain_Oring__iso__imp__img__domain,axiom,
! [R3: partia2670972154091845814t_unit,H3: list_a > list_a,S: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R3 @ S ) )
=> ( domain6553523120543210313t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H3 @ ( zero_l4142658623432671053t_unit @ R3 ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_848_domain_Oring__iso__imp__img__domain,axiom,
! [R3: partia7496981018696276118t_unit,H3: set_list_a > a,S: partia2175431115845679010xt_a_b] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( member_set_list_a_a @ H3 @ ( ring_i8122894263081988538it_a_b @ R3 @ S ) )
=> ( domain_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H3 @ ( zero_s2910681146719230829t_unit @ R3 ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_849_domain_Oring__iso__imp__img__domain,axiom,
! [R3: partia7496981018696276118t_unit,H3: set_list_a > list_a,S: partia2670972154091845814t_unit] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( member5910328476188217884list_a @ H3 @ ( ring_i8566987394125245378t_unit @ R3 @ S ) )
=> ( domain6553523120543210313t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H3 @ ( zero_s2910681146719230829t_unit @ R3 ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_850_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_851_associated__polynomials__imp__same__length,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
=> ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q ) ) ) ) ) ) ).
% associated_polynomials_imp_same_length
thf(fact_852_univ__poly__a__inv__length,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
= ( size_size_list_a @ P ) ) ) ) ).
% univ_poly_a_inv_length
thf(fact_853_univ__poly__a__inv__def_H,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
= ( map_a_a @ ( a_inv_a_b @ r ) @ P ) ) ) ) ).
% univ_poly_a_inv_def'
thf(fact_854_alg__mult__def,axiom,
! [P: list_a,X2: a] :
( ( ( P = nil_a )
=> ( ( polyno4422430861927485590lt_a_b @ r @ P @ X2 )
= zero_zero_nat ) )
& ( ( P != nil_a )
=> ( ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno4422430861927485590lt_a_b @ r @ P @ X2 )
= ( order_Greatest_nat
@ ^ [N2: nat] : ( 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 @ X2 ) @ nil_a ) ) @ N2 ) @ P ) ) ) )
& ( ~ ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno4422430861927485590lt_a_b @ r @ P @ X2 )
= zero_zero_nat ) ) ) ) ) ).
% alg_mult_def
thf(fact_855_semiring_Onat__pow__zero,axiom,
! [R3: partia2175431115845679010xt_a_b,N: nat] :
( ( semiring_a_b @ R3 )
=> ( ( N != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ R3 @ ( zero_a_b @ R3 ) @ N )
= ( zero_a_b @ R3 ) ) ) ) ).
% semiring.nat_pow_zero
thf(fact_856_semiring_Onat__pow__zero,axiom,
! [R3: partia2670972154091845814t_unit,N: nat] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( N != zero_zero_nat )
=> ( ( pow_li1142815632869257134it_nat @ R3 @ ( zero_l4142658623432671053t_unit @ R3 ) @ N )
= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ) ).
% semiring.nat_pow_zero
thf(fact_857_semiring_Onat__pow__zero,axiom,
! [R3: partia7496981018696276118t_unit,N: nat] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( ( N != zero_zero_nat )
=> ( ( pow_se8252319793075206062it_nat @ R3 @ ( zero_s2910681146719230829t_unit @ R3 ) @ N )
= ( zero_s2910681146719230829t_unit @ R3 ) ) ) ) ).
% semiring.nat_pow_zero
thf(fact_858_domain_Ouniv__poly__a__inv__length,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P ) )
= ( size_size_list_a @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_length
thf(fact_859_domain_Ouniv__poly__a__inv__length,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( size_s349497388124573686list_a @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P ) )
= ( size_s349497388124573686list_a @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_length
thf(fact_860_domain_Oassociated__polynomials__imp__same__length,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P @ Q )
=> ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q ) ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_length
thf(fact_861_domain_Oassociated__polynomials__imp__same__length,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P @ Q )
=> ( ( size_s349497388124573686list_a @ P )
= ( size_s349497388124573686list_a @ Q ) ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_length
thf(fact_862_domain_Ouniv__poly__a__inv__def_H,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R3 @ K ) @ P )
= ( map_a_a @ ( a_inv_a_b @ R3 ) @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_def'
thf(fact_863_domain_Ouniv__poly__a__inv__def_H,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P )
= ( map_list_a_list_a @ ( a_inv_8944721093294617173t_unit @ R3 ) @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_def'
thf(fact_864_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_865_ring__iso__memE_I1_J,axiom,
! [H3: list_a > list_a,R3: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X2: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R3 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_list_a @ ( H3 @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_866_ring__iso__memE_I1_J,axiom,
! [H3: list_a > a,R3: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X2: list_a] :
( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R3 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_a @ ( H3 @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_867_ring__iso__memE_I1_J,axiom,
! [H3: list_a > list_list_a,R3: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,X2: list_a] :
( ( member6714375691612171394list_a @ H3 @ ( ring_i7582117978422105628t_unit @ R3 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_list_list_a @ ( H3 @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_868_ring__iso__memE_I1_J,axiom,
! [H3: a > list_a,R3: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X2: a] :
( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R3 @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_list_a @ ( H3 @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_869_ring__iso__memE_I1_J,axiom,
! [H3: a > a,R3: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X2: a] :
( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R3 @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_a @ ( H3 @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_870_ring__iso__memE_I1_J,axiom,
! [H3: a > list_list_a,R3: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,X2: a] :
( ( member_a_list_list_a @ H3 @ ( ring_i4464730343205239444t_unit @ R3 @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_list_list_a @ ( H3 @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_871_ring__iso__memE_I1_J,axiom,
! [H3: list_list_a > list_a,R3: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X2: list_list_a] :
( ( member7168557129179038582list_a @ H3 @ ( ring_i4611353245267337884t_unit @ R3 @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( member_list_a @ ( H3 @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_872_ring__iso__memE_I1_J,axiom,
! [H3: list_list_a > a,R3: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X2: list_list_a] :
( ( member_list_list_a_a @ H3 @ ( ring_i5684343068699926420it_a_b @ R3 @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( member_a @ ( H3 @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_873_ring__iso__memE_I1_J,axiom,
! [H3: list_list_a > list_list_a,R3: partia2956882679547061052t_unit,S: partia2956882679547061052t_unit,X2: list_list_a] :
( ( member8231385768148312316list_a @ H3 @ ( ring_i6186174840089424918t_unit @ R3 @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( member_list_list_a @ ( H3 @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_874_ring__iso__memE_I4_J,axiom,
! [H3: a > a,R3: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R3 @ S ) )
=> ( ( H3 @ ( one_a_ring_ext_a_b @ R3 ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_875_ring__iso__memE_I4_J,axiom,
! [H3: a > list_a,R3: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R3 @ S ) )
=> ( ( H3 @ ( one_a_ring_ext_a_b @ R3 ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_876_ring__iso__memE_I4_J,axiom,
! [H3: a > set_list_a,R3: partia2175431115845679010xt_a_b,S: partia7496981018696276118t_unit] :
( ( member_a_set_list_a @ H3 @ ( ring_i5325512697602418746t_unit @ R3 @ S ) )
=> ( ( H3 @ ( one_a_ring_ext_a_b @ R3 ) )
= ( one_se1127990129394575805t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_877_ring__iso__memE_I4_J,axiom,
! [H3: list_a > a,R3: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R3 @ S ) )
=> ( ( H3 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_878_ring__iso__memE_I4_J,axiom,
! [H3: list_a > list_a,R3: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit] :
( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R3 @ S ) )
=> ( ( H3 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_879_ring__iso__memE_I4_J,axiom,
! [H3: list_a > set_list_a,R3: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit] :
( ( member4263473470251683292list_a @ H3 @ ( ring_i6716952747341498306t_unit @ R3 @ S ) )
=> ( ( H3 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_se1127990129394575805t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_880_ring__iso__memE_I4_J,axiom,
! [H3: set_list_a > a,R3: partia7496981018696276118t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_set_list_a_a @ H3 @ ( ring_i8122894263081988538it_a_b @ R3 @ S ) )
=> ( ( H3 @ ( one_se1127990129394575805t_unit @ R3 ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_881_ring__iso__memE_I4_J,axiom,
! [H3: set_list_a > list_a,R3: partia7496981018696276118t_unit,S: partia2670972154091845814t_unit] :
( ( member5910328476188217884list_a @ H3 @ ( ring_i8566987394125245378t_unit @ R3 @ S ) )
=> ( ( H3 @ ( one_se1127990129394575805t_unit @ R3 ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_882_ring__iso__memE_I4_J,axiom,
! [H3: set_list_a > set_list_a,R3: partia7496981018696276118t_unit,S: partia7496981018696276118t_unit] :
( ( member5068272912271824380list_a @ H3 @ ( ring_i6425008796027319266t_unit @ R3 @ S ) )
=> ( ( H3 @ ( one_se1127990129394575805t_unit @ R3 ) )
= ( one_se1127990129394575805t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_883_ring__iso__memE_I3_J,axiom,
! [H3: list_a > list_a,R3: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R3 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R3 @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_884_ring__iso__memE_I3_J,axiom,
! [H3: list_a > a,R3: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X2: list_a,Y: list_a] :
( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R3 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R3 @ X2 @ Y ) )
= ( add_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_885_ring__iso__memE_I3_J,axiom,
! [H3: a > list_a,R3: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X2: a,Y: a] :
( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R3 @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H3 @ ( add_a_b @ R3 @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_886_ring__iso__memE_I3_J,axiom,
! [H3: a > a,R3: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R3 @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H3 @ ( add_a_b @ R3 @ X2 @ Y ) )
= ( add_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_887_ring__iso__memE_I3_J,axiom,
! [H3: list_list_a > list_a,R3: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X2: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H3 @ ( ring_i4611353245267337884t_unit @ R3 @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( H3 @ ( add_li174743652000525320t_unit @ R3 @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_888_ring__iso__memE_I3_J,axiom,
! [H3: list_list_a > a,R3: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X2: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H3 @ ( ring_i5684343068699926420it_a_b @ R3 @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( H3 @ ( add_li174743652000525320t_unit @ R3 @ X2 @ Y ) )
= ( add_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_889_ring__iso__memE_I2_J,axiom,
! [H3: list_a > list_a,R3: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R3 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R3 @ X2 @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_890_ring__iso__memE_I2_J,axiom,
! [H3: list_a > a,R3: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X2: list_a,Y: list_a] :
( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R3 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R3 @ X2 @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_891_ring__iso__memE_I2_J,axiom,
! [H3: a > list_a,R3: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X2: a,Y: a] :
( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R3 @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R3 @ X2 @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_892_ring__iso__memE_I2_J,axiom,
! [H3: a > a,R3: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R3 @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R3 @ X2 @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_893_ring__iso__memE_I2_J,axiom,
! [H3: list_list_a > list_a,R3: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X2: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H3 @ ( ring_i4611353245267337884t_unit @ R3 @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( H3 @ ( mult_l4853965630390486993t_unit @ R3 @ X2 @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_894_ring__iso__memE_I2_J,axiom,
! [H3: list_list_a > a,R3: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X2: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H3 @ ( ring_i5684343068699926420it_a_b @ R3 @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( H3 @ ( mult_l4853965630390486993t_unit @ R3 @ X2 @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_895_pirreducible__hom,axiom,
! [H3: list_list_a > list_a,R3: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,F: list_list_list_a] :
( ( member7168557129179038582list_a @ H3 @ ( ring_i4611353245267337884t_unit @ R3 @ S ) )
=> ( ( domain7810152921033798211t_unit @ R3 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( ( member5342144027231129785list_a @ F @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ F )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) @ ( map_li1646474281249396926list_a @ H3 @ F ) ) ) ) ) ) ) ).
% pirreducible_hom
thf(fact_896_pirreducible__hom,axiom,
! [H3: list_list_a > list_list_a,R3: partia2956882679547061052t_unit,S: partia2956882679547061052t_unit,F: list_list_list_a] :
( ( member8231385768148312316list_a @ H3 @ ( ring_i6186174840089424918t_unit @ R3 @ S ) )
=> ( ( domain7810152921033798211t_unit @ R3 )
=> ( ( domain7810152921033798211t_unit @ S )
=> ( ( member5342144027231129785list_a @ F @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ F )
= ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ S @ ( partia2464479390973590831t_unit @ S ) ) @ ( map_li8713736314956022724list_a @ H3 @ F ) ) ) ) ) ) ) ).
% pirreducible_hom
thf(fact_897_pirreducible__hom,axiom,
! [H3: list_a > list_a,R3: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,F: list_list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R3 @ S ) )
=> ( ( domain6553523120543210313t_unit @ R3 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ F )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) @ ( map_list_a_list_a @ H3 @ F ) ) ) ) ) ) ) ).
% pirreducible_hom
thf(fact_898_pirreducible__hom,axiom,
! [H3: list_a > list_list_a,R3: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,F: list_list_a] :
( ( member6714375691612171394list_a @ H3 @ ( ring_i7582117978422105628t_unit @ R3 @ S ) )
=> ( ( domain6553523120543210313t_unit @ R3 )
=> ( ( domain7810152921033798211t_unit @ S )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ F )
= ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ S @ ( partia2464479390973590831t_unit @ S ) ) @ ( map_li5729356230488778442list_a @ H3 @ F ) ) ) ) ) ) ) ).
% pirreducible_hom
thf(fact_899_pirreducible__hom,axiom,
! [H3: list_list_a > a,R3: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,F: list_list_list_a] :
( ( member_list_list_a_a @ H3 @ ( ring_i5684343068699926420it_a_b @ R3 @ S ) )
=> ( ( domain7810152921033798211t_unit @ R3 )
=> ( ( domain_a_b @ S )
=> ( ( member5342144027231129785list_a @ F @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ F )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) @ ( map_list_list_a_a @ H3 @ F ) ) ) ) ) ) ) ).
% pirreducible_hom
thf(fact_900_pirreducible__hom,axiom,
! [H3: list_a > a,R3: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,F: list_list_a] :
( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R3 @ S ) )
=> ( ( domain6553523120543210313t_unit @ R3 )
=> ( ( domain_a_b @ S )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ F )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) @ ( map_list_a_a @ H3 @ F ) ) ) ) ) ) ) ).
% pirreducible_hom
thf(fact_901_pirreducible__hom,axiom,
! [H3: a > list_a,R3: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,F: list_a] :
( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R3 @ S ) )
=> ( ( domain_a_b @ R3 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ F )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) @ ( map_a_list_a @ H3 @ F ) ) ) ) ) ) ) ).
% pirreducible_hom
thf(fact_902_pirreducible__hom,axiom,
! [H3: a > list_list_a,R3: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,F: list_a] :
( ( member_a_list_list_a @ H3 @ ( ring_i4464730343205239444t_unit @ R3 @ S ) )
=> ( ( domain_a_b @ R3 )
=> ( ( domain7810152921033798211t_unit @ S )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ F )
= ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ S @ ( partia2464479390973590831t_unit @ S ) ) @ ( map_a_list_list_a @ H3 @ F ) ) ) ) ) ) ) ).
% pirreducible_hom
thf(fact_903_pirreducible__hom,axiom,
! [H3: a > a,R3: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,F: list_a] :
( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R3 @ S ) )
=> ( ( domain_a_b @ R3 )
=> ( ( domain_a_b @ S )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ F )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) @ ( map_a_a @ H3 @ F ) ) ) ) ) ) ) ).
% pirreducible_hom
thf(fact_904_carrier__hom,axiom,
! [F: list_list_list_a,R3: partia2956882679547061052t_unit,H3: list_list_a > list_list_a,S: partia2956882679547061052t_unit] :
( ( member5342144027231129785list_a @ F @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( member8231385768148312316list_a @ H3 @ ( ring_i6186174840089424918t_unit @ R3 @ S ) )
=> ( ( domain7810152921033798211t_unit @ R3 )
=> ( ( domain7810152921033798211t_unit @ S )
=> ( member5342144027231129785list_a @ ( map_li8713736314956022724list_a @ H3 @ F ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ S @ ( partia2464479390973590831t_unit @ S ) ) ) ) ) ) ) ) ).
% carrier_hom
thf(fact_905_carrier__hom,axiom,
! [F: list_list_list_a,R3: partia2956882679547061052t_unit,H3: list_list_a > a,S: partia2175431115845679010xt_a_b] :
( ( member5342144027231129785list_a @ F @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( member_list_list_a_a @ H3 @ ( ring_i5684343068699926420it_a_b @ R3 @ S ) )
=> ( ( domain7810152921033798211t_unit @ R3 )
=> ( ( domain_a_b @ S )
=> ( member_list_a @ ( map_list_list_a_a @ H3 @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ) ).
% carrier_hom
thf(fact_906_carrier__hom,axiom,
! [F: list_list_list_a,R3: partia2956882679547061052t_unit,H3: list_list_a > list_a,S: partia2670972154091845814t_unit] :
( ( member5342144027231129785list_a @ F @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( member7168557129179038582list_a @ H3 @ ( ring_i4611353245267337884t_unit @ R3 @ S ) )
=> ( ( domain7810152921033798211t_unit @ R3 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( member_list_list_a @ ( map_li1646474281249396926list_a @ H3 @ F ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) ) ) ) ) ) ) ).
% carrier_hom
thf(fact_907_carrier__hom,axiom,
! [F: list_a,R3: partia2175431115845679010xt_a_b,H3: a > list_list_a,S: partia2956882679547061052t_unit] :
( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( member_a_list_list_a @ H3 @ ( ring_i4464730343205239444t_unit @ R3 @ S ) )
=> ( ( domain_a_b @ R3 )
=> ( ( domain7810152921033798211t_unit @ S )
=> ( member5342144027231129785list_a @ ( map_a_list_list_a @ H3 @ F ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ S @ ( partia2464479390973590831t_unit @ S ) ) ) ) ) ) ) ) ).
% carrier_hom
thf(fact_908_carrier__hom,axiom,
! [F: list_a,R3: partia2175431115845679010xt_a_b,H3: a > a,S: partia2175431115845679010xt_a_b] :
( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R3 @ S ) )
=> ( ( domain_a_b @ R3 )
=> ( ( domain_a_b @ S )
=> ( member_list_a @ ( map_a_a @ H3 @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ) ).
% carrier_hom
thf(fact_909_carrier__hom,axiom,
! [F: list_a,R3: partia2175431115845679010xt_a_b,H3: a > list_a,S: partia2670972154091845814t_unit] :
( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R3 @ S ) )
=> ( ( domain_a_b @ R3 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( member_list_list_a @ ( map_a_list_a @ H3 @ F ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) ) ) ) ) ) ) ).
% carrier_hom
thf(fact_910_carrier__hom,axiom,
! [F: list_list_a,R3: partia2670972154091845814t_unit,H3: list_a > list_list_a,S: partia2956882679547061052t_unit] :
( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( member6714375691612171394list_a @ H3 @ ( ring_i7582117978422105628t_unit @ R3 @ S ) )
=> ( ( domain6553523120543210313t_unit @ R3 )
=> ( ( domain7810152921033798211t_unit @ S )
=> ( member5342144027231129785list_a @ ( map_li5729356230488778442list_a @ H3 @ F ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ S @ ( partia2464479390973590831t_unit @ S ) ) ) ) ) ) ) ) ).
% carrier_hom
thf(fact_911_carrier__hom,axiom,
! [F: list_list_a,R3: partia2670972154091845814t_unit,H3: list_a > a,S: partia2175431115845679010xt_a_b] :
( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R3 @ S ) )
=> ( ( domain6553523120543210313t_unit @ R3 )
=> ( ( domain_a_b @ S )
=> ( member_list_a @ ( map_list_a_a @ H3 @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ) ).
% carrier_hom
thf(fact_912_carrier__hom,axiom,
! [F: list_list_a,R3: partia2670972154091845814t_unit,H3: list_a > list_a,S: partia2670972154091845814t_unit] :
( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R3 @ S ) )
=> ( ( domain6553523120543210313t_unit @ R3 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( member_list_list_a @ ( map_list_a_list_a @ H3 @ F ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) ) ) ) ) ) ) ).
% carrier_hom
thf(fact_913_alg__mult__gt__zero__iff__is__root,axiom,
! [P: list_a,X2: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P @ X2 ) )
= ( polyno4133073214067823460ot_a_b @ r @ P @ X2 ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_914_lift__iso__to__poly__ring,axiom,
! [H3: list_a > list_a,R3: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit] :
( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R3 @ S ) )
=> ( ( domain6553523120543210313t_unit @ R3 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( member8231385768148312316list_a @ ( map_list_a_list_a @ H3 ) @ ( ring_i6186174840089424918t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) ) ) ) ) ) ).
% lift_iso_to_poly_ring
thf(fact_915_lift__iso__to__poly__ring,axiom,
! [H3: list_a > a,R3: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R3 @ S ) )
=> ( ( domain6553523120543210313t_unit @ R3 )
=> ( ( domain_a_b @ S )
=> ( member7168557129179038582list_a @ ( map_list_a_a @ H3 ) @ ( ring_i4611353245267337884t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ).
% lift_iso_to_poly_ring
thf(fact_916_lift__iso__to__poly__ring,axiom,
! [H3: list_a > list_list_a,R3: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit] :
( ( member6714375691612171394list_a @ H3 @ ( ring_i7582117978422105628t_unit @ R3 @ S ) )
=> ( ( domain6553523120543210313t_unit @ R3 )
=> ( ( domain7810152921033798211t_unit @ S )
=> ( member8941091444255470082list_a @ ( map_li5729356230488778442list_a @ H3 ) @ ( ring_i5500208736156103184t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ ( univ_p2250591967980070728t_unit @ S @ ( partia2464479390973590831t_unit @ S ) ) ) ) ) ) ) ).
% lift_iso_to_poly_ring
thf(fact_917_lift__iso__to__poly__ring,axiom,
! [H3: a > list_a,R3: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R3 @ S ) )
=> ( ( domain_a_b @ R3 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( member6714375691612171394list_a @ ( map_a_list_a @ H3 ) @ ( ring_i7582117978422105628t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) ) ) ) ) ) ).
% lift_iso_to_poly_ring
thf(fact_918_lift__iso__to__poly__ring,axiom,
! [H3: a > a,R3: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R3 @ S ) )
=> ( ( domain_a_b @ R3 )
=> ( ( domain_a_b @ S )
=> ( member_list_a_list_a @ ( map_a_a @ H3 ) @ ( ring_i7414513579304222626t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ).
% lift_iso_to_poly_ring
thf(fact_919_lift__iso__to__poly__ring,axiom,
! [H3: a > list_list_a,R3: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit] :
( ( member_a_list_list_a @ H3 @ ( ring_i4464730343205239444t_unit @ R3 @ S ) )
=> ( ( domain_a_b @ R3 )
=> ( ( domain7810152921033798211t_unit @ S )
=> ( member7649053982783128200list_a @ ( map_a_list_list_a @ H3 ) @ ( ring_i8780440220419004694t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ ( univ_p2250591967980070728t_unit @ S @ ( partia2464479390973590831t_unit @ S ) ) ) ) ) ) ) ).
% lift_iso_to_poly_ring
thf(fact_920_lift__iso__to__poly__ring,axiom,
! [H3: list_list_a > list_a,R3: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit] :
( ( member7168557129179038582list_a @ H3 @ ( ring_i4611353245267337884t_unit @ R3 @ S ) )
=> ( ( domain7810152921033798211t_unit @ R3 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( member2200549709344965110list_a @ ( map_li1646474281249396926list_a @ H3 ) @ ( ring_i2841386107341831824t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) ) ) ) ) ) ).
% lift_iso_to_poly_ring
thf(fact_921_lift__iso__to__poly__ring,axiom,
! [H3: list_list_a > a,R3: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_list_list_a_a @ H3 @ ( ring_i5684343068699926420it_a_b @ R3 @ S ) )
=> ( ( domain7810152921033798211t_unit @ R3 )
=> ( ( domain_a_b @ S )
=> ( member3330872541585647984list_a @ ( map_list_list_a_a @ H3 ) @ ( ring_i3639271393462710806t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ).
% lift_iso_to_poly_ring
thf(fact_922_lift__iso__to__poly__ring,axiom,
! [H3: list_list_a > list_list_a,R3: partia2956882679547061052t_unit,S: partia2956882679547061052t_unit] :
( ( member8231385768148312316list_a @ H3 @ ( ring_i6186174840089424918t_unit @ R3 @ S ) )
=> ( ( domain7810152921033798211t_unit @ R3 )
=> ( ( domain7810152921033798211t_unit @ S )
=> ( member8456684049276653052list_a @ ( map_li8713736314956022724list_a @ H3 ) @ ( ring_i6414939324658939786t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ ( univ_p2250591967980070728t_unit @ S @ ( partia2464479390973590831t_unit @ S ) ) ) ) ) ) ) ).
% lift_iso_to_poly_ring
thf(fact_923_Group_Onat__pow__0,axiom,
! [G2: partia8223610829204095565t_unit,X2: a] :
( ( pow_a_1875594501834816709it_nat @ G2 @ X2 @ zero_zero_nat )
= ( one_a_Product_unit @ G2 ) ) ).
% Group.nat_pow_0
thf(fact_924_Group_Onat__pow__0,axiom,
! [G2: partia2175431115845679010xt_a_b,X2: a] :
( ( pow_a_1026414303147256608_b_nat @ G2 @ X2 @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ G2 ) ) ).
% Group.nat_pow_0
thf(fact_925_Group_Onat__pow__0,axiom,
! [G2: partia2670972154091845814t_unit,X2: list_a] :
( ( pow_li1142815632869257134it_nat @ G2 @ X2 @ zero_zero_nat )
= ( one_li8328186300101108157t_unit @ G2 ) ) ).
% Group.nat_pow_0
thf(fact_926_Group_Onat__pow__0,axiom,
! [G2: partia7496981018696276118t_unit,X2: set_list_a] :
( ( pow_se8252319793075206062it_nat @ G2 @ X2 @ zero_zero_nat )
= ( one_se1127990129394575805t_unit @ G2 ) ) ).
% Group.nat_pow_0
thf(fact_927_Units__pow__closed,axiom,
! [X2: a,D: nat] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ D ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_pow_closed
thf(fact_928_nat__pow__consistent,axiom,
! [X2: a,N: nat,H: set_a] :
( ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N )
= ( pow_a_1026414303147256608_b_nat
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r )
@ X2
@ N ) ) ).
% nat_pow_consistent
thf(fact_929_nat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( zero_a_b @ r ) @ N )
= ( zero_a_b @ r ) ) ) ).
% nat_pow_zero
thf(fact_930_pow__non__zero,axiom,
! [X2: a,N: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X2
!= ( zero_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N )
!= ( zero_a_b @ r ) ) ) ) ).
% pow_non_zero
thf(fact_931_pow__mult__distrib,axiom,
! [X2: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X2 ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_932_nat__pow__distrib,axiom,
! [X2: a,Y: a,N: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ).
% nat_pow_distrib
thf(fact_933_nat__pow__comm,axiom,
! [X2: a,N: nat,M: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ M ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ M ) @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_934_group__commutes__pow,axiom,
! [X2: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X2 ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_935_map__norm__in__poly__ring__carrier,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_list_a @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ).
% map_norm_in_poly_ring_carrier
thf(fact_936_map__in__poly__ring__carrier,axiom,
! [P: list_a,F: a > list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( F @ A4 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [A4: a] :
( ( A4
!= ( zero_a_b @ r ) )
=> ( ( F @ A4 )
!= nil_a ) )
=> ( member_list_list_a @ ( map_a_list_a @ F @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).
% map_in_poly_ring_carrier
thf(fact_937_nat__pow__closed,axiom,
! [X2: a,N: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% nat_pow_closed
thf(fact_938_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_939_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_940_local_Onat__pow__0,axiom,
! [X2: a] :
( ( pow_a_1026414303147256608_b_nat @ r @ X2 @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ r ) ) ).
% local.nat_pow_0
thf(fact_941_filter__preserves__multiset,axiom,
! [M2: a > nat,P2: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] : ( ord_less_nat @ zero_zero_nat @ ( M2 @ X ) ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X: a] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P2 @ X ) @ ( M2 @ X ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_942_filter__preserves__multiset,axiom,
! [M2: nat > nat,P2: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( M2 @ X ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P2 @ X ) @ ( M2 @ X ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_943_filter__preserves__multiset,axiom,
! [M2: complex > nat,P2: complex > $o] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( M2 @ X ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P2 @ X ) @ ( M2 @ X ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_944_filter__preserves__multiset,axiom,
! [M2: list_a > nat,P2: list_a > $o] :
( ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X: list_a] : ( ord_less_nat @ zero_zero_nat @ ( M2 @ X ) ) ) )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X: list_a] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P2 @ X ) @ ( M2 @ X ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_945_set__mset__def,axiom,
( set_mset_nat
= ( ^ [M4: multiset_nat] :
( collect_nat
@ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( count_nat @ M4 @ X ) ) ) ) ) ).
% set_mset_def
thf(fact_946_set__mset__def,axiom,
( set_mset_complex
= ( ^ [M4: multiset_complex] :
( collect_complex
@ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( count_complex @ M4 @ X ) ) ) ) ) ).
% set_mset_def
thf(fact_947_set__mset__def,axiom,
( set_mset_list_a
= ( ^ [M4: multiset_list_a] :
( collect_list_a
@ ^ [X: list_a] : ( ord_less_nat @ zero_zero_nat @ ( count_list_a @ M4 @ X ) ) ) ) ) ).
% set_mset_def
thf(fact_948_set__mset__def,axiom,
( set_mset_a
= ( ^ [M4: multiset_a] :
( collect_a
@ ^ [X: a] : ( ord_less_nat @ zero_zero_nat @ ( count_a @ M4 @ X ) ) ) ) ) ).
% set_mset_def
thf(fact_949_count__induct,axiom,
! [Y: nat > nat,P2: ( nat > nat ) > $o] :
( ( member_nat_nat @ Y
@ ( collect_nat_nat
@ ^ [F2: nat > nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( F2 @ X ) ) ) ) ) )
=> ( ! [X3: multiset_nat] : ( P2 @ ( count_nat @ X3 ) )
=> ( P2 @ Y ) ) ) ).
% count_induct
thf(fact_950_count__induct,axiom,
! [Y: complex > nat,P2: ( complex > nat ) > $o] :
( ( member_complex_nat @ Y
@ ( collect_complex_nat
@ ^ [F2: complex > nat] :
( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( F2 @ X ) ) ) ) ) )
=> ( ! [X3: multiset_complex] : ( P2 @ ( count_complex @ X3 ) )
=> ( P2 @ Y ) ) ) ).
% count_induct
thf(fact_951_count__induct,axiom,
! [Y: list_a > nat,P2: ( list_a > nat ) > $o] :
( ( member_list_a_nat @ Y
@ ( collect_list_a_nat
@ ^ [F2: list_a > nat] :
( finite_finite_list_a
@ ( collect_list_a
@ ^ [X: list_a] : ( ord_less_nat @ zero_zero_nat @ ( F2 @ X ) ) ) ) ) )
=> ( ! [X3: multiset_list_a] : ( P2 @ ( count_list_a @ X3 ) )
=> ( P2 @ Y ) ) ) ).
% count_induct
thf(fact_952_count__induct,axiom,
! [Y: a > nat,P2: ( a > nat ) > $o] :
( ( member_a_nat @ Y
@ ( collect_a_nat
@ ^ [F2: a > nat] :
( finite_finite_a
@ ( collect_a
@ ^ [X: a] : ( ord_less_nat @ zero_zero_nat @ ( F2 @ X ) ) ) ) ) )
=> ( ! [X3: multiset_a] : ( P2 @ ( count_a @ X3 ) )
=> ( P2 @ Y ) ) ) ).
% count_induct
thf(fact_953_count__cases,axiom,
! [Y: nat > nat] :
( ( member_nat_nat @ Y
@ ( collect_nat_nat
@ ^ [F2: nat > nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( F2 @ X ) ) ) ) ) )
=> ~ ! [X3: multiset_nat] :
( Y
!= ( count_nat @ X3 ) ) ) ).
% count_cases
thf(fact_954_count__cases,axiom,
! [Y: complex > nat] :
( ( member_complex_nat @ Y
@ ( collect_complex_nat
@ ^ [F2: complex > nat] :
( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( F2 @ X ) ) ) ) ) )
=> ~ ! [X3: multiset_complex] :
( Y
!= ( count_complex @ X3 ) ) ) ).
% count_cases
thf(fact_955_count__cases,axiom,
! [Y: list_a > nat] :
( ( member_list_a_nat @ Y
@ ( collect_list_a_nat
@ ^ [F2: list_a > nat] :
( finite_finite_list_a
@ ( collect_list_a
@ ^ [X: list_a] : ( ord_less_nat @ zero_zero_nat @ ( F2 @ X ) ) ) ) ) )
=> ~ ! [X3: multiset_list_a] :
( Y
!= ( count_list_a @ X3 ) ) ) ).
% count_cases
thf(fact_956_count__cases,axiom,
! [Y: a > nat] :
( ( member_a_nat @ Y
@ ( collect_a_nat
@ ^ [F2: a > nat] :
( finite_finite_a
@ ( collect_a
@ ^ [X: a] : ( ord_less_nat @ zero_zero_nat @ ( F2 @ X ) ) ) ) ) )
=> ~ ! [X3: multiset_a] :
( Y
!= ( count_a @ X3 ) ) ) ).
% count_cases
thf(fact_957_count,axiom,
! [X2: multiset_nat] :
( member_nat_nat @ ( count_nat @ X2 )
@ ( collect_nat_nat
@ ^ [F2: nat > nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( F2 @ X ) ) ) ) ) ) ).
% count
thf(fact_958_count,axiom,
! [X2: multiset_complex] :
( member_complex_nat @ ( count_complex @ X2 )
@ ( collect_complex_nat
@ ^ [F2: complex > nat] :
( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( F2 @ X ) ) ) ) ) ) ).
% count
thf(fact_959_count,axiom,
! [X2: multiset_list_a] :
( member_list_a_nat @ ( count_list_a @ X2 )
@ ( collect_list_a_nat
@ ^ [F2: list_a > nat] :
( finite_finite_list_a
@ ( collect_list_a
@ ^ [X: list_a] : ( ord_less_nat @ zero_zero_nat @ ( F2 @ X ) ) ) ) ) ) ).
% count
thf(fact_960_count,axiom,
! [X2: multiset_a] :
( member_a_nat @ ( count_a @ X2 )
@ ( collect_a_nat
@ ^ [F2: a > nat] :
( finite_finite_a
@ ( collect_a
@ ^ [X: a] : ( ord_less_nat @ zero_zero_nat @ ( F2 @ X ) ) ) ) ) ) ).
% count
thf(fact_961_domain_Omap__norm__in__poly__ring__carrier,axiom,
! [R3: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R3 )
=> ( ( subring_a_b @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) )
=> ( member_list_list_a @ ( map_a_list_a @ ( poly_of_const_a_b @ R3 ) @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ R3 @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K ) ) ) ) ) ) ) ) ).
% domain.map_norm_in_poly_ring_carrier
thf(fact_962_domain_Omap__norm__in__poly__ring__carrier,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subrin6918843898125473962t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( member5342144027231129785list_a @ ( map_li5729356230488778442list_a @ ( poly_o8716471131768098070t_unit @ R3 ) @ P ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) ) ) ) ) ) ) ).
% domain.map_norm_in_poly_ring_carrier
thf(fact_963_domain_Omap__in__poly__ring__carrier,axiom,
! [R3: partia7496981018696276118t_unit,P: list_set_list_a,F: set_list_a > list_set_list_a] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) ) )
=> ( ! [A4: set_list_a] :
( ( member_set_list_a @ A4 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( member5524387281408368019list_a @ ( F @ A4 ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) ) ) )
=> ( ! [A4: set_list_a] :
( ( A4
!= ( zero_s2910681146719230829t_unit @ R3 ) )
=> ( ( F @ A4 )
!= nil_set_list_a ) )
=> ( member352051402189872281list_a @ ( map_se1776605471917444810list_a @ F @ P ) @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) ) ) ) ) ) ) ) ) ).
% domain.map_in_poly_ring_carrier
thf(fact_964_domain_Omap__in__poly__ring__carrier,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a,F: list_list_a > list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ! [A4: list_list_a] :
( ( member_list_list_a @ A4 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( member5342144027231129785list_a @ ( F @ A4 ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) ) )
=> ( ! [A4: list_list_a] :
( ( A4
!= ( zero_l347298301471573063t_unit @ R3 ) )
=> ( ( F @ A4 )
!= nil_list_list_a ) )
=> ( member6842060177613954879list_a @ ( map_li5227692475714150986list_a @ F @ P ) @ ( partia2412307164297199803t_unit @ ( univ_p3766428211910075458t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) ) ) ) ) ) ) ) ).
% domain.map_in_poly_ring_carrier
thf(fact_965_domain_Omap__in__poly__ring__carrier,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a,F: a > list_a] :
( ( domain_a_b @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_list_a @ ( F @ A4 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) ) )
=> ( ! [A4: a] :
( ( A4
!= ( zero_a_b @ R3 ) )
=> ( ( F @ A4 )
!= nil_a ) )
=> ( member_list_list_a @ ( map_a_list_a @ F @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) ) ) ) ) ) ) ) ).
% domain.map_in_poly_ring_carrier
thf(fact_966_domain_Omap__in__poly__ring__carrier,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a,F: list_a > list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ! [A4: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_list_list_a @ ( F @ A4 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) ) )
=> ( ! [A4: list_a] :
( ( A4
!= ( zero_l4142658623432671053t_unit @ R3 ) )
=> ( ( F @ A4 )
!= nil_list_a ) )
=> ( member5342144027231129785list_a @ ( map_li5729356230488778442list_a @ F @ P ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) ) ) ) ) ) ) ) ).
% domain.map_in_poly_ring_carrier
thf(fact_967_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a,X2: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno1672195411705137432t_unit @ R3 @ P @ X2 ) )
= ( polyno5142720416380192742t_unit @ R3 @ P @ X2 ) ) ) ) ).
% domain.alg_mult_gt_zero_iff_is_root
thf(fact_968_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a,X2: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4259638811958763678t_unit @ R3 @ P @ X2 ) )
= ( polyno6951661231331188332t_unit @ R3 @ P @ X2 ) ) ) ) ).
% domain.alg_mult_gt_zero_iff_is_root
thf(fact_969_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
! [R3: partia2175431115845679010xt_a_b,P: list_a,X2: a] :
( ( domain_a_b @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ R3 @ P @ X2 ) )
= ( polyno4133073214067823460ot_a_b @ R3 @ P @ X2 ) ) ) ) ).
% domain.alg_mult_gt_zero_iff_is_root
thf(fact_970_domain_Opow__non__zero,axiom,
! [R3: partia7496981018696276118t_unit,X2: set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R3 )
=> ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( X2
!= ( zero_s2910681146719230829t_unit @ R3 ) )
=> ( ( pow_se8252319793075206062it_nat @ R3 @ X2 @ N )
!= ( zero_s2910681146719230829t_unit @ R3 ) ) ) ) ) ).
% domain.pow_non_zero
thf(fact_971_domain_Opow__non__zero,axiom,
! [R3: partia2670972154091845814t_unit,X2: list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( X2
!= ( zero_l4142658623432671053t_unit @ R3 ) )
=> ( ( pow_li1142815632869257134it_nat @ R3 @ X2 @ N )
!= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ) ) ).
% domain.pow_non_zero
thf(fact_972_domain_Opow__non__zero,axiom,
! [R3: partia2175431115845679010xt_a_b,X2: a,N: nat] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( X2
!= ( zero_a_b @ R3 ) )
=> ( ( pow_a_1026414303147256608_b_nat @ R3 @ X2 @ N )
!= ( zero_a_b @ R3 ) ) ) ) ) ).
% domain.pow_non_zero
thf(fact_973_domain_Opow__non__zero,axiom,
! [R3: partia2956882679547061052t_unit,X2: list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( X2
!= ( zero_l347298301471573063t_unit @ R3 ) )
=> ( ( pow_li488931774710091566it_nat @ R3 @ X2 @ N )
!= ( zero_l347298301471573063t_unit @ R3 ) ) ) ) ) ).
% domain.pow_non_zero
thf(fact_974_domain_Odiv__sum,axiom,
! [R3: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( factor1757716651909850160t_unit @ R3 @ A @ B )
=> ( ( factor1757716651909850160t_unit @ R3 @ A @ C )
=> ( factor1757716651909850160t_unit @ R3 @ A @ ( add_li7652885771158616974t_unit @ R3 @ B @ C ) ) ) ) ) ) ) ) ).
% domain.div_sum
thf(fact_975_domain_Odiv__sum,axiom,
! [R3: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( factor8216151070175719842xt_a_b @ R3 @ A @ B )
=> ( ( factor8216151070175719842xt_a_b @ R3 @ A @ C )
=> ( factor8216151070175719842xt_a_b @ R3 @ A @ ( add_a_b @ R3 @ B @ C ) ) ) ) ) ) ) ) ).
% domain.div_sum
thf(fact_976_domain_Odiv__sum,axiom,
! [R3: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,C: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( factor6954119973539764400t_unit @ R3 @ A @ B )
=> ( ( factor6954119973539764400t_unit @ R3 @ A @ C )
=> ( factor6954119973539764400t_unit @ R3 @ A @ ( add_li174743652000525320t_unit @ R3 @ B @ C ) ) ) ) ) ) ) ) ).
% domain.div_sum
thf(fact_977_domain_Odiv__sum__iff,axiom,
! [R3: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( factor1757716651909850160t_unit @ R3 @ A @ B )
=> ( ( factor1757716651909850160t_unit @ R3 @ A @ ( add_li7652885771158616974t_unit @ R3 @ B @ C ) )
= ( factor1757716651909850160t_unit @ R3 @ A @ C ) ) ) ) ) ) ) ).
% domain.div_sum_iff
thf(fact_978_domain_Odiv__sum__iff,axiom,
! [R3: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( factor8216151070175719842xt_a_b @ R3 @ A @ B )
=> ( ( factor8216151070175719842xt_a_b @ R3 @ A @ ( add_a_b @ R3 @ B @ C ) )
= ( factor8216151070175719842xt_a_b @ R3 @ A @ C ) ) ) ) ) ) ) ).
% domain.div_sum_iff
thf(fact_979_domain_Odiv__sum__iff,axiom,
! [R3: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,C: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( factor6954119973539764400t_unit @ R3 @ A @ B )
=> ( ( factor6954119973539764400t_unit @ R3 @ A @ ( add_li174743652000525320t_unit @ R3 @ B @ C ) )
= ( factor6954119973539764400t_unit @ R3 @ A @ C ) ) ) ) ) ) ) ).
% domain.div_sum_iff
thf(fact_980_domain_Odiv__neg,axiom,
! [R3: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( factor1757716651909850160t_unit @ R3 @ A @ B )
=> ( factor1757716651909850160t_unit @ R3 @ A @ ( a_inv_8944721093294617173t_unit @ R3 @ B ) ) ) ) ) ) ).
% domain.div_neg
thf(fact_981_domain_Odiv__neg,axiom,
! [R3: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( factor8216151070175719842xt_a_b @ R3 @ A @ B )
=> ( factor8216151070175719842xt_a_b @ R3 @ A @ ( a_inv_a_b @ R3 @ B ) ) ) ) ) ) ).
% domain.div_neg
thf(fact_982_domain_Odiv__neg,axiom,
! [R3: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( factor6954119973539764400t_unit @ R3 @ A @ B )
=> ( factor6954119973539764400t_unit @ R3 @ A @ ( a_inv_7033018035630854991t_unit @ R3 @ B ) ) ) ) ) ) ).
% domain.div_neg
thf(fact_983_Units__hom,axiom,
! [H3: list_a > a,R3: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X2: list_a] :
( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R3 @ S ) )
=> ( ( domain6553523120543210313t_unit @ R3 )
=> ( ( domain_a_b @ S )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ R3 ) )
= ( member_a @ ( H3 @ X2 ) @ ( units_a_ring_ext_a_b @ S ) ) ) ) ) ) ) ).
% Units_hom
thf(fact_984_Units__hom,axiom,
! [H3: list_a > list_a,R3: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X2: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R3 @ S ) )
=> ( ( domain6553523120543210313t_unit @ R3 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ R3 ) )
= ( member_list_a @ ( H3 @ X2 ) @ ( units_2932844235741507942t_unit @ S ) ) ) ) ) ) ) ).
% Units_hom
thf(fact_985_Units__hom,axiom,
! [H3: a > a,R3: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X2: a] :
( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R3 @ S ) )
=> ( ( domain_a_b @ R3 )
=> ( ( domain_a_b @ S )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ R3 ) )
= ( member_a @ ( H3 @ X2 ) @ ( units_a_ring_ext_a_b @ S ) ) ) ) ) ) ) ).
% Units_hom
thf(fact_986_Units__hom,axiom,
! [H3: a > list_a,R3: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X2: a] :
( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R3 @ S ) )
=> ( ( domain_a_b @ R3 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ R3 ) )
= ( member_list_a @ ( H3 @ X2 ) @ ( units_2932844235741507942t_unit @ S ) ) ) ) ) ) ) ).
% Units_hom
thf(fact_987_Units__hom,axiom,
! [H3: list_list_a > a,R3: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X2: list_list_a] :
( ( member_list_list_a_a @ H3 @ ( ring_i5684343068699926420it_a_b @ R3 @ S ) )
=> ( ( domain7810152921033798211t_unit @ R3 )
=> ( ( domain_a_b @ S )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ X2 @ ( units_4903515905731149798t_unit @ R3 ) )
= ( member_a @ ( H3 @ X2 ) @ ( units_a_ring_ext_a_b @ S ) ) ) ) ) ) ) ).
% Units_hom
thf(fact_988_Units__hom,axiom,
! [H3: list_list_a > list_a,R3: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X2: list_list_a] :
( ( member7168557129179038582list_a @ H3 @ ( ring_i4611353245267337884t_unit @ R3 @ S ) )
=> ( ( domain7810152921033798211t_unit @ R3 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ X2 @ ( units_4903515905731149798t_unit @ R3 ) )
= ( member_list_a @ ( H3 @ X2 ) @ ( units_2932844235741507942t_unit @ S ) ) ) ) ) ) ) ).
% Units_hom
thf(fact_989_divides__hom,axiom,
! [H3: list_a > list_a,R3: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R3 @ S ) )
=> ( ( domain6553523120543210313t_unit @ R3 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( factor1757716651909850160t_unit @ R3 @ X2 @ Y )
= ( factor1757716651909850160t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ) ) ).
% divides_hom
thf(fact_990_divides__hom,axiom,
! [H3: list_a > a,R3: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X2: list_a,Y: list_a] :
( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R3 @ S ) )
=> ( ( domain6553523120543210313t_unit @ R3 )
=> ( ( domain_a_b @ S )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( factor1757716651909850160t_unit @ R3 @ X2 @ Y )
= ( factor8216151070175719842xt_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ) ) ).
% divides_hom
thf(fact_991_divides__hom,axiom,
! [H3: a > list_a,R3: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X2: a,Y: a] :
( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R3 @ S ) )
=> ( ( domain_a_b @ R3 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( factor8216151070175719842xt_a_b @ R3 @ X2 @ Y )
= ( factor1757716651909850160t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ) ) ).
% divides_hom
thf(fact_992_divides__hom,axiom,
! [H3: a > a,R3: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R3 @ S ) )
=> ( ( domain_a_b @ R3 )
=> ( ( domain_a_b @ S )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( factor8216151070175719842xt_a_b @ R3 @ X2 @ Y )
= ( factor8216151070175719842xt_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ) ) ).
% divides_hom
thf(fact_993_divides__hom,axiom,
! [H3: list_list_a > list_a,R3: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X2: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H3 @ ( ring_i4611353245267337884t_unit @ R3 @ S ) )
=> ( ( domain7810152921033798211t_unit @ R3 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( factor6954119973539764400t_unit @ R3 @ X2 @ Y )
= ( factor1757716651909850160t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ) ) ).
% divides_hom
thf(fact_994_divides__hom,axiom,
! [H3: list_list_a > a,R3: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X2: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H3 @ ( ring_i5684343068699926420it_a_b @ R3 @ S ) )
=> ( ( domain7810152921033798211t_unit @ R3 )
=> ( ( domain_a_b @ S )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( factor6954119973539764400t_unit @ R3 @ X2 @ Y )
= ( factor8216151070175719842xt_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y ) ) ) ) ) ) ) ) ).
% divides_hom
thf(fact_995_units__of__pow,axiom,
! [X2: a,N: nat] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( units_8174867845824275201xt_a_b @ r ) @ X2 @ N )
= ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) ) ) ).
% units_of_pow
thf(fact_996_boundD__carrier,axiom,
! [N: nat,F: nat > a,M: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_a @ ( F @ M ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_997_length__greater__0__conv,axiom,
! [Xs2: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs2 ) )
= ( Xs2 != nil_a ) ) ).
% length_greater_0_conv
thf(fact_998_order__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_a_ring_ext_a_b @ r ) )
= ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% order_gt_0_iff_finite
thf(fact_999_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_1000_map__ident,axiom,
( ( map_a_a
@ ^ [X: a] : X )
= ( ^ [Xs3: list_a] : Xs3 ) ) ).
% map_ident
thf(fact_1001_list_Omap__disc__iff,axiom,
! [F: a > a,A: list_a] :
( ( ( map_a_a @ F @ A )
= nil_a )
= ( A = nil_a ) ) ).
% list.map_disc_iff
thf(fact_1002_list_Omap__disc__iff,axiom,
! [F: a > list_a,A: list_a] :
( ( ( map_a_list_a @ F @ A )
= nil_list_a )
= ( A = nil_a ) ) ).
% list.map_disc_iff
thf(fact_1003_Nil__is__map__conv,axiom,
! [F: a > a,Xs2: list_a] :
( ( nil_a
= ( map_a_a @ F @ Xs2 ) )
= ( Xs2 = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_1004_Nil__is__map__conv,axiom,
! [F: a > list_a,Xs2: list_a] :
( ( nil_list_a
= ( map_a_list_a @ F @ Xs2 ) )
= ( Xs2 = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_1005_map__is__Nil__conv,axiom,
! [F: a > a,Xs2: list_a] :
( ( ( map_a_a @ F @ Xs2 )
= nil_a )
= ( Xs2 = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_1006_map__is__Nil__conv,axiom,
! [F: a > list_a,Xs2: list_a] :
( ( ( map_a_list_a @ F @ Xs2 )
= nil_list_a )
= ( Xs2 = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_1007_units__of__units,axiom,
! [G2: partia2670972154091845814t_unit] :
( ( units_8735880885477018085t_unit @ ( units_6477118173342999439t_unit @ G2 ) )
= ( units_2932844235741507942t_unit @ G2 ) ) ).
% units_of_units
thf(fact_1008_units__of__units,axiom,
! [G2: partia2175431115845679010xt_a_b] :
( ( units_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G2 ) )
= ( units_a_ring_ext_a_b @ G2 ) ) ).
% units_of_units
thf(fact_1009_length__0__conv,axiom,
! [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_a ) ) ).
% length_0_conv
thf(fact_1010_units__of__mult,axiom,
! [G2: partia2670972154091845814t_unit] :
( ( mult_l6995149843440949818t_unit @ ( units_6477118173342999439t_unit @ G2 ) )
= ( mult_l7073676228092353617t_unit @ G2 ) ) ).
% units_of_mult
thf(fact_1011_units__of__mult,axiom,
! [G2: partia2175431115845679010xt_a_b] :
( ( mult_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G2 ) )
= ( mult_a_ring_ext_a_b @ G2 ) ) ).
% units_of_mult
thf(fact_1012_units__of__carrier,axiom,
! [G2: partia2670972154091845814t_unit] :
( ( partia7074150537345710456t_unit @ ( units_6477118173342999439t_unit @ G2 ) )
= ( units_2932844235741507942t_unit @ G2 ) ) ).
% units_of_carrier
thf(fact_1013_units__of__carrier,axiom,
! [G2: partia2175431115845679010xt_a_b] :
( ( partia6735698275553448452t_unit @ ( units_8174867845824275201xt_a_b @ G2 ) )
= ( units_a_ring_ext_a_b @ G2 ) ) ).
% units_of_carrier
thf(fact_1014_not__Cons__self2,axiom,
! [X2: a,Xs2: list_a] :
( ( cons_a @ X2 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_1015_list_Omap__ident,axiom,
! [T3: list_a] :
( ( map_a_a
@ ^ [X: a] : X
@ T3 )
= T3 ) ).
% list.map_ident
thf(fact_1016_list__nonempty__induct,axiom,
! [Xs2: list_a,P2: list_a > $o] :
( ( Xs2 != nil_a )
=> ( ! [X3: a] : ( P2 @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs: list_a] :
( ( Xs != nil_a )
=> ( ( P2 @ Xs )
=> ( P2 @ ( cons_a @ X3 @ Xs ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_1017_list__induct2_H,axiom,
! [P2: list_a > list_a > $o,Xs2: list_a,Ys2: list_a] :
( ( P2 @ nil_a @ nil_a )
=> ( ! [X3: a,Xs: list_a] : ( P2 @ ( cons_a @ X3 @ Xs ) @ nil_a )
=> ( ! [Y5: a,Ys: list_a] : ( P2 @ nil_a @ ( cons_a @ Y5 @ Ys ) )
=> ( ! [X3: a,Xs: list_a,Y5: a,Ys: list_a] :
( ( P2 @ Xs @ Ys )
=> ( P2 @ ( cons_a @ X3 @ Xs ) @ ( cons_a @ Y5 @ Ys ) ) )
=> ( P2 @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_1018_neq__Nil__conv,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
= ( ? [Y3: a,Ys3: list_a] :
( Xs2
= ( cons_a @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1019_remdups__adj_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ( ! [X3: a] :
( X2
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y5: a,Xs: list_a] :
( X2
!= ( cons_a @ X3 @ ( cons_a @ Y5 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1020_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_1021_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_1022_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_1023_transpose_Ocases,axiom,
! [X2: list_list_a] :
( ( X2 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X2
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X3: a,Xs: list_a,Xss: list_list_a] :
( X2
!= ( cons_list_a @ ( cons_a @ X3 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_1024_map__eq__Cons__conv,axiom,
! [F: a > list_a,Xs2: list_a,Y: list_a,Ys2: list_list_a] :
( ( ( map_a_list_a @ F @ Xs2 )
= ( cons_list_a @ Y @ Ys2 ) )
= ( ? [Z3: a,Zs: list_a] :
( ( Xs2
= ( cons_a @ Z3 @ Zs ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_a_list_a @ F @ Zs )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1025_map__eq__Cons__conv,axiom,
! [F: a > a,Xs2: list_a,Y: a,Ys2: list_a] :
( ( ( map_a_a @ F @ Xs2 )
= ( cons_a @ Y @ Ys2 ) )
= ( ? [Z3: a,Zs: list_a] :
( ( Xs2
= ( cons_a @ Z3 @ Zs ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_a_a @ F @ Zs )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1026_Cons__eq__map__conv,axiom,
! [X2: list_a,Xs2: list_list_a,F: a > list_a,Ys2: list_a] :
( ( ( cons_list_a @ X2 @ Xs2 )
= ( map_a_list_a @ F @ Ys2 ) )
= ( ? [Z3: a,Zs: list_a] :
( ( Ys2
= ( cons_a @ Z3 @ Zs ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs2
= ( map_a_list_a @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1027_Cons__eq__map__conv,axiom,
! [X2: a,Xs2: list_a,F: a > a,Ys2: list_a] :
( ( ( cons_a @ X2 @ Xs2 )
= ( map_a_a @ F @ Ys2 ) )
= ( ? [Z3: a,Zs: list_a] :
( ( Ys2
= ( cons_a @ Z3 @ Zs ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs2
= ( map_a_a @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1028_map__eq__Cons__D,axiom,
! [F: a > list_a,Xs2: list_a,Y: list_a,Ys2: list_list_a] :
( ( ( map_a_list_a @ F @ Xs2 )
= ( cons_list_a @ Y @ Ys2 ) )
=> ? [Z4: a,Zs2: list_a] :
( ( Xs2
= ( cons_a @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_a_list_a @ F @ Zs2 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_1029_map__eq__Cons__D,axiom,
! [F: a > a,Xs2: list_a,Y: a,Ys2: list_a] :
( ( ( map_a_a @ F @ Xs2 )
= ( cons_a @ Y @ Ys2 ) )
=> ? [Z4: a,Zs2: list_a] :
( ( Xs2
= ( cons_a @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_a_a @ F @ Zs2 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_1030_Cons__eq__map__D,axiom,
! [X2: list_a,Xs2: list_list_a,F: a > list_a,Ys2: list_a] :
( ( ( cons_list_a @ X2 @ Xs2 )
= ( map_a_list_a @ F @ Ys2 ) )
=> ? [Z4: a,Zs2: list_a] :
( ( Ys2
= ( cons_a @ Z4 @ Zs2 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs2
= ( map_a_list_a @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1031_Cons__eq__map__D,axiom,
! [X2: a,Xs2: list_a,F: a > a,Ys2: list_a] :
( ( ( cons_a @ X2 @ Xs2 )
= ( map_a_a @ F @ Ys2 ) )
=> ? [Z4: a,Zs2: list_a] :
( ( Ys2
= ( cons_a @ Z4 @ Zs2 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs2
= ( map_a_a @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1032_list_Osimps_I9_J,axiom,
! [F: a > list_a,X21: a,X22: list_a] :
( ( map_a_list_a @ F @ ( cons_a @ X21 @ X22 ) )
= ( cons_list_a @ ( F @ X21 ) @ ( map_a_list_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1033_list_Osimps_I9_J,axiom,
! [F: a > a,X21: a,X22: list_a] :
( ( map_a_a @ F @ ( cons_a @ X21 @ X22 ) )
= ( cons_a @ ( F @ X21 ) @ ( map_a_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1034_list_Osimps_I8_J,axiom,
! [F: a > a] :
( ( map_a_a @ F @ nil_a )
= nil_a ) ).
% list.simps(8)
thf(fact_1035_list_Osimps_I8_J,axiom,
! [F: a > list_a] :
( ( map_a_list_a @ F @ nil_a )
= nil_list_a ) ).
% list.simps(8)
thf(fact_1036_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1037_list__induct4,axiom,
! [Xs2: list_a,Ys2: list_a,Zs3: list_a,Ws: list_a,P2: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs: list_a,Y5: a,Ys: list_a,Z4: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs @ Ys @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs ) @ ( cons_a @ Y5 @ Ys ) @ ( cons_a @ Z4 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1038_list__induct3,axiom,
! [Xs2: list_a,Ys2: list_a,Zs3: list_a,P2: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs: list_a,Y5: a,Ys: list_a,Z4: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs @ Ys @ Zs2 )
=> ( P2 @ ( cons_a @ X3 @ Xs ) @ ( cons_a @ Y5 @ Ys ) @ ( cons_a @ Z4 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys2 @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1039_list__induct2,axiom,
! [Xs2: list_a,Ys2: list_a,P2: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P2 @ nil_a @ nil_a )
=> ( ! [X3: a,Xs: list_a,Y5: a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ Xs @ Ys )
=> ( P2 @ ( cons_a @ X3 @ Xs ) @ ( cons_a @ Y5 @ Ys ) ) ) )
=> ( P2 @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_1040_impossible__Cons,axiom,
! [Xs2: list_a,Ys2: list_a,X2: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ ( size_size_list_a @ Ys2 ) )
=> ( Xs2
!= ( cons_a @ X2 @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_1041_shuffles_Ocases,axiom,
! [X2: produc9164743771328383783list_a] :
( ! [Ys: list_a] :
( X2
!= ( produc6837034575241423639list_a @ nil_a @ Ys ) )
=> ( ! [Xs: list_a] :
( X2
!= ( produc6837034575241423639list_a @ Xs @ nil_a ) )
=> ~ ! [X3: a,Xs: list_a,Y5: a,Ys: list_a] :
( X2
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs ) @ ( cons_a @ Y5 @ Ys ) ) ) ) ) ).
% shuffles.cases
thf(fact_1042_norm__map__in__poly__ring__carrier,axiom,
! [P: list_a,F: a > list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( F @ A4 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( map_a_list_a @ F @ P ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% norm_map_in_poly_ring_carrier
thf(fact_1043_num__roots__le__deg,axiom,
! [D: nat] :
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( D != zero_zero_nat )
=> ( ord_less_eq_nat
@ ( finite_card_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( pow_a_1026414303147256608_b_nat @ r @ X @ D )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
@ D ) ) ) ).
% num_roots_le_deg
thf(fact_1044_nunit__factors,axiom,
! [A: a,As: list_a] :
( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( factor5638265376665762323xt_a_b @ r @ As @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ As ) ) ) ) ).
% nunit_factors
thf(fact_1045_freshmans__dream,axiom,
! [X2: a,Y: a] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( add_a_b @ r @ X2 @ Y ) @ ( ring_char_a_b @ r ) )
= ( add_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ ( ring_char_a_b @ r ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ ( ring_char_a_b @ r ) ) ) ) ) ) ) ).
% freshmans_dream
thf(fact_1046_char__consistent,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( ( ring_char_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r ) )
= ( ring_char_a_b @ r ) ) ) ).
% char_consistent
thf(fact_1047_finite__carr__imp__char__ge__0,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) ) ) ).
% finite_carr_imp_char_ge_0
thf(fact_1048_units__power__order__eq__one,axiom,
! [A: a] :
( ( finite_finite_a @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ A @ ( finite_card_a @ ( units_a_ring_ext_a_b @ r ) ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% units_power_order_eq_one
thf(fact_1049_card_Oempty,axiom,
( ( finite_card_complex @ bot_bot_set_complex )
= zero_zero_nat ) ).
% card.empty
thf(fact_1050_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_1051_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_1052_card_Oempty,axiom,
( ( finite_card_list_a @ bot_bot_set_list_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_1053_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1054_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1055_card_Oinfinite,axiom,
! [A2: set_complex] :
( ~ ( finite3207457112153483333omplex @ A2 )
=> ( ( finite_card_complex @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1056_card_Oinfinite,axiom,
! [A2: set_list_a] :
( ~ ( finite_finite_list_a @ A2 )
=> ( ( finite_card_list_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1057_card__0__eq,axiom,
! [A2: set_complex] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( ( finite_card_complex @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_complex ) ) ) ).
% card_0_eq
thf(fact_1058_card__0__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_1059_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_1060_card__0__eq,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( ( finite_card_list_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_list_a ) ) ) ).
% card_0_eq
thf(fact_1061_ring_Onormalize_Ocong,axiom,
normal637505603836502915t_unit = normal637505603836502915t_unit ).
% ring.normalize.cong
thf(fact_1062_ring_Onormalize_Ocong,axiom,
normalize_a_b = normalize_a_b ).
% ring.normalize.cong
thf(fact_1063_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N )
& ( ord_less_eq_set_nat @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1064_infinite__arbitrarily__large,axiom,
! [A2: set_complex,N: nat] :
( ~ ( finite3207457112153483333omplex @ A2 )
=> ? [B7: set_complex] :
( ( finite3207457112153483333omplex @ B7 )
& ( ( finite_card_complex @ B7 )
= N )
& ( ord_le211207098394363844omplex @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1065_infinite__arbitrarily__large,axiom,
! [A2: set_list_a,N: nat] :
( ~ ( finite_finite_list_a @ A2 )
=> ? [B7: set_list_a] :
( ( finite_finite_list_a @ B7 )
& ( ( finite_card_list_a @ B7 )
= N )
& ( ord_le8861187494160871172list_a @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1066_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B7: set_a] :
( ( finite_finite_a @ B7 )
& ( ( finite_card_a @ B7 )
= N )
& ( ord_less_eq_set_a @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1067_card__subset__eq,axiom,
! [B5: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B5 ) )
=> ( A2 = B5 ) ) ) ) ).
% card_subset_eq
thf(fact_1068_card__subset__eq,axiom,
! [B5: set_complex,A2: set_complex] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B5 )
=> ( ( ( finite_card_complex @ A2 )
= ( finite_card_complex @ B5 ) )
=> ( A2 = B5 ) ) ) ) ).
% card_subset_eq
thf(fact_1069_card__subset__eq,axiom,
! [B5: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B5 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B5 )
=> ( ( ( finite_card_list_a @ A2 )
= ( finite_card_list_a @ B5 ) )
=> ( A2 = B5 ) ) ) ) ).
% card_subset_eq
thf(fact_1070_card__subset__eq,axiom,
! [B5: set_a,A2: set_a] :
( ( finite_finite_a @ B5 )
=> ( ( ord_less_eq_set_a @ A2 @ B5 )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B5 ) )
=> ( A2 = B5 ) ) ) ) ).
% card_subset_eq
thf(fact_1071_card__eq__0__iff,axiom,
! [A2: set_complex] :
( ( ( finite_card_complex @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_complex )
| ~ ( finite3207457112153483333omplex @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_1072_card__eq__0__iff,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_a )
| ~ ( finite_finite_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_1073_card__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_1074_card__eq__0__iff,axiom,
! [A2: set_list_a] :
( ( ( finite_card_list_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_list_a )
| ~ ( finite_finite_list_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_1075_card__ge__0__finite,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ( finite_finite_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_1076_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_1077_card__ge__0__finite,axiom,
! [A2: set_complex] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A2 ) )
=> ( finite3207457112153483333omplex @ A2 ) ) ).
% card_ge_0_finite
thf(fact_1078_card__ge__0__finite,axiom,
! [A2: set_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_a @ A2 ) )
=> ( finite_finite_list_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_1079_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_nat,C3: nat] :
( ! [G4: set_nat] :
( ( ord_less_eq_set_nat @ G4 @ F3 )
=> ( ( finite_finite_nat @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G4 ) @ C3 ) ) )
=> ( ( finite_finite_nat @ F3 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F3 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1080_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_complex,C3: nat] :
( ! [G4: set_complex] :
( ( ord_le211207098394363844omplex @ G4 @ F3 )
=> ( ( finite3207457112153483333omplex @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_complex @ G4 ) @ C3 ) ) )
=> ( ( finite3207457112153483333omplex @ F3 )
& ( ord_less_eq_nat @ ( finite_card_complex @ F3 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1081_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_list_a,C3: nat] :
( ! [G4: set_list_a] :
( ( ord_le8861187494160871172list_a @ G4 @ F3 )
=> ( ( finite_finite_list_a @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ G4 ) @ C3 ) ) )
=> ( ( finite_finite_list_a @ F3 )
& ( ord_less_eq_nat @ ( finite_card_list_a @ F3 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1082_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_a,C3: nat] :
( ! [G4: set_a] :
( ( ord_less_eq_set_a @ G4 @ F3 )
=> ( ( finite_finite_a @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G4 ) @ C3 ) ) )
=> ( ( finite_finite_a @ F3 )
& ( ord_less_eq_nat @ ( finite_card_a @ F3 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1083_a__card__cosets__equal,axiom,
! [C: set_a,H: set_a] :
( ( member_set_a @ C @ ( a_RCOSETS_a_b @ r @ H ) )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_card_a @ C )
= ( finite_card_a @ H ) ) ) ) ) ).
% a_card_cosets_equal
thf(fact_1084_freshmans__dream__ext,axiom,
! [X2: a,Y: a,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( add_a_b @ r @ X2 @ Y ) @ ( power_power_nat @ ( ring_char_a_b @ r ) @ M ) )
= ( add_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ ( power_power_nat @ ( ring_char_a_b @ r ) @ M ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ ( power_power_nat @ ( ring_char_a_b @ r ) @ M ) ) ) ) ) ) ) ).
% freshmans_dream_ext
thf(fact_1085_normalize_Osimps_I1_J,axiom,
( ( normalize_a_b @ r @ nil_a )
= nil_a ) ).
% normalize.simps(1)
thf(fact_1086_normalize__length__le,axiom,
! [P: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) @ ( size_size_list_a @ P ) ) ).
% normalize_length_le
thf(fact_1087_poly__of__const__def,axiom,
( ( poly_of_const_a_b @ r )
= ( ^ [K4: a] : ( normalize_a_b @ r @ ( cons_a @ K4 @ nil_a ) ) ) ) ).
% poly_of_const_def
thf(fact_1088_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1089_normalize__consistent,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ( normalize_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) )
= ( normalize_a_b @ r ) ) ) ).
% normalize_consistent
thf(fact_1090_frobenius__hom,axiom,
! [M: nat,K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) )
=> ( ( M
= ( power_power_nat @ ( ring_char_a_b @ r ) @ K2 ) )
=> ( ring_h661254511236296859_b_a_b @ r @ r
@ ^ [X: a] : ( pow_a_1026414303147256608_b_nat @ r @ X @ M ) ) ) ) ).
% frobenius_hom
thf(fact_1091_nth__root__nat__aux2_I1_J,axiom,
! [K2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [M5: nat] : ( ord_less_eq_nat @ ( power_power_nat @ M5 @ K2 ) @ N ) ) ) ) ).
% nth_root_nat_aux2(1)
thf(fact_1092_nth__root__nat__aux2_I2_J,axiom,
! [K2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( collect_nat
@ ^ [M5: nat] : ( ord_less_eq_nat @ ( power_power_nat @ M5 @ K2 ) @ N ) )
!= bot_bot_set_nat ) ) ).
% nth_root_nat_aux2(2)
thf(fact_1093_finite__nth__roots,axiom,
! [N: nat,C: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= C ) ) ) ) ).
% finite_nth_roots
thf(fact_1094_up__one__closed,axiom,
( member_nat_a
@ ^ [N2: nat] : ( if_a @ ( N2 = zero_zero_nat ) @ ( one_a_ring_ext_a_b @ r ) @ ( zero_a_b @ r ) )
@ ( up_a_b @ r ) ) ).
% up_one_closed
thf(fact_1095_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1096_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1097_up__add__closed,axiom,
! [P: nat > a,Q: nat > a] :
( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I2: nat] : ( add_a_b @ r @ ( P @ I2 ) @ ( Q @ I2 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_add_closed
thf(fact_1098_up__a__inv__closed,axiom,
! [P: nat > a] :
( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I2: nat] : ( a_inv_a_b @ r @ ( P @ I2 ) )
@ ( up_a_b @ r ) ) ) ).
% up_a_inv_closed
thf(fact_1099_up__minus__closed,axiom,
! [P: nat > a,Q: nat > a] :
( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I2: nat] : ( a_minus_a_b @ r @ ( P @ I2 ) @ ( Q @ I2 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_minus_closed
thf(fact_1100_up__smult__closed,axiom,
! [A: a,P: nat > a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I2: nat] : ( mult_a_ring_ext_a_b @ r @ A @ ( P @ I2 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_smult_closed
thf(fact_1101_bound__upD,axiom,
! [F: nat > a] :
( ( member_nat_a @ F @ ( up_a_b @ r ) )
=> ? [N3: nat] : ( bound_a @ ( zero_a_b @ r ) @ N3 @ F ) ) ).
% bound_upD
thf(fact_1102_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K2: nat,B: nat] :
( ( P2 @ K2 )
=> ( ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1103_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1104_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1105_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1106_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_1107_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1108_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1109_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1110_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1111_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1112_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1113_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1114_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1115_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
| ( M5 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1116_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1117_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N2: nat] :
( ( ord_less_eq_nat @ M5 @ N2 )
& ( M5 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1118_card__nth__roots,axiom,
! [C: complex,N: nat] :
( ( C != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= C ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_1119_GreatestI__nat,axiom,
! [P2: nat > $o,K2: nat,B: nat] :
( ( P2 @ K2 )
=> ( ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B ) )
=> ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).
% GreatestI_nat
thf(fact_1120_Greatest__le__nat,axiom,
! [P2: nat > $o,K2: nat,B: nat] :
( ( P2 @ K2 )
=> ( ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B ) )
=> ( ord_less_eq_nat @ K2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).
% Greatest_le_nat
thf(fact_1121_GreatestI__ex__nat,axiom,
! [P2: nat > $o,B: nat] :
( ? [X_12: nat] : ( P2 @ X_12 )
=> ( ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B ) )
=> ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_1122_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1123_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_1124_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_1125_ring__irreducibleE_I2_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( irredu6211895646901577903xt_a_b @ r @ R2 ) ) ) ).
% ring_irreducibleE(2)
thf(fact_1126_normalize__in__carrier,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( normalize_a_b @ r @ P ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% normalize_in_carrier
thf(fact_1127_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_1128_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_1129_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_1130_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_1131_exp__base__closed,axiom,
! [X2: a,N: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X2 @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% exp_base_closed
thf(fact_1132_poly__mult__const_H_I2_J,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 ) )
=> ( ( poly_mult_a_b @ r @ P @ ( cons_a @ A @ nil_a ) )
= ( normalize_a_b @ r @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ A ) @ P ) ) ) ) ) ).
% poly_mult_const'(2)
thf(fact_1133_poly__mult__const_H_I1_J,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 ) )
=> ( ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ P )
= ( normalize_a_b @ r @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ A ) @ P ) ) ) ) ) ).
% poly_mult_const'(1)
thf(fact_1134_poly__mult_Osimps_I1_J,axiom,
! [P23: list_a] :
( ( poly_mult_a_b @ r @ nil_a @ P23 )
= nil_a ) ).
% poly_mult.simps(1)
thf(fact_1135_poly__mult__comm,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P12 @ P23 )
= ( poly_mult_a_b @ r @ P23 @ P12 ) ) ) ) ).
% poly_mult_comm
thf(fact_1136_poly__mult__in__carrier,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_mult_a_b @ r @ P12 @ P23 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_mult_in_carrier
thf(fact_1137_poly__mult__zero_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ nil_a )
= nil_a ) ) ).
% poly_mult_zero(2)
thf(fact_1138_poly__mult__zero_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ nil_a @ P )
= nil_a ) ) ).
% poly_mult_zero(1)
thf(fact_1139_poly__mult__normalize,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P12 @ P23 )
= ( poly_mult_a_b @ r @ ( normalize_a_b @ r @ P12 ) @ P23 ) ) ) ) ).
% poly_mult_normalize
thf(fact_1140_poly__mult__semiassoc,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 ) )
=> ( ( poly_mult_a_b @ r @ ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ P ) @ Q )
= ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ ( poly_mult_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% poly_mult_semiassoc
thf(fact_1141_const__term__simprules_I2_J,axiom,
! [P: list_a,Q: list_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 ) )
=> ( ( const_term_a_b @ r @ ( poly_mult_a_b @ r @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ).
% const_term_simprules(2)
thf(fact_1142_poly__mult__one_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_mult_one'(1)
thf(fact_1143_poly__mult__one_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_mult_one'(2)
thf(fact_1144_poly__mult__consistent,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ( poly_mult_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) )
= ( poly_mult_a_b @ r ) ) ) ).
% poly_mult_consistent
thf(fact_1145_poly__mult__append__zero,axiom,
! [P: list_a,Q: list_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 ) )
=> ( ( poly_mult_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Q )
= ( normalize_a_b @ r @ ( append_a @ ( poly_mult_a_b @ r @ P @ Q ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_mult_append_zero
thf(fact_1146_poly__mult__monom__assoc,axiom,
! [P: list_a,Q: list_a,A: a,N: nat] :
( ( 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 ) )
=> ( ( poly_mult_a_b @ r @ ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P ) @ Q )
= ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ ( poly_mult_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% poly_mult_monom_assoc
thf(fact_1147_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_1148_local_Onormalize__idem,axiom,
! [P: list_a,Q: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( normalize_a_b @ r @ P ) @ Q ) )
= ( normalize_a_b @ r @ ( append_a @ P @ Q ) ) ) ).
% local.normalize_idem
thf(fact_1149_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_1150_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_1151_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_1152_poly__mult__monom_H,axiom,
! [P: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P )
= ( normalize_a_b @ r @ ( append_a @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ A ) @ P ) @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ) ) ) ).
% poly_mult_monom'
thf(fact_1153_poly__mult__var_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( var_a_b @ r ) @ P )
= ( normalize_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ).
% poly_mult_var'(1)
thf(fact_1154_poly__mult__var_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ ( var_a_b @ r ) )
= ( normalize_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ).
% poly_mult_var'(2)
thf(fact_1155_pderiv__var,axiom,
! [K: set_a] :
( ( formal4452980811800949548iv_a_b @ r @ ( var_a_b @ r ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% pderiv_var
thf(fact_1156_var__closed_I1_J,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( member_list_a @ ( var_a_b @ r ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).
% var_closed(1)
thf(fact_1157_normalize__replicate__zero,axiom,
! [N: nat,P: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) )
= ( normalize_a_b @ r @ P ) ) ).
% normalize_replicate_zero
thf(fact_1158_local_Omonom__def,axiom,
! [A: a,N: nat] :
( ( monom_a_b @ r @ A @ N )
= ( cons_a @ A @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ).
% local.monom_def
thf(fact_1159_var__pow__closed,axiom,
! [K: set_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).
% var_pow_closed
thf(fact_1160_unitary__monom__eq__var__pow,axiom,
! [K: set_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( monom_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) ) ) ).
% unitary_monom_eq_var_pow
thf(fact_1161_poly__mult__replicate__zero_I1_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= nil_a ) ) ).
% poly_mult_replicate_zero(1)
thf(fact_1162_poly__mult__replicate__zero_I2_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= nil_a ) ) ).
% poly_mult_replicate_zero(2)
thf(fact_1163_poly__mult__prepend__replicate__zero,axiom,
! [P12: list_a,P23: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P12 @ P23 )
= ( poly_mult_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P12 ) @ P23 ) ) ) ) ).
% poly_mult_prepend_replicate_zero
thf(fact_1164_poly__mult__var,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( P = nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ ( var_a_b @ r ) )
= nil_a ) )
& ( ( P != nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ ( var_a_b @ r ) )
= ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ).
% poly_mult_var
thf(fact_1165_eval__rewrite,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( P
= ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ K ) @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ P ) @ ( var_a_b @ r ) ) ) ) ) ).
% eval_rewrite
thf(fact_1166_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_1167_poly__add__append__replicate,axiom,
! [P: list_a,Q: list_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 ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P @ ( replicate_a @ ( size_size_list_a @ Q ) @ ( zero_a_b @ r ) ) ) @ Q )
= ( normalize_a_b @ r @ ( append_a @ P @ Q ) ) ) ) ) ).
% poly_add_append_replicate
thf(fact_1168_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_1169_eval__var,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X2 )
= X2 ) ) ).
% eval_var
thf(fact_1170_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_1171_poly__add__in__carrier,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_add_a_b @ r @ P12 @ P23 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_add_in_carrier
thf(fact_1172_poly__add__comm,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P12 @ P23 )
= ( poly_add_a_b @ r @ P23 @ P12 ) ) ) ) ).
% poly_add_comm
thf(fact_1173_eval__in__carrier,axiom,
! [P: list_a,X2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier
thf(fact_1174_eval__is__hom,axiom,
! [K: set_a,A: a] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a_a
@ ^ [P4: list_a] : ( eval_a_b @ r @ P4 @ A )
@ ( ring_h2895973938487309444it_a_b @ ( univ_poly_a_b @ r @ K ) @ r ) ) ) ) ).
% eval_is_hom
thf(fact_1175_eval__cring__hom,axiom,
! [K: set_a,A: a] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ring_h1547129875642963619it_a_b @ ( univ_poly_a_b @ r @ K ) @ r
@ ^ [P4: list_a] : ( eval_a_b @ r @ P4 @ A ) ) ) ) ).
% eval_cring_hom
thf(fact_1176_poly__mult__l__distr_H,axiom,
! [P12: list_a,P23: list_a,P32: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P32 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_add_a_b @ r @ P12 @ P23 ) @ P32 )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P12 @ P32 ) @ ( poly_mult_a_b @ r @ P23 @ P32 ) ) ) ) ) ) ).
% poly_mult_l_distr'
thf(fact_1177_poly__mult__r__distr_H,axiom,
! [P12: list_a,P23: list_a,P32: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P32 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P12 @ ( poly_add_a_b @ r @ P23 @ P32 ) )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P12 @ P23 ) @ ( poly_mult_a_b @ r @ P12 @ P32 ) ) ) ) ) ) ).
% poly_mult_r_distr'
thf(fact_1178_poly__add__normalize_I3_J,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P12 @ P23 )
= ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P12 ) @ ( normalize_a_b @ r @ P23 ) ) ) ) ) ).
% poly_add_normalize(3)
thf(fact_1179_poly__add__normalize_I2_J,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P12 @ P23 )
= ( poly_add_a_b @ r @ P12 @ ( normalize_a_b @ r @ P23 ) ) ) ) ) ).
% poly_add_normalize(2)
thf(fact_1180_poly__add__normalize__aux,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P12 @ P23 )
= ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P12 ) @ P23 ) ) ) ) ).
% poly_add_normalize_aux
thf(fact_1181_eval__normalize,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 ) )
=> ( ( eval_a_b @ r @ ( normalize_a_b @ r @ P ) @ A )
= ( eval_a_b @ r @ P @ A ) ) ) ) ).
% eval_normalize
thf(fact_1182_is__root__def,axiom,
! [P: list_a,X2: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P @ X2 )
= ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P @ X2 )
= ( zero_a_b @ r ) )
& ( P != nil_a ) ) ) ).
% is_root_def
thf(fact_1183_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_1184_eval__poly__add,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 @ ( poly_add_a_b @ r @ P @ Q ) @ A )
= ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_poly_add
thf(fact_1185_eval__poly__mult,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 @ ( poly_mult_a_b @ r @ P @ Q ) @ A )
= ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_poly_mult
thf(fact_1186_poly__add__zero_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P @ nil_a )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_zero'(1)
thf(fact_1187_poly__add__zero_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ nil_a @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_zero'(2)
thf(fact_1188_const__term__simprules_I3_J,axiom,
! [P: list_a,Q: list_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 ) )
=> ( ( const_term_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) )
= ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ).
% const_term_simprules(3)
thf(fact_1189_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_1190_eval__poly__add__aux,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 ) )
=> ( ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) @ A )
= ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ) ).
% eval_poly_add_aux
thf(fact_1191_poly__add__replicate__zero_H_I1_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_replicate_zero'(1)
thf(fact_1192_poly__add__replicate__zero_H_I2_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_replicate_zero'(2)
thf(fact_1193_eval__replicate,axiom,
! [P: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) @ A )
= ( eval_a_b @ r @ P @ A ) ) ) ) ).
% eval_replicate
thf(fact_1194_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_1195_poly__add__append__zero,axiom,
! [P: list_a,Q: list_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 ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ ( append_a @ Q @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
= ( normalize_a_b @ r @ ( append_a @ ( poly_add_a_b @ r @ P @ Q ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_add_append_zero
thf(fact_1196_poly__add__consistent,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ( poly_add_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) )
= ( poly_add_a_b @ r ) ) ) ).
% poly_add_consistent
thf(fact_1197_eval__consistent,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ( eval_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) )
= ( eval_a_b @ r ) ) ) ).
% eval_consistent
thf(fact_1198_eval__as__unique__hom,axiom,
! [K: set_a,X2: a,H3: list_a > a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ r @ H3 )
=> ( ! [K3: a] :
( ( member_a @ K3 @ K )
=> ( ( H3 @ ( cons_a @ K3 @ nil_a ) )
= K3 ) )
=> ( ( ( H3 @ ( var_a_b @ r ) )
= X2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( H3 @ P )
= ( eval_a_b @ r @ P @ X2 ) ) ) ) ) ) ) ) ).
% eval_as_unique_hom
thf(fact_1199_normalize__trick,axiom,
! [P: list_a] :
( P
= ( append_a @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) ) @ ( zero_a_b @ r ) ) @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_trick
thf(fact_1200_eval__ring__hom,axiom,
! [K: set_a,A: a] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ r
@ ^ [P4: list_a] : ( eval_a_b @ r @ P4 @ A ) ) ) ) ).
% eval_ring_hom
thf(fact_1201_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1202_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1203_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1204_eq__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1205_le__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1206_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1207_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1208_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1209_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1210_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1211_less__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1212_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1213_long__dividesI,axiom,
! [B: list_a,R2: list_a,P: list_a,Q: list_a] :
( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ B ) @ R2 ) )
=> ( ( ( R2 = nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) )
=> ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B @ R2 ) ) ) ) ) ) ).
% long_dividesI
thf(fact_1214_subfield__long__division__theorem__shell,axiom,
! [K: set_a,P: list_a,B: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( B
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ? [Q3: list_a,R: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ B @ Q3 ) @ R ) )
& ( ( R
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% subfield_long_division_theorem_shell
thf(fact_1215_degree__var,axiom,
( ( minus_minus_nat @ ( size_size_list_a @ ( var_a_b @ r ) ) @ one_one_nat )
= one_one_nat ) ).
% degree_var
thf(fact_1216_canonical__embedding__ring__hom,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ring_h5357930050666032198t_unit
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r )
@ ( univ_poly_a_b @ r @ K )
@ ( poly_of_const_a_b @ r ) ) ) ).
% canonical_embedding_ring_hom
thf(fact_1217_degree__one,axiom,
! [K: set_a] :
( ( minus_minus_nat @ ( size_size_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ K ) ) ) @ one_one_nat )
= zero_zero_nat ) ).
% degree_one
thf(fact_1218_degree__one__imp__pirreducible,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).
% degree_one_imp_pirreducible
thf(fact_1219_univ__poly__a__inv__degree,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) @ one_one_nat )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ).
% univ_poly_a_inv_degree
thf(fact_1220_var__pow__degree,axiom,
! [K: set_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) ) @ one_one_nat )
= N ) ) ).
% var_pow_degree
thf(fact_1221_pderiv__const,axiom,
! [X2: list_a,K: set_a] :
( ( ( minus_minus_nat @ ( size_size_list_a @ X2 ) @ one_one_nat )
= zero_zero_nat )
=> ( ( formal4452980811800949548iv_a_b @ r @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).
% pderiv_const
thf(fact_1222_finite__poly_I1_J,axiom,
! [K: set_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( finite_finite_a @ K )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [F2: list_a] :
( ( member_list_a @ F2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( ( minus_minus_nat @ ( size_size_list_a @ F2 ) @ one_one_nat )
= N ) ) ) ) ) ) ).
% finite_poly(1)
thf(fact_1223_degree__oneE,axiom,
! [P: list_a,K: set_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A4: a] :
( ( member_a @ A4 @ K )
=> ( ( A4
!= ( zero_a_b @ r ) )
=> ! [B4: a] :
( ( member_a @ B4 @ K )
=> ( P
!= ( cons_a @ A4 @ ( cons_a @ B4 @ nil_a ) ) ) ) ) ) ) ) ).
% degree_oneE
thf(fact_1224_degree__zero__imp__not__is__root,axiom,
! [P: list_a,X2: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4133073214067823460ot_a_b @ r @ P @ X2 ) ) ) ).
% degree_zero_imp_not_is_root
thf(fact_1225_pmod__const_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
=> ( ( polynomial_pmod_a_b @ r @ P @ Q )
= P ) ) ) ) ) ).
% pmod_const(2)
thf(fact_1226_pirreducible__degree,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% pirreducible_degree
thf(fact_1227_finite__poly_I2_J,axiom,
! [K: set_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( finite_finite_a @ K )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [F2: list_a] :
( ( member_list_a @ F2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ F2 ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% finite_poly(2)
thf(fact_1228_pdivides__imp__degree__le,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).
% pdivides_imp_degree_le
thf(fact_1229_pmod__degree,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( ( polynomial_pmod_a_b @ r @ P @ Q )
= nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).
% pmod_degree
thf(fact_1230_univ__poly__units_H,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
= ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( P != nil_a )
& ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ).
% univ_poly_units'
thf(fact_1231_pmod__const_I1_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
=> ( ( polynomial_pdiv_a_b @ r @ P @ Q )
= nil_a ) ) ) ) ) ).
% pmod_const(1)
thf(fact_1232_degree__zero__imp__empty__roots,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polynomial_roots_a_b @ r @ P )
= zero_zero_multiset_a ) ) ) ).
% degree_zero_imp_empty_roots
thf(fact_1233_pirreducible__roots,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ( ( polynomial_roots_a_b @ r @ P )
= zero_zero_multiset_a ) ) ) ) ).
% pirreducible_roots
thf(fact_1234_nat__pow__eone,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X2 @ one_one_nat )
= X2 ) ) ).
% nat_pow_eone
thf(fact_1235_univ__poly__is__euclidean,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ring_e7478897652244013592t_unit @ ( univ_poly_a_b @ r @ K )
@ ^ [P4: list_a] : ( minus_minus_nat @ ( size_size_list_a @ P4 ) @ one_one_nat ) ) ) ).
% univ_poly_is_euclidean
thf(fact_1236_degree__zero__imp__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).
% degree_zero_imp_splitted
thf(fact_1237_rupture__char,axiom,
! [K: set_a,F: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat ) )
=> ( ( ring_c6053888738502451990t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ F ) )
= ( ring_char_a_b @ r ) ) ) ) ) ).
% rupture_char
thf(fact_1238_rupture__one__not__zero,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) )
=> ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) )
!= ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) ) ) ) ) ) ).
% rupture_one_not_zero
thf(fact_1239_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z3: complex] :
( ( power_power_complex @ Z3 @ N )
= one_one_complex ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_1240_rupture__order,axiom,
! [K: set_a,F: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat ) )
=> ( ( order_1351569949434154782t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ F ) )
= ( power_power_nat @ ( finite_card_a @ K ) @ ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat ) ) ) ) ) ) ).
% rupture_order
thf(fact_1241_monom__eq__var__pow,axiom,
! [K: set_a,A: a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( monom_a_b @ r @ A @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ A @ nil_a ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) ) ) ) ) ).
% monom_eq_var_pow
thf(fact_1242_associated__polynomials__iff,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
= ( ? [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
& ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ X @ nil_a ) @ Q ) ) ) ) ) ) ) ) ).
% associated_polynomials_iff
thf(fact_1243_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_1244_carrier__one__not__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) ) ) ).
% carrier_one_not_zero
thf(fact_1245_carrier__one__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% carrier_one_zero
thf(fact_1246_one__zeroD,axiom,
( ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) )
=> ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% one_zeroD
thf(fact_1247_one__zeroI,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% one_zeroI
thf(fact_1248_subfield__m__inv__simprule,axiom,
! [K: set_a,K2: a,A: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ A ) @ K )
=> ( member_a @ A @ K ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_1249_mult__divides,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ B ) ) ) ) ) ).
% mult_divides
thf(fact_1250_finite__domain__units,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% finite_domain_units
thf(fact_1251_euclidean__domainI,axiom,
! [Phi: a > nat] :
( ! [A4: a,B4: a] :
( ( member_a @ A4 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B4 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ? [Q4: a,R5: a] :
( ( member_a @ Q4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ R5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( A4
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ B4 @ Q4 ) @ R5 ) )
& ( ( R5
= ( zero_a_b @ r ) )
| ( ord_less_nat @ ( Phi @ R5 ) @ ( Phi @ B4 ) ) ) ) ) )
=> ( ring_e8745995371659049232in_a_b @ r @ Phi ) ) ).
% euclidean_domainI
thf(fact_1252_ring__irreducibleI,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ! [A4: a,B4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ r @ A4 @ B4 ) )
=> ( ( member_a @ A4 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B4 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ r @ R2 ) ) ) ) ).
% ring_irreducibleI
thf(fact_1253_subfieldI,axiom,
! [K: set_a] :
( ( subcring_a_b @ K @ r )
=> ( ( ( units_a_ring_ext_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) )
= ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( subfield_a_b @ K @ r ) ) ) ).
% subfieldI
thf(fact_1254_poly__add__monom,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( poly_add_a_b @ r @ ( monom_a_b @ r @ A @ ( size_size_list_a @ P ) ) @ P )
= ( cons_a @ A @ P ) ) ) ) ).
% poly_add_monom
thf(fact_1255_primeideal__iff__prime,axiom,
! [P: a] :
( ( member_a @ P @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P ) @ r )
= ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% primeideal_iff_prime
thf(fact_1256_poly__mult__monom,axiom,
! [P: list_a,A: a,N: nat] :
( ( polynomial_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ P )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( ( P = nil_a )
=> ( ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P )
= nil_a ) )
& ( ( P != nil_a )
=> ( ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P )
= ( append_a @ ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ P ) @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ) ) ) ) ).
% poly_mult_monom
thf(fact_1257_normalize__polynomial,axiom,
! [K: set_a,P: list_a] :
( ( polynomial_a_b @ r @ K @ P )
=> ( ( normalize_a_b @ r @ P )
= P ) ) ).
% normalize_polynomial
thf(fact_1258_polynomial__incl,axiom,
! [K: set_a,P: list_a] :
( ( polynomial_a_b @ r @ K @ P )
=> ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K ) ) ).
% polynomial_incl
thf(fact_1259_poly__mult__closed,axiom,
! [K: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P12 )
=> ( ( polynomial_a_b @ r @ K @ P23 )
=> ( polynomial_a_b @ r @ K @ ( poly_mult_a_b @ r @ P12 @ P23 ) ) ) ) ) ).
% poly_mult_closed
thf(fact_1260_poly__add__closed,axiom,
! [K: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P12 )
=> ( ( polynomial_a_b @ r @ K @ P23 )
=> ( polynomial_a_b @ r @ K @ ( poly_add_a_b @ r @ P12 @ P23 ) ) ) ) ) ).
% poly_add_closed
thf(fact_1261_var__closed_I2_J,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( polynomial_a_b @ r @ K @ ( var_a_b @ r ) ) ) ).
% var_closed(2)
thf(fact_1262_normalize__gives__polynomial,axiom,
! [P: list_a,K: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K )
=> ( polynomial_a_b @ r @ K @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_gives_polynomial
thf(fact_1263_eval__poly__in__carrier,axiom,
! [K: set_a,P: list_a,X2: a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% eval_poly_in_carrier
thf(fact_1264_poly__mult__integral,axiom,
! [K: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P12 )
=> ( ( polynomial_a_b @ r @ K @ P23 )
=> ( ( ( poly_mult_a_b @ r @ P12 @ P23 )
= nil_a )
=> ( ( P12 = nil_a )
| ( P23 = nil_a ) ) ) ) ) ) ).
% poly_mult_integral
thf(fact_1265_poly__add__zero_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_add_a_b @ r @ nil_a @ P )
= P ) ) ) ).
% poly_add_zero(2)
thf(fact_1266_poly__add__zero_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_add_a_b @ r @ P @ nil_a )
= P ) ) ) ).
% poly_add_zero(1)
thf(fact_1267_poly__mult__r__distr,axiom,
! [K: set_a,P12: list_a,P23: list_a,P32: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P12 )
=> ( ( polynomial_a_b @ r @ K @ P23 )
=> ( ( polynomial_a_b @ r @ K @ P32 )
=> ( ( poly_mult_a_b @ r @ P12 @ ( poly_add_a_b @ r @ P23 @ P32 ) )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P12 @ P23 ) @ ( poly_mult_a_b @ r @ P12 @ P32 ) ) ) ) ) ) ) ).
% poly_mult_r_distr
thf(fact_1268_poly__mult__l__distr,axiom,
! [K: set_a,P12: list_a,P23: list_a,P32: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P12 )
=> ( ( polynomial_a_b @ r @ K @ P23 )
=> ( ( polynomial_a_b @ r @ K @ P32 )
=> ( ( poly_mult_a_b @ r @ ( poly_add_a_b @ r @ P12 @ P23 ) @ P32 )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P12 @ P32 ) @ ( poly_mult_a_b @ r @ P23 @ P32 ) ) ) ) ) ) ) ).
% poly_mult_l_distr
thf(fact_1269_zeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroprimeideal
thf(fact_1270_poly__mult__is__polynomial,axiom,
! [K: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ K )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ K )
=> ( polynomial_a_b @ r @ K @ ( poly_mult_a_b @ r @ P12 @ P23 ) ) ) ) ) ).
% poly_mult_is_polynomial
thf(fact_1271_poly__add__is__polynomial,axiom,
! [K: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ K )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ K )
=> ( polynomial_a_b @ r @ K @ ( poly_add_a_b @ r @ P12 @ P23 ) ) ) ) ) ).
% poly_add_is_polynomial
thf(fact_1272_poly__add__replicate__zero_I2_J,axiom,
! [K: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_add_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= P ) ) ) ).
% poly_add_replicate_zero(2)
thf(fact_1273_poly__add__replicate__zero_I1_J,axiom,
! [K: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_add_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= P ) ) ) ).
% poly_add_replicate_zero(1)
thf(fact_1274_var__carr,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( member_list_a @ ( var_a_b @ r ) @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) @ bot_bot_set_list_a ) ) ) ) ).
% var_carr
thf(fact_1275_pdivides__iff,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( polynomial_a_b @ r @ K @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) ) ) ) ) ).
% pdivides_iff
thf(fact_1276_domain__eq__zeroprimeideal,axiom,
( ( domain_a_b @ r )
= ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ) ) ).
% domain_eq_zeroprimeideal
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X2: a,Y: a] :
( ( if_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X2: a,Y: a] :
( ( if_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ? [X: list_a] :
( ( member_list_a @ X
@ ( partia5361259788508890537t_unit
@ ( univ_poly_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r )
@ ( partia707051561876973205xt_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r ) ) ) ) )
& ( ( mult_l7073676228092353617t_unit
@ ( univ_poly_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r )
@ ( partia707051561876973205xt_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r ) ) )
@ f
@ X )
= g ) ) )
!= ( ~ ? [X: list_a] :
( ( member_list_a @ X
@ ( partia5361259788508890537t_unit
@ ( univ_poly_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r )
@ ( partia707051561876973205xt_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r ) ) ) ) )
& ( g
= ( mult_l7073676228092353617t_unit
@ ( univ_poly_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r )
@ ( partia707051561876973205xt_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r ) ) )
@ f
@ X ) ) ) ) ) ).
%------------------------------------------------------------------------------