TPTP Problem File: SLH0383^1.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/0006_Formal_Polynomial_Derivatives/prob_00167_005222__18257938_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1512 ( 452 unt; 228 typ; 0 def)
% Number of atoms : 3923 (1425 equ; 0 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 16785 ( 219 ~; 51 |; 100 &;14474 @)
% ( 0 <=>;1941 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 29 ( 28 usr)
% Number of type conns : 475 ( 475 >; 0 *; 0 +; 0 <<)
% Number of symbols : 203 ( 200 usr; 13 con; 0-4 aty)
% Number of variables : 3046 ( 66 ^;2948 !; 32 ?;3046 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:21:56.283
%------------------------------------------------------------------------------
% Could-be-implicit typings (28)
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% Explicit typings (200)
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ring_hom_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > set_a_a ).
thf(sy_c_Ring_Osemiring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
semiri2265994252334843677t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring_Osemiring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
semiri2871908745932252451t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
semiring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_p715737262848045090t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_p8098905331641078952t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001tf__a_001tf__b,type,
ring_p8803135361686045600in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r360171070648044744t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r932985474545269838t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_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_001tf__a_001tf__b,type,
ring_ring_prime_a_b: partia2175431115845679010xt_a_b > a > $o ).
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_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
insert_list_list_a: list_list_a > set_list_list_a > set_list_list_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_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subcri7763218559781929323t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001tf__a_001tf__b,type,
subcring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubdomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subdom7821232466298058046t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_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_001tf__a_001tf__b,type,
subring_a_b: set_a > partia2175431115845679010xt_a_b > $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_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_Mtf__a_J,type,
member_list_a_a: ( list_a > a ) > 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_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_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_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_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_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 ).
% Relevant facts (1278)
thf(fact_0_p_Ominus__eq,axiom,
! [X: list_a,Y: list_a] :
( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y ) ) ) ).
% p.minus_eq
thf(fact_1_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_2_calculation,axiom,
( ( formal4452980811800949548iv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( formal4452980811800949548iv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ ( formal4452980811800949548iv_a_b @ r @ f ) @ ( formal4452980811800949548iv_a_b @ r @ f ) ) ) ) ).
% calculation
thf(fact_3_assms_I1_J,axiom,
subring_a_b @ k @ r ).
% assms(1)
thf(fact_4_assms_I2_J,axiom,
member_list_a @ f @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).
% assms(2)
thf(fact_5__092_060open_062pderiv_A_I_092_060ominus_062_092_060_094bsub_062K_A_091X_093_092_060_094esub_062_Af_J_A_092_060oplus_062_092_060_094bsub_062K_A_091X_093_092_060_094esub_062_A_092_060zero_062_092_060_094bsub_062K_A_091X_093_092_060_094esub_062_A_061_Apderiv_A_I_092_060ominus_062_092_060_094bsub_062K_A_091X_093_092_060_094esub_062_Af_J_A_092_060oplus_062_092_060_094bsub_062K_A_091X_093_092_060_094esub_062_A_Ipderiv_Af_A_092_060ominus_062_092_060_094bsub_062K_A_091X_093_092_060_094esub_062_Apderiv_Af_J_092_060close_062,axiom,
( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( formal4452980811800949548iv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( formal4452980811800949548iv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ ( formal4452980811800949548iv_a_b @ r @ f ) @ ( formal4452980811800949548iv_a_b @ r @ f ) ) ) ) ).
% \<open>pderiv (\<ominus>\<^bsub>K [X]\<^esub> f) \<oplus>\<^bsub>K [X]\<^esub> \<zero>\<^bsub>K [X]\<^esub> = pderiv (\<ominus>\<^bsub>K [X]\<^esub> f) \<oplus>\<^bsub>K [X]\<^esub> (pderiv f \<ominus>\<^bsub>K [X]\<^esub> pderiv f)\<close>
thf(fact_6__092_060open_062pderiv_A_I_092_060ominus_062_092_060_094bsub_062K_A_091X_093_092_060_094esub_062_Af_J_A_061_Apderiv_A_I_092_060ominus_062_092_060_094bsub_062K_A_091X_093_092_060_094esub_062_Af_J_A_092_060oplus_062_092_060_094bsub_062K_A_091X_093_092_060_094esub_062_A_092_060zero_062_092_060_094bsub_062K_A_091X_093_092_060_094esub_062_092_060close_062,axiom,
( ( formal4452980811800949548iv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( formal4452980811800949548iv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% \<open>pderiv (\<ominus>\<^bsub>K [X]\<^esub> f) = pderiv (\<ominus>\<^bsub>K [X]\<^esub> f) \<oplus>\<^bsub>K [X]\<^esub> \<zero>\<^bsub>K [X]\<^esub>\<close>
thf(fact_7_p_Osemiring__axioms,axiom,
semiri2871908745932252451t_unit @ ( univ_poly_a_b @ r @ k ) ).
% p.semiring_axioms
thf(fact_8_p_Oa__transpose__inv,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y )
= Z )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ Z )
= Y ) ) ) ) ) ).
% p.a_transpose_inv
thf(fact_9_p_Oadd_Oinv__mult__group,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) ) ) ) ) ).
% p.add.inv_mult_group
thf(fact_10_p_Oadd_Oinv__solve__left,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( 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 ) ) )
=> ( ( A
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ B ) @ C ) )
= ( C
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ A ) ) ) ) ) ) ).
% p.add.inv_solve_left
thf(fact_11_p_Oadd_Oinv__solve__left_H,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( 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 ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ B ) @ C )
= A )
= ( C
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ A ) ) ) ) ) ) ).
% p.add.inv_solve_left'
thf(fact_12_p_Oadd_Oinv__solve__right,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( 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 ) ) )
=> ( ( A
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ C ) ) )
= ( B
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ C ) ) ) ) ) ) ).
% p.add.inv_solve_right
thf(fact_13_p_Oadd_Oinv__solve__right_H,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( 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 ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ C ) )
= A )
= ( B
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ C ) ) ) ) ) ) ).
% p.add.inv_solve_right'
thf(fact_14_p_Ominus__add,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y ) ) ) ) ) ).
% p.minus_add
thf(fact_15_p_Or__neg1,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) )
= Y ) ) ) ).
% p.r_neg1
thf(fact_16_p_Or__neg2,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ Y ) )
= Y ) ) ) ).
% p.r_neg2
thf(fact_17_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_18_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_19_p_Oadd_Or__cancel,axiom,
! [A: list_a,C: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ C )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ C ) )
=> ( ( 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 ) ) )
=> ( A = B ) ) ) ) ) ).
% p.add.r_cancel
thf(fact_20_p_Oadd_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Z ) ) ) ) ) ) ).
% p.add.m_lcomm
thf(fact_21_p_Oadd_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X ) ) ) ) ).
% p.add.m_comm
thf(fact_22_p_Oadd_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ Z ) ) ) ) ) ) ).
% p.add.m_assoc
thf(fact_23_p_Oadd_Ol__cancel,axiom,
! [C: list_a,A: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ C @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ C @ B ) )
=> ( ( 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 ) ) )
=> ( A = B ) ) ) ) ) ).
% p.add.l_cancel
thf(fact_24_pderiv__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 ) ) ) ) ) ) ).
% pderiv_add
thf(fact_25_p_Ominus__unique,axiom,
! [Y: list_a,X: list_a,Y2: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% p.minus_unique
thf(fact_26_p_Oadd_Or__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.add.r_inv_ex
thf(fact_27_p_Oadd_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.add.one_unique
thf(fact_28_p_Oadd_Ol__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.add.l_inv_ex
thf(fact_29_p_Oadd_Oinv__comm,axiom,
! [X: list_a,Y: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).
% p.add.inv_comm
thf(fact_30_p_Osum__zero__eq__neg,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( X
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y ) ) ) ) ) ).
% p.sum_zero_eq_neg
thf(fact_31_p_Or__neg,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.r_neg
thf(fact_32_p_Ominus__equality,axiom,
! [Y: list_a,X: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X )
= Y ) ) ) ) ).
% p.minus_equality
thf(fact_33_p_Ol__neg,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.l_neg
thf(fact_34_p_Ozero__closed,axiom,
member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).
% p.zero_closed
thf(fact_35_p_Oadd_Oright__cancel,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% p.add.right_cancel
thf(fact_36_p_Oadd_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.add.m_closed
thf(fact_37_p_Ominus__minus,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) )
= X ) ) ).
% p.minus_minus
thf(fact_38_p_Oadd_Oinv__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.add.inv_closed
thf(fact_39_p_Ominus__zero,axiom,
( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.minus_zero
thf(fact_40_p_Ominus__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.minus_closed
thf(fact_41_p_Or__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= X ) ) ).
% p.r_zero
thf(fact_42_p_Ol__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ X )
= X ) ) ).
% p.l_zero
thf(fact_43_p_Oadd_Or__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ X ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).
% p.add.r_cancel_one'
thf(fact_44_p_Oadd_Or__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ X )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).
% p.add.r_cancel_one
thf(fact_45_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A: list_list_a,P: list_list_a > $o] :
( ( member_list_list_a @ A @ ( collect_list_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A2: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_51_p_Oadd_Ol__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ A ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).
% p.add.l_cancel_one'
thf(fact_52_p_Oadd_Ol__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ A )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).
% p.add.l_cancel_one
thf(fact_53_p_Oadd_Oinv__eq__1__iff,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= ( X
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.add.inv_eq_1_iff
thf(fact_54_p_Or__right__minus__eq,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= ( A = B ) ) ) ) ).
% p.r_right_minus_eq
thf(fact_55_domain_Opderiv__carr,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,F: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member_list_list_a @ ( formal6075833236969493044t_unit @ R @ F ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ).
% domain.pderiv_carr
thf(fact_56_domain_Opderiv__carr,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,F: list_a] :
( ( domain_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ).
% domain.pderiv_carr
thf(fact_57_domain_Opderiv__add,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,F: list_list_a,G: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ G @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( formal6075833236969493044t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ F @ G ) )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( formal6075833236969493044t_unit @ R @ F ) @ ( formal6075833236969493044t_unit @ R @ G ) ) ) ) ) ) ) ).
% domain.pderiv_add
thf(fact_58_domain_Opderiv__add,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,F: list_a,G: list_a] :
( ( domain_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% domain.pderiv_add
thf(fact_59_domain_Opderiv__zero,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( formal6075833236969493044t_unit @ R @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.pderiv_zero
thf(fact_60_domain_Opderiv__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( formal4452980811800949548iv_a_b @ R @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.pderiv_zero
thf(fact_61_p_Oonepideal,axiom,
princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( univ_poly_a_b @ r @ k ) ).
% p.onepideal
thf(fact_62_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a,Q: list_list_list_a,P2: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
=> ( ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 @ Q )
= ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P2 @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_63_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Q: list_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( 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 ) @ P2 @ Q )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_64_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 @ Q )
= ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_65_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
=> ( ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 )
= ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P2 ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_66_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_67_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_68_p_Ocarrier__is__subcring,axiom,
subcri7763218559781929323t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( univ_poly_a_b @ r @ k ) ).
% p.carrier_is_subcring
thf(fact_69_univ__poly__a__minus__consistent,axiom,
! [K: set_a,Q: list_a,P2: 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 ) @ P2 @ Q )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) ) ) ) ).
% univ_poly_a_minus_consistent
thf(fact_70_p_Oadd_Oint__pow__inv,axiom,
! [X: list_a,I: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ X ) ) ) ) ).
% p.add.int_pow_inv
thf(fact_71_p_Oadd_Oint__pow__mult__distrib,axiom,
! [X: list_a,Y: list_a,I: int] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ X ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ Y ) ) ) ) ) ) ).
% p.add.int_pow_mult_distrib
thf(fact_72_p_Oadd_Oint__pow__distrib,axiom,
! [X: list_a,Y: list_a,I: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ X ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ Y ) ) ) ) ) ).
% p.add.int_pow_distrib
thf(fact_73_univ__poly__a__inv__consistent,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ) ).
% univ_poly_a_inv_consistent
thf(fact_74_p_Or__minus,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) ) ) ) ) ).
% p.r_minus
thf(fact_75_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_76_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_77_p_Ocarrier__is__subring,axiom,
subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( univ_poly_a_b @ r @ k ) ).
% p.carrier_is_subring
thf(fact_78_p_OsubcringI_H,axiom,
! [H: set_list_a] :
( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ k ) )
=> ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.subcringI'
thf(fact_79_p_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ Z ) ) ) ) ) ) ).
% p.m_assoc
thf(fact_80_p_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X ) ) ) ) ).
% p.m_comm
thf(fact_81_p_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ Z ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Z ) ) ) ) ) ) ).
% p.m_lcomm
thf(fact_82_p_OsubcringI,axiom,
! [H: set_list_a] :
( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ k ) )
=> ( ! [H1: list_a,H2: list_a] :
( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ H1 @ H2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ H2 @ H1 ) ) ) )
=> ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.subcringI
thf(fact_83_p_Ol__distr,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ Z ) ) ) ) ) ) ).
% p.l_distr
thf(fact_84_p_Or__distr,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Z @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Z @ Y ) ) ) ) ) ) ).
% p.r_distr
thf(fact_85_p_Ol__minus,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ Y )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) ) ) ) ) ).
% p.l_minus
thf(fact_86_p_Oadd__pow__ldistr__int,axiom,
! [A: list_a,B: list_a,K2: int] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ K2 @ A ) @ B )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ K2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B ) ) ) ) ) ).
% p.add_pow_ldistr_int
thf(fact_87_p_Oadd__pow__rdistr__int,axiom,
! [A: list_a,B: list_a,K2: int] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ K2 @ B ) )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ K2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B ) ) ) ) ) ).
% p.add_pow_rdistr_int
thf(fact_88_p_Ocarrier__polynomial__shell,axiom,
! [K: set_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
=> ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ) ).
% p.carrier_polynomial_shell
thf(fact_89_carrier__polynomial__shell,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% carrier_polynomial_shell
thf(fact_90_p_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.m_closed
thf(fact_91_p_Oadd_Oint__pow__closed,axiom,
! [X: list_a,I: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.add.int_pow_closed
thf(fact_92_p_Oadd_Oint__pow__one,axiom,
! [Z: int] :
( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ Z @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.add.int_pow_one
thf(fact_93_p_Ol__null,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 ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.l_null
thf(fact_94_p_Or__null,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 ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.r_null
thf(fact_95_domain_Ouniv__poly__is__domain,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_domain
thf(fact_96_domain_Ouniv__poly__is__domain,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( domain7810152921033798211t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_domain
thf(fact_97_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_98_p_Omonoid__cancelI,axiom,
( ! [A3: list_a,B2: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ C2 @ A3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ C2 @ B2 ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: list_a,B2: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A3 @ C2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ B2 @ C2 ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.monoid_cancelI
thf(fact_99_is__root__poly__mult__imp__is__root,axiom,
! [P2: list_a,Q: list_a,X: a] :
( ( member_list_a @ P2 @ ( 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 ) ) @ P2 @ Q ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_100_p_Oline__extension__mem__iff,axiom,
! [U: list_a,K: set_list_a,A: list_a,E: set_list_a] :
( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ A @ E ) )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ K )
& ? [Y3: list_a] :
( ( member_list_a @ Y3 @ E )
& ( U
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X3 @ A ) @ Y3 ) ) ) ) ) ) ).
% p.line_extension_mem_iff
thf(fact_101_p_Oadd_Oint__pow__diff,axiom,
! [X: list_a,N: int,M: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ ( minus_minus_int @ N @ M ) @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ N @ X ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ M @ X ) ) ) ) ) ).
% p.add.int_pow_diff
thf(fact_102_p_Oadd_Oint__pow__mult,axiom,
! [X: list_a,I: int,J: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ ( plus_plus_int @ I @ J ) @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ X ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ J @ X ) ) ) ) ).
% p.add.int_pow_mult
thf(fact_103_p_Oadd_Oint__pow__neg,axiom,
! [X: list_a,I: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ ( uminus_uminus_int @ I ) @ X )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ X ) ) ) ) ).
% p.add.int_pow_neg
thf(fact_104_p_Oadd_Oint__pow__pow,axiom,
! [X: list_a,M: int,N: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ M @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ N @ X ) )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ ( times_times_int @ N @ M ) @ X ) ) ) ).
% p.add.int_pow_pow
thf(fact_105_p_Oproperfactor__prod__l,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( proper8313688649498433056t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B )
=> ( ( 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 ) ) )
=> ( proper8313688649498433056t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ C @ B ) ) ) ) ) ) ).
% p.properfactor_prod_l
thf(fact_106_p_Oproperfactor__prod__r,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( proper8313688649498433056t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B )
=> ( ( 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 ) ) )
=> ( proper8313688649498433056t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ C ) ) ) ) ) ) ).
% p.properfactor_prod_r
thf(fact_107_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_108_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_109_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_list_a,Q: list_list_list_a,X: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( polyno5142720416380192742t_unit @ R @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P2 @ Q ) @ X )
=> ( ( polyno5142720416380192742t_unit @ R @ P2 @ X )
| ( polyno5142720416380192742t_unit @ R @ Q @ X ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_110_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,Q: list_a,X: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( 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 ) ) @ P2 @ Q ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P2 @ X )
| ( polyno4133073214067823460ot_a_b @ R @ Q @ X ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_111_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,Q: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 @ Q ) @ X )
=> ( ( polyno6951661231331188332t_unit @ R @ P2 @ X )
| ( polyno6951661231331188332t_unit @ R @ Q @ X ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_112_p_Oline__extension__in__carrier,axiom,
! [K: set_list_a,A: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ A @ E ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).
% p.line_extension_in_carrier
thf(fact_113_p_Ocoeff__add,axiom,
! [K: set_list_a,F: list_list_a,G: list_list_a,I: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
=> ( ( member_list_list_a @ G @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
=> ( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ F @ G ) @ I )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ k ) @ F @ I ) @ ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ k ) @ G @ I ) ) ) ) ) ) ).
% p.coeff_add
thf(fact_114_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_115_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_116_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_117_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_118_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_119_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_120_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_121_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_122_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_123_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_124_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_125_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_126_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_127_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_128_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_129_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_130_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_131_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_132_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_133_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_134_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_135_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_136_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_137_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_138_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_139_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_140_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_141_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_142_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_143_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_144_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_145_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_146_ring_Ocoeff_Ocong,axiom,
coeff_6360649920519955023t_unit = coeff_6360649920519955023t_unit ).
% ring.coeff.cong
thf(fact_147_ring_Ocoeff_Ocong,axiom,
coeff_a_b = coeff_a_b ).
% ring.coeff.cong
thf(fact_148_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_149_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_150_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_151_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_152_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_153_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_154_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_155_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_156_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_157_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_158_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_159_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_160_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_161_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_162_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_163_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_164_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_165_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_166_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_167_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_168_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_169_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_170_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_171_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_172_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_173_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_174_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_175_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_176_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_177_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_178_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_179_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_180_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_181_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_182_add__le__add__imp__diff__le,axiom,
! [I: int,K2: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_183_add__le__add__imp__diff__le,axiom,
! [I: nat,K2: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_184_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_185_add__le__imp__le__diff,axiom,
! [I: int,K2: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_186_add__le__imp__le__diff,axiom,
! [I: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_187_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_188_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_189_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_190_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_191_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_192_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_193_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_194_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_195_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_196_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_197_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_198_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_199_group__add__class_Oadd_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% group_add_class.add.right_cancel
thf(fact_200_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_201_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_202_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_203_group__cancel_Oadd2,axiom,
! [B4: int,K2: int,B: int,A: int] :
( ( B4
= ( plus_plus_int @ K2 @ B ) )
=> ( ( plus_plus_int @ A @ B4 )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_204_group__cancel_Oadd2,axiom,
! [B4: nat,K2: nat,B: nat,A: nat] :
( ( B4
= ( plus_plus_nat @ K2 @ B ) )
=> ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_205_group__cancel_Oadd1,axiom,
! [A2: int,K2: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_206_group__cancel_Oadd1,axiom,
! [A2: nat,K2: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_207_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_int @ I @ K2 )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_208_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I @ K2 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_209_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_210_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_211_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_212_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_213_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B3: int] : ( times_times_int @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_214_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_215_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_216_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_217_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_218_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_219_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_220_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_221_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_222_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_223_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_224_ring_Ois__root_Ocong,axiom,
polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).
% ring.is_root.cong
thf(fact_225_ring_Ois__root_Ocong,axiom,
polyno6951661231331188332t_unit = polyno6951661231331188332t_unit ).
% ring.is_root.cong
thf(fact_226_domain_Ocoeff__range,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,F: list_a,I: nat] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_a @ ( coeff_a_b @ R @ F @ I ) @ K ) ) ) ) ).
% domain.coeff_range
thf(fact_227_domain_Ocoeff__range,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,F: list_list_a,I: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member_list_a @ ( coeff_6360649920519955023t_unit @ R @ F @ I ) @ K ) ) ) ) ).
% domain.coeff_range
thf(fact_228_combine__common__factor,axiom,
! [A: int,E2: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_229_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_230_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_231_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_232_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_233_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_234_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_235_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_236_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_237_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_238_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_239_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_240_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_241_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_242_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_243_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_244_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_245_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_246_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_247_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_248_group__cancel_Osub1,axiom,
! [A2: int,K2: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K2 @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_249_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_250_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_251_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_252_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_253_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_254_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_255_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_256_group__cancel_Oneg1,axiom,
! [A2: int,K2: int,A: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_257_minus__mult__commute,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_258_square__eq__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ A )
= ( times_times_int @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_259_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_260_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_261_eq__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_262_eq__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_263_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_264_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_265_group__cancel_Osub2,axiom,
! [B4: int,K2: int,B: int,A: int] :
( ( B4
= ( plus_plus_int @ K2 @ B ) )
=> ( ( minus_minus_int @ A @ B4 )
= ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_266_p_Omonom__coeff,axiom,
! [A: list_a,N: nat] :
( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ k ) @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ N ) )
= ( ^ [I2: nat] : ( if_list_a @ ( I2 = N ) @ A @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.monom_coeff
thf(fact_267_p_Oadd_Oone__in__subset,axiom,
! [H: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( H != bot_bot_set_list_a )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 ) @ H ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ H )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Xa ) @ H ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ H ) ) ) ) ) ).
% p.add.one_in_subset
thf(fact_268_p_Oa__lcos__m__assoc,axiom,
! [M2: set_list_a,G: list_a,H3: list_a] :
( ( ord_le8861187494160871172list_a @ M2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ H3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ G @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ H3 @ M2 ) )
= ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ G @ H3 ) @ M2 ) ) ) ) ) ).
% p.a_lcos_m_assoc
thf(fact_269_p_Oa__lcos__mult__one,axiom,
! [M2: set_list_a] :
( ( ord_le8861187494160871172list_a @ M2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ M2 )
= M2 ) ) ).
% p.a_lcos_mult_one
thf(fact_270_p_OsubringI,axiom,
! [H: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ H )
=> ( ! [H4: list_a] :
( ( member_list_a @ H4 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ H4 ) @ H ) )
=> ( ! [H1: list_a,H2: list_a] :
( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ H1 @ H2 ) @ H ) ) )
=> ( ! [H1: list_a,H2: list_a] :
( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ H1 @ H2 ) @ H ) ) )
=> ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ) ).
% p.subringI
thf(fact_271_p_Ocarrier__is__subalgebra,axiom,
! [K: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( embedd1768981623711841426t_unit @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.carrier_is_subalgebra
thf(fact_272_p_Osubalgebra__in__carrier,axiom,
! [K: set_list_a,V: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ k ) )
=> ( ord_le8861187494160871172list_a @ V @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.subalgebra_in_carrier
thf(fact_273_p_Oa__l__coset__subset__G,axiom,
! [H: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ H ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.a_l_coset_subset_G
thf(fact_274_p_Ogenideal__self,axiom,
! [S: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ord_le8861187494160871172list_a @ S @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ S ) ) ) ).
% p.genideal_self
thf(fact_275_coeff__range,axiom,
! [K: set_a,F: list_a,I: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_a @ ( coeff_a_b @ r @ F @ I ) @ K ) ) ) ).
% coeff_range
thf(fact_276_p_Ocarrier__not__empty,axiom,
( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) )
!= bot_bot_set_list_a ) ).
% p.carrier_not_empty
thf(fact_277_p_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ U @ X2 )
= X2 ) )
=> ( U
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.one_unique
thf(fact_278_p_Oinv__unique,axiom,
! [Y: list_a,X: list_a,Y2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% p.inv_unique
thf(fact_279_p_Osubset__Idl__subset,axiom,
! [I3: set_list_a,H: set_list_a] :
( ( ord_le8861187494160871172list_a @ I3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ord_le8861187494160871172list_a @ H @ I3 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ H ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ I3 ) ) ) ) ).
% p.subset_Idl_subset
thf(fact_280_p_Oone__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).
% p.one_closed
thf(fact_281_p_Or__one,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 ) @ X @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= X ) ) ).
% p.r_one
thf(fact_282_p_Ol__one,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 ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ X )
= X ) ) ).
% p.l_one
thf(fact_283_ring_Omonom_Ocong,axiom,
monom_7446464087056152608t_unit = monom_7446464087056152608t_unit ).
% ring.monom.cong
thf(fact_284_ring_Omonom_Ocong,axiom,
monom_a_b = monom_a_b ).
% ring.monom.cong
thf(fact_285_p_OsubdomainI,axiom,
! [H: set_list_a] :
( ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ k ) )
=> ( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ! [H1: list_a,H2: list_a] :
( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ H1 @ H2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( H1
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
| ( H2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) )
=> ( subdom7821232466298058046t_unit @ H @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.subdomainI
thf(fact_286_p_Ogenideal__one,axiom,
( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) )
= ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.genideal_one
thf(fact_287_p_OIdl__subset__ideal_H,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) )
= ( member_list_a @ A @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ) ) ).
% p.Idl_subset_ideal'
thf(fact_288_p_Oone__zeroI,axiom,
( ( ( 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 ) )
=> ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.one_zeroI
thf(fact_289_p_Oone__zeroD,axiom,
( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( 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 ) ) ) ).
% p.one_zeroD
thf(fact_290_p_Ocarrier__one__zero,axiom,
( ( ( 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 ) )
= ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.carrier_one_zero
thf(fact_291_p_Ocarrier__one__not__zero,axiom,
( ( ( 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 ) )
= ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.carrier_one_not_zero
thf(fact_292_p_Ogenideal__self_H,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ I @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ I @ bot_bot_set_list_a ) ) ) ) ).
% p.genideal_self'
thf(fact_293_p_Ogenideal__zero,axiom,
( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) ).
% p.genideal_zero
thf(fact_294_p_Ozeropideal,axiom,
princi8786919440553033881t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ k ) ).
% p.zeropideal
thf(fact_295_p_Odomain__eq__zeroprimeideal,axiom,
( ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ k ) )
= ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.domain_eq_zeroprimeideal
thf(fact_296_p_Ozeroprimeideal__domainI,axiom,
( ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ k ) )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.zeroprimeideal_domainI
thf(fact_297_singleton__insert__inj__eq,axiom,
! [B: list_a,A: list_a,A2: set_list_a] :
( ( ( insert_list_a @ B @ bot_bot_set_list_a )
= ( insert_list_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_298_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_299_singleton__insert__inj__eq_H,axiom,
! [A: list_a,A2: set_list_a,B: list_a] :
( ( ( insert_list_a @ A @ A2 )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_300_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_301_subset__Compl__singleton,axiom,
! [A2: set_list_list_a,B: list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ ( uminus4049073354455507169list_a @ ( insert_list_list_a @ B @ bot_bo1875519244922727510list_a ) ) )
= ( ~ ( member_list_list_a @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_302_subset__Compl__singleton,axiom,
! [A2: set_list_a,B: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( uminus7925729386456332763list_a @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) )
= ( ~ ( member_list_a @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_303_subset__Compl__singleton,axiom,
! [A2: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ ( uminus_uminus_set_a @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_304_p_Oset__add__zero,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ A2 )
= A2 ) ) ).
% p.set_add_zero
thf(fact_305_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_306_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_307_subsetI,axiom,
! [A2: set_list_list_a,B4: set_list_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A2 )
=> ( member_list_list_a @ X2 @ B4 ) )
=> ( ord_le8488217952732425610list_a @ A2 @ B4 ) ) ).
% subsetI
thf(fact_308_subsetI,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( member_list_a @ X2 @ B4 ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B4 ) ) ).
% subsetI
thf(fact_309_subsetI,axiom,
! [A2: set_a,B4: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B4 ) )
=> ( ord_less_eq_set_a @ A2 @ B4 ) ) ).
% subsetI
thf(fact_310_subset__antisym,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B4 )
=> ( ( ord_le8861187494160871172list_a @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% subset_antisym
thf(fact_311_subset__antisym,axiom,
! [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% subset_antisym
thf(fact_312_empty__iff,axiom,
! [C: list_list_a] :
~ ( member_list_list_a @ C @ bot_bo1875519244922727510list_a ) ).
% empty_iff
thf(fact_313_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_314_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_315_all__not__in__conv,axiom,
! [A2: set_list_list_a] :
( ( ! [X3: list_list_a] :
~ ( member_list_list_a @ X3 @ A2 ) )
= ( A2 = bot_bo1875519244922727510list_a ) ) ).
% all_not_in_conv
thf(fact_316_all__not__in__conv,axiom,
! [A2: set_list_a] :
( ( ! [X3: list_a] :
~ ( member_list_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_317_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_318_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_319_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_320_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_321_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_322_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_323_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_324_p_Osetadd__subset__G,axiom,
! [H: set_list_a,K: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ H @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.setadd_subset_G
thf(fact_325_p_Oset__add__comm,axiom,
! [I3: set_list_a,J2: set_list_a] :
( ( ord_le8861187494160871172list_a @ I3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ord_le8861187494160871172list_a @ J2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ I3 @ J2 )
= ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ J2 @ I3 ) ) ) ) ).
% p.set_add_comm
thf(fact_326_p_Oset__add__closed,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ord_le8861187494160871172list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ A2 @ B4 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.set_add_closed
thf(fact_327_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_328_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_329_empty__subsetI,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% empty_subsetI
thf(fact_330_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_331_insert__subset,axiom,
! [X: list_list_a,A2: set_list_list_a,B4: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ ( insert_list_list_a @ X @ A2 ) @ B4 )
= ( ( member_list_list_a @ X @ B4 )
& ( ord_le8488217952732425610list_a @ A2 @ B4 ) ) ) ).
% insert_subset
thf(fact_332_insert__subset,axiom,
! [X: list_a,A2: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X @ A2 ) @ B4 )
= ( ( member_list_a @ X @ B4 )
& ( ord_le8861187494160871172list_a @ A2 @ B4 ) ) ) ).
% insert_subset
thf(fact_333_insert__subset,axiom,
! [X: a,A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B4 )
= ( ( member_a @ X @ B4 )
& ( ord_less_eq_set_a @ A2 @ B4 ) ) ) ).
% insert_subset
thf(fact_334_singletonI,axiom,
! [A: list_list_a] : ( member_list_list_a @ A @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) ).
% singletonI
thf(fact_335_singletonI,axiom,
! [A: list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_336_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_337_Diff__empty,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% Diff_empty
thf(fact_338_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_339_empty__Diff,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ bot_bot_set_list_a @ A2 )
= bot_bot_set_list_a ) ).
% empty_Diff
thf(fact_340_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_341_Diff__cancel,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ A2 )
= bot_bot_set_list_a ) ).
% Diff_cancel
thf(fact_342_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_343_Compl__subset__Compl__iff,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( uminus7925729386456332763list_a @ A2 ) @ ( uminus7925729386456332763list_a @ B4 ) )
= ( ord_le8861187494160871172list_a @ B4 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_344_Compl__subset__Compl__iff,axiom,
! [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( uminus_uminus_set_a @ B4 ) )
= ( ord_less_eq_set_a @ B4 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_345_Compl__anti__mono,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B4 )
=> ( ord_le8861187494160871172list_a @ ( uminus7925729386456332763list_a @ B4 ) @ ( uminus7925729386456332763list_a @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_346_Compl__anti__mono,axiom,
! [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B4 ) @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_347_Diff__eq__empty__iff,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ( ( minus_646659088055828811list_a @ A2 @ B4 )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A2 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_348_Diff__eq__empty__iff,axiom,
! [A2: set_a,B4: set_a] :
( ( ( minus_minus_set_a @ A2 @ B4 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_349_insert__Diff__single,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
= ( insert_list_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_350_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_351_in__mono,axiom,
! [A2: set_list_list_a,B4: set_list_list_a,X: list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ B4 )
=> ( ( member_list_list_a @ X @ A2 )
=> ( member_list_list_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_352_in__mono,axiom,
! [A2: set_list_a,B4: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B4 )
=> ( ( member_list_a @ X @ A2 )
=> ( member_list_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_353_in__mono,axiom,
! [A2: set_a,B4: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_354_subsetD,axiom,
! [A2: set_list_list_a,B4: set_list_list_a,C: list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ B4 )
=> ( ( member_list_list_a @ C @ A2 )
=> ( member_list_list_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_355_subsetD,axiom,
! [A2: set_list_a,B4: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B4 )
=> ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_356_subsetD,axiom,
! [A2: set_a,B4: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_357_equalityE,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ( A2 = B4 )
=> ~ ( ( ord_le8861187494160871172list_a @ A2 @ B4 )
=> ~ ( ord_le8861187494160871172list_a @ B4 @ A2 ) ) ) ).
% equalityE
thf(fact_358_equalityE,axiom,
! [A2: set_a,B4: set_a] :
( ( A2 = B4 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B4 )
=> ~ ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ).
% equalityE
thf(fact_359_subset__eq,axiom,
( ord_le8488217952732425610list_a
= ( ^ [A5: set_list_list_a,B5: set_list_list_a] :
! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A5 )
=> ( member_list_list_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_360_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B5: set_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ A5 )
=> ( member_list_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_361_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_362_equalityD1,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ( A2 = B4 )
=> ( ord_le8861187494160871172list_a @ A2 @ B4 ) ) ).
% equalityD1
thf(fact_363_equalityD1,axiom,
! [A2: set_a,B4: set_a] :
( ( A2 = B4 )
=> ( ord_less_eq_set_a @ A2 @ B4 ) ) ).
% equalityD1
thf(fact_364_equalityD2,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ( A2 = B4 )
=> ( ord_le8861187494160871172list_a @ B4 @ A2 ) ) ).
% equalityD2
thf(fact_365_equalityD2,axiom,
! [A2: set_a,B4: set_a] :
( ( A2 = B4 )
=> ( ord_less_eq_set_a @ B4 @ A2 ) ) ).
% equalityD2
thf(fact_366_subset__iff,axiom,
( ord_le8488217952732425610list_a
= ( ^ [A5: set_list_list_a,B5: set_list_list_a] :
! [T: list_list_a] :
( ( member_list_list_a @ T @ A5 )
=> ( member_list_list_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_367_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B5: set_list_a] :
! [T: list_a] :
( ( member_list_a @ T @ A5 )
=> ( member_list_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_368_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_369_subset__refl,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_370_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_371_Collect__mono,axiom,
! [P: list_a > $o,Q2: list_a > $o] :
( ! [X2: list_a] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_372_Collect__mono,axiom,
! [P: a > $o,Q2: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_373_subset__trans,axiom,
! [A2: set_list_a,B4: set_list_a,C4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B4 )
=> ( ( ord_le8861187494160871172list_a @ B4 @ C4 )
=> ( ord_le8861187494160871172list_a @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_374_subset__trans,axiom,
! [A2: set_a,B4: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C4 )
=> ( ord_less_eq_set_a @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_375_set__eq__subset,axiom,
( ( ^ [Y4: set_list_a,Z2: set_list_a] : ( Y4 = Z2 ) )
= ( ^ [A5: set_list_a,B5: set_list_a] :
( ( ord_le8861187494160871172list_a @ A5 @ B5 )
& ( ord_le8861187494160871172list_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_376_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_377_Collect__mono__iff,axiom,
! [P: list_a > $o,Q2: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q2 ) )
= ( ! [X3: list_a] :
( ( P @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_378_Collect__mono__iff,axiom,
! [P: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q2 ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_379_double__diff,axiom,
! [A2: set_list_a,B4: set_list_a,C4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B4 )
=> ( ( ord_le8861187494160871172list_a @ B4 @ C4 )
=> ( ( minus_646659088055828811list_a @ B4 @ ( minus_646659088055828811list_a @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_380_double__diff,axiom,
! [A2: set_a,B4: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C4 )
=> ( ( minus_minus_set_a @ B4 @ ( minus_minus_set_a @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_381_Diff__subset,axiom,
! [A2: set_list_a,B4: set_list_a] : ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ B4 ) @ A2 ) ).
% Diff_subset
thf(fact_382_Diff__subset,axiom,
! [A2: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B4 ) @ A2 ) ).
% Diff_subset
thf(fact_383_Diff__mono,axiom,
! [A2: set_list_a,C4: set_list_a,D2: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ C4 )
=> ( ( ord_le8861187494160871172list_a @ D2 @ B4 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ B4 ) @ ( minus_646659088055828811list_a @ C4 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_384_Diff__mono,axiom,
! [A2: set_a,C4: set_a,D2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ C4 )
=> ( ( ord_less_eq_set_a @ D2 @ B4 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B4 ) @ ( minus_minus_set_a @ C4 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_385_emptyE,axiom,
! [A: list_list_a] :
~ ( member_list_list_a @ A @ bot_bo1875519244922727510list_a ) ).
% emptyE
thf(fact_386_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_387_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_388_equals0D,axiom,
! [A2: set_list_list_a,A: list_list_a] :
( ( A2 = bot_bo1875519244922727510list_a )
=> ~ ( member_list_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_389_equals0D,axiom,
! [A2: set_list_a,A: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_390_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_391_equals0I,axiom,
! [A2: set_list_list_a] :
( ! [Y5: list_list_a] :
~ ( member_list_list_a @ Y5 @ A2 )
=> ( A2 = bot_bo1875519244922727510list_a ) ) ).
% equals0I
thf(fact_392_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y5: list_a] :
~ ( member_list_a @ Y5 @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_393_equals0I,axiom,
! [A2: set_a] :
( ! [Y5: a] :
~ ( member_a @ Y5 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_394_ex__in__conv,axiom,
! [A2: set_list_list_a] :
( ( ? [X3: list_list_a] : ( member_list_list_a @ X3 @ A2 ) )
= ( A2 != bot_bo1875519244922727510list_a ) ) ).
% ex_in_conv
thf(fact_395_ex__in__conv,axiom,
! [A2: set_list_a] :
( ( ? [X3: list_a] : ( member_list_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_396_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_397_domain_Ovar__closed_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( member_list_a @ ( var_a_b @ R ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ).
% domain.var_closed(1)
thf(fact_398_domain_Ovar__closed_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( member_list_list_a @ ( var_li8453953174693405341t_unit @ R ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ).
% domain.var_closed(1)
thf(fact_399_domain_Opderiv__var,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( formal6075833236969493044t_unit @ R @ ( var_li8453953174693405341t_unit @ R ) )
= ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.pderiv_var
thf(fact_400_domain_Opderiv__var,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( formal4452980811800949548iv_a_b @ R @ ( var_a_b @ R ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.pderiv_var
thf(fact_401_subset__Compl__self__eq,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( uminus7925729386456332763list_a @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% subset_Compl_self_eq
thf(fact_402_subset__Compl__self__eq,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( uminus_uminus_set_a @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_403_subset__Diff__insert,axiom,
! [A2: set_list_list_a,B4: set_list_list_a,X: list_list_a,C4: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ ( minus_5335179877275218001list_a @ B4 @ ( insert_list_list_a @ X @ C4 ) ) )
= ( ( ord_le8488217952732425610list_a @ A2 @ ( minus_5335179877275218001list_a @ B4 @ C4 ) )
& ~ ( member_list_list_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_404_subset__Diff__insert,axiom,
! [A2: set_list_a,B4: set_list_a,X: list_a,C4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( minus_646659088055828811list_a @ B4 @ ( insert_list_a @ X @ C4 ) ) )
= ( ( ord_le8861187494160871172list_a @ A2 @ ( minus_646659088055828811list_a @ B4 @ C4 ) )
& ~ ( member_list_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_405_subset__Diff__insert,axiom,
! [A2: set_a,B4: set_a,X: a,C4: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B4 @ ( insert_a @ X @ C4 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B4 @ C4 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_406_subset__insertI2,axiom,
! [A2: set_list_a,B4: set_list_a,B: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B4 )
=> ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_407_subset__insertI2,axiom,
! [A2: set_a,B4: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_408_subset__insertI,axiom,
! [B4: set_list_a,A: list_a] : ( ord_le8861187494160871172list_a @ B4 @ ( insert_list_a @ A @ B4 ) ) ).
% subset_insertI
thf(fact_409_subset__insertI,axiom,
! [B4: set_a,A: a] : ( ord_less_eq_set_a @ B4 @ ( insert_a @ A @ B4 ) ) ).
% subset_insertI
thf(fact_410_subset__insert,axiom,
! [X: list_list_a,A2: set_list_list_a,B4: set_list_list_a] :
( ~ ( member_list_list_a @ X @ A2 )
=> ( ( ord_le8488217952732425610list_a @ A2 @ ( insert_list_list_a @ X @ B4 ) )
= ( ord_le8488217952732425610list_a @ A2 @ B4 ) ) ) ).
% subset_insert
thf(fact_411_subset__insert,axiom,
! [X: list_a,A2: set_list_a,B4: set_list_a] :
( ~ ( member_list_a @ X @ A2 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ B4 ) )
= ( ord_le8861187494160871172list_a @ A2 @ B4 ) ) ) ).
% subset_insert
thf(fact_412_subset__insert,axiom,
! [X: a,A2: set_a,B4: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B4 ) )
= ( ord_less_eq_set_a @ A2 @ B4 ) ) ) ).
% subset_insert
thf(fact_413_insert__mono,axiom,
! [C4: set_list_a,D2: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ C4 @ D2 )
=> ( ord_le8861187494160871172list_a @ ( insert_list_a @ A @ C4 ) @ ( insert_list_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_414_insert__mono,axiom,
! [C4: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C4 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C4 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_415_singleton__inject,axiom,
! [A: list_a,B: list_a] :
( ( ( insert_list_a @ A @ bot_bot_set_list_a )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_416_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_417_insert__not__empty,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ A2 )
!= bot_bot_set_list_a ) ).
% insert_not_empty
thf(fact_418_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_419_doubleton__eq__iff,axiom,
! [A: list_a,B: list_a,C: list_a,D: list_a] :
( ( ( insert_list_a @ A @ ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( insert_list_a @ C @ ( insert_list_a @ D @ bot_bot_set_list_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_420_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_421_singleton__iff,axiom,
! [B: list_list_a,A: list_list_a] :
( ( member_list_list_a @ B @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_422_singleton__iff,axiom,
! [B: list_a,A: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_423_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_424_singletonD,axiom,
! [B: list_list_a,A: list_list_a] :
( ( member_list_list_a @ B @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_425_singletonD,axiom,
! [B: list_a,A: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_426_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_427_Diff__insert__absorb,axiom,
! [X: list_list_a,A2: set_list_list_a] :
( ~ ( member_list_list_a @ X @ A2 )
=> ( ( minus_5335179877275218001list_a @ ( insert_list_list_a @ X @ A2 ) @ ( insert_list_list_a @ X @ bot_bo1875519244922727510list_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_428_Diff__insert__absorb,axiom,
! [X: list_a,A2: set_list_a] :
( ~ ( member_list_a @ X @ A2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A2 ) @ ( insert_list_a @ X @ bot_bot_set_list_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_429_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_430_Diff__insert2,axiom,
! [A2: set_list_a,A: list_a,B4: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B4 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_431_Diff__insert2,axiom,
! [A2: set_a,A: a,B4: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B4 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_432_Compl__insert,axiom,
! [X: list_a,A2: set_list_a] :
( ( uminus7925729386456332763list_a @ ( insert_list_a @ X @ A2 ) )
= ( minus_646659088055828811list_a @ ( uminus7925729386456332763list_a @ A2 ) @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ).
% Compl_insert
thf(fact_433_Compl__insert,axiom,
! [X: a,A2: set_a] :
( ( uminus_uminus_set_a @ ( insert_a @ X @ A2 ) )
= ( minus_minus_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% Compl_insert
thf(fact_434_insert__Diff,axiom,
! [A: list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ A @ A2 )
=> ( ( insert_list_list_a @ A @ ( minus_5335179877275218001list_a @ A2 @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_435_insert__Diff,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_436_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_437_Diff__insert,axiom,
! [A2: set_list_a,A: list_a,B4: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B4 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ B4 ) @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ).
% Diff_insert
thf(fact_438_Diff__insert,axiom,
! [A2: set_a,A: a,B4: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B4 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B4 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_439_subset__insert__iff,axiom,
! [A2: set_list_list_a,X: list_list_a,B4: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ ( insert_list_list_a @ X @ B4 ) )
= ( ( ( member_list_list_a @ X @ A2 )
=> ( ord_le8488217952732425610list_a @ ( minus_5335179877275218001list_a @ A2 @ ( insert_list_list_a @ X @ bot_bo1875519244922727510list_a ) ) @ B4 ) )
& ( ~ ( member_list_list_a @ X @ A2 )
=> ( ord_le8488217952732425610list_a @ A2 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_440_subset__insert__iff,axiom,
! [A2: set_list_a,X: list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ B4 ) )
= ( ( ( member_list_a @ X @ A2 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B4 ) )
& ( ~ ( member_list_a @ X @ A2 )
=> ( ord_le8861187494160871172list_a @ A2 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_441_subset__insert__iff,axiom,
! [A2: set_a,X: a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B4 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B4 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_442_Diff__single__insert,axiom,
! [A2: set_list_a,X: list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B4 )
=> ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_443_Diff__single__insert,axiom,
! [A2: set_a,X: a,B4: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B4 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_444_subset__singleton__iff,axiom,
! [X4: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( ( X4 = bot_bot_set_list_a )
| ( X4
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_445_subset__singleton__iff,axiom,
! [X4: set_a,A: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_446_subset__singletonD,axiom,
! [A2: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) )
=> ( ( A2 = bot_bot_set_list_a )
| ( A2
= ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ) ).
% subset_singletonD
thf(fact_447_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_448_p_Omaximalideal__prime,axiom,
! [I3: set_list_a] :
( ( maxima6585700282301356660t_unit @ I3 @ ( univ_poly_a_b @ r @ k ) )
=> ( primei6309817859076077608t_unit @ I3 @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.maximalideal_prime
thf(fact_449_domain_Ovar__carr,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% domain.var_carr
thf(fact_450_domain_Ovar__carr,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( member_list_list_a @ ( var_li8453953174693405341t_unit @ R ) @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) @ bot_bo1875519244922727510list_a ) ) ) ) ) ).
% domain.var_carr
thf(fact_451_p_Oadd__additive__subgroups,axiom,
! [H: set_list_a,K: set_list_a] :
( ( additi4714453376129182166t_unit @ H @ ( univ_poly_a_b @ r @ k ) )
=> ( ( additi4714453376129182166t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( additi4714453376129182166t_unit @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ H @ K ) @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.add_additive_subgroups
thf(fact_452_p_Osubfield__m__inv__simprule,axiom,
! [K: set_list_a,K2: list_a,A: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ K2 @ A ) @ K )
=> ( member_list_a @ A @ K ) ) ) ) ) ).
% p.subfield_m_inv_simprule
thf(fact_453_var__pow__carr,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 ) @ ( 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_pow_carr
thf(fact_454_semiring_Oone__zeroD,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_455_semiring_Oone__zeroD,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( 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 ) ) ) ) ).
% semiring.one_zeroD
thf(fact_456_semiring_Oone__zeroD,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( ( one_li8234411390022467901t_unit @ R )
= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( partia2464479390973590831t_unit @ R )
= ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_457_semiring_Oone__zeroI,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
=> ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_458_semiring_Oone__zeroI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( 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 ) ) ) ) ).
% semiring.one_zeroI
thf(fact_459_semiring_Oone__zeroI,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( ( partia2464479390973590831t_unit @ R )
= ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) )
=> ( ( one_li8234411390022467901t_unit @ R )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_460_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_461_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_462_monom__coeff,axiom,
! [A: a,N: nat] :
( ( coeff_a_b @ r @ ( monom_a_b @ r @ A @ N ) )
= ( ^ [I2: nat] : ( if_a @ ( I2 = N ) @ A @ ( zero_a_b @ r ) ) ) ) ).
% monom_coeff
thf(fact_463_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_464_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_465_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_466_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_467_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_468_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_469_p_Osubring__props_I2_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ K ) ) ).
% p.subring_props(2)
thf(fact_470_p_Osubring__props_I7_J,axiom,
! [K: set_list_a,H12: list_a,H22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( member_list_a @ H12 @ K )
=> ( ( member_list_a @ H22 @ K )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ H12 @ H22 ) @ K ) ) ) ) ).
% p.subring_props(7)
thf(fact_471_p_Onat__pow__pow,axiom,
! [X: list_a,N: nat,M: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) @ M )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ ( times_times_nat @ N @ M ) ) ) ) ).
% p.nat_pow_pow
thf(fact_472_p_Osubring__props_I6_J,axiom,
! [K: set_list_a,H12: list_a,H22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( member_list_a @ H12 @ K )
=> ( ( member_list_a @ H22 @ K )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ H12 @ H22 ) @ K ) ) ) ) ).
% p.subring_props(6)
thf(fact_473_p_Osubring__props_I4_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( K != bot_bot_set_list_a ) ) ).
% p.subring_props(4)
thf(fact_474_p_Osubring__props_I5_J,axiom,
! [K: set_list_a,H3: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( member_list_a @ H3 @ K )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ H3 ) @ K ) ) ) ).
% p.subring_props(5)
thf(fact_475_p_Osubring__props_I3_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ K ) ) ).
% p.subring_props(3)
thf(fact_476_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_477_p_Osubring__props_I1_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.subring_props(1)
thf(fact_478_p_Onat__pow__mult,axiom,
! [X: list_a,N: nat,M: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ M ) )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ ( plus_plus_nat @ N @ M ) ) ) ) ).
% p.nat_pow_mult
thf(fact_479_p_Ogroup__commutes__pow,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) ) ) ) ) ) ).
% p.group_commutes_pow
thf(fact_480_p_Onat__pow__comm,axiom,
! [X: list_a,N: nat,M: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ M ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ M ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) ) ) ) ).
% p.nat_pow_comm
thf(fact_481_p_Onat__pow__distrib,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ Y @ N ) ) ) ) ) ).
% p.nat_pow_distrib
thf(fact_482_p_Opow__mult__distrib,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ Y @ N ) ) ) ) ) ) ).
% p.pow_mult_distrib
thf(fact_483_p_Oline__extension__smult__closed,axiom,
! [K: set_list_a,E: set_list_a,A: list_a,K2: list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ! [K3: list_a,V2: list_a] :
( ( member_list_a @ K3 @ K )
=> ( ( member_list_a @ V2 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ K3 @ V2 ) @ E ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ K2 @ K )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ A @ E ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ K2 @ U ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ A @ E ) ) ) ) ) ) ) ) ).
% p.line_extension_smult_closed
thf(fact_484_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_485_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_486_p_Onat__pow__closed,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.nat_pow_closed
thf(fact_487_p_Onat__pow__one,axiom,
! [N: nat] :
( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ N )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.nat_pow_one
thf(fact_488_domain_Opow__non__zero,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( X
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( pow_li1142815632869257134it_nat @ R @ X @ N )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.pow_non_zero
thf(fact_489_domain_Opow__non__zero,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,N: nat] :
( ( domain_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( X
!= ( zero_a_b @ R ) )
=> ( ( pow_a_1026414303147256608_b_nat @ R @ X @ N )
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.pow_non_zero
thf(fact_490_domain_Opow__non__zero,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( X
!= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( pow_li488931774710091566it_nat @ R @ X @ N )
!= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ).
% domain.pow_non_zero
thf(fact_491_domain_Ovar__pow__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% domain.var_pow_closed
thf(fact_492_domain_Ovar__pow__closed,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( member_list_list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( var_li8453953174693405341t_unit @ R ) @ N ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ).
% domain.var_pow_closed
thf(fact_493_domain_Ounitary__monom__eq__var__pow,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( monom_7446464087056152608t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ N )
= ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( var_li8453953174693405341t_unit @ R ) @ N ) ) ) ) ).
% domain.unitary_monom_eq_var_pow
thf(fact_494_domain_Ounitary__monom__eq__var__pow,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% domain.unitary_monom_eq_var_pow
thf(fact_495_domain_Ovar__pow__carr,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K ) @ ( var_a_b @ R ) @ N ) @ ( 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 ) ) ) ) ) ).
% domain.var_pow_carr
thf(fact_496_domain_Ovar__pow__carr,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( member_list_list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( var_li8453953174693405341t_unit @ R ) @ N ) @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) @ bot_bo1875519244922727510list_a ) ) ) ) ) ).
% domain.var_pow_carr
thf(fact_497_ring__hom__closed,axiom,
! [H3: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H3 @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_498_ring__hom__closed,axiom,
! [H3: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a] :
( ( member_list_a_a @ H3 @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H3 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_499_ring__hom__closed,axiom,
! [H3: list_a > list_list_a,R: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,X: list_a] :
( ( member6714375691612171394list_a @ H3 @ ( ring_h8002040739877300486t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( H3 @ X ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_500_ring__hom__closed,axiom,
! [H3: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a] :
( ( member_a_list_a @ H3 @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H3 @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_501_ring__hom__closed,axiom,
! [H3: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H3 @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H3 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_502_ring__hom__closed,axiom,
! [H3: a > list_list_a,R: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,X: a] :
( ( member_a_list_list_a @ H3 @ ( ring_h6858658657455840382t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_list_a @ ( H3 @ X ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_503_ring__hom__closed,axiom,
! [H3: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X: list_list_a] :
( ( member7168557129179038582list_a @ H3 @ ( ring_h5031276006722532742t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_a @ ( H3 @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_504_ring__hom__closed,axiom,
! [H3: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X: list_list_a] :
( ( member_list_list_a_a @ H3 @ ( ring_h8078271382950527358it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_a @ ( H3 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_505_ring__hom__closed,axiom,
! [H3: list_list_a > list_list_a,R: partia2956882679547061052t_unit,S: partia2956882679547061052t_unit,X: list_list_a] :
( ( member8231385768148312316list_a @ H3 @ ( ring_h8129544334414776832t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( H3 @ X ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_506_ring__hom__one,axiom,
! [H3: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit] :
( ( member_list_a_list_a @ H3 @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( H3 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_507_ring__hom__one,axiom,
! [H3: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_list_a_a @ H3 @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( H3 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_hom_one
thf(fact_508_ring__hom__one,axiom,
! [H3: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H3 @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( H3 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_509_ring__hom__one,axiom,
! [H3: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H3 @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( H3 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_hom_one
thf(fact_510_domain_Ozero__not__one,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( zero_l4142658623432671053t_unit @ R )
!= ( one_li8328186300101108157t_unit @ R ) ) ) ).
% domain.zero_not_one
thf(fact_511_domain_Ozero__not__one,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( zero_a_b @ R )
!= ( one_a_ring_ext_a_b @ R ) ) ) ).
% domain.zero_not_one
thf(fact_512_domain_Oone__not__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% domain.one_not_zero
thf(fact_513_domain_Oone__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% domain.one_not_zero
thf(fact_514_ring__hom__add,axiom,
! [H3: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_515_ring__hom__add,axiom,
! [H3: list_a > list_list_a,R: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,X: list_a,Y: list_a] :
( ( member6714375691612171394list_a @ H3 @ ( ring_h8002040739877300486t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li174743652000525320t_unit @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_516_ring__hom__add,axiom,
! [H3: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H3 @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_517_ring__hom__add,axiom,
! [H3: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H3 @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( add_a_b @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_518_ring__hom__add,axiom,
! [H3: a > list_list_a,R: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,X: a,Y: a] :
( ( member_a_list_list_a @ H3 @ ( ring_h6858658657455840382t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( add_a_b @ R @ X @ Y ) )
= ( add_li174743652000525320t_unit @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_519_ring__hom__add,axiom,
! [H3: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H3 @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_520_ring__hom__add,axiom,
! [H3: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H3 @ ( ring_h5031276006722532742t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_521_ring__hom__add,axiom,
! [H3: list_list_a > list_list_a,R: partia2956882679547061052t_unit,S: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( member8231385768148312316list_a @ H3 @ ( ring_h8129544334414776832t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
= ( add_li174743652000525320t_unit @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_522_ring__hom__add,axiom,
! [H3: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H3 @ ( ring_h8078271382950527358it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_523_ring__hom__mult,axiom,
! [H3: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_524_ring__hom__mult,axiom,
! [H3: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H3 @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_525_ring__hom__mult,axiom,
! [H3: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H3 @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_526_ring__hom__mult,axiom,
! [H3: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H3 @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_527_ring__hom__mult,axiom,
! [H3: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H3 @ ( ring_h5031276006722532742t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_528_ring__hom__mult,axiom,
! [H3: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H3 @ ( ring_h8078271382950527358it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_529_a__minus__def,axiom,
( a_minu3984020753470702548t_unit
= ( ^ [R2: partia2670972154091845814t_unit,X3: list_a,Y3: list_a] : ( add_li7652885771158616974t_unit @ R2 @ X3 @ ( a_inv_8944721093294617173t_unit @ R2 @ Y3 ) ) ) ) ).
% a_minus_def
thf(fact_530_a__minus__def,axiom,
( a_minu2241224857956505934t_unit
= ( ^ [R2: partia2956882679547061052t_unit,X3: list_list_a,Y3: list_list_a] : ( add_li174743652000525320t_unit @ R2 @ X3 @ ( a_inv_7033018035630854991t_unit @ R2 @ Y3 ) ) ) ) ).
% a_minus_def
thf(fact_531_a__minus__def,axiom,
( a_minus_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,X3: a,Y3: a] : ( add_a_b @ R2 @ X3 @ ( a_inv_a_b @ R2 @ Y3 ) ) ) ) ).
% a_minus_def
thf(fact_532_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_533_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_534_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_535_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_536_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_537_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_538_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_539_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_540_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_541_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_542_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y )
= ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_543_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X @ Y )
= ( add_li174743652000525320t_unit @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_544_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ R @ Y @ ( add_li7652885771158616974t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_545_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_546_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) )
= ( add_li174743652000525320t_unit @ R @ Y @ ( add_li174743652000525320t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_547_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_548_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_549_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_550_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ R @ X @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_551_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_552_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) @ Z )
= ( mult_l4853965630390486993t_unit @ R @ X @ ( mult_l4853965630390486993t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_553_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_554_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_555_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( member_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_556_domain_Ointegral__iff,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
= ( ( A
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_557_domain_Ointegral__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_558_domain_Ointegral__iff,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B )
= ( zero_l347298301471573063t_unit @ R ) )
= ( ( A
= ( zero_l347298301471573063t_unit @ R ) )
| ( B
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_559_domain_Om__rcancel,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ B @ A )
= ( mult_l7073676228092353617t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_560_domain_Om__rcancel,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_561_domain_Om__rcancel,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,C: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( A
!= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ B @ A )
= ( mult_l4853965630390486993t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_562_domain_Om__lcancel,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( mult_l7073676228092353617t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_563_domain_Om__lcancel,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_564_domain_Om__lcancel,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,C: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( A
!= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B )
= ( mult_l4853965630390486993t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_565_domain_Ointegral,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( A
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_566_domain_Ointegral,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( ( 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 ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_567_domain_Ointegral,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B )
= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( A
= ( zero_l347298301471573063t_unit @ R ) )
| ( B
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_568_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_569_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_570_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_571_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_572_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_573_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X @ ( zero_l347298301471573063t_unit @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_574_semiring_Or__null,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.r_null
thf(fact_575_semiring_Or__null,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.r_null
thf(fact_576_semiring_Or__null,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ X @ ( zero_l347298301471573063t_unit @ R ) )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.r_null
thf(fact_577_semiring_Ol__null,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.l_null
thf(fact_578_semiring_Ol__null,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.l_null
thf(fact_579_semiring_Ol__null,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.l_null
thf(fact_580_semiring_Or__distr,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ Z @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_581_semiring_Or__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( 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 @ X @ Y ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_582_semiring_Or__distr,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ Z @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
= ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ Z @ X ) @ ( mult_l4853965630390486993t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_583_semiring_Ol__distr,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_584_semiring_Ol__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_585_semiring_Ol__distr,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Z ) @ ( mult_l4853965630390486993t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_586_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_587_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_588_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( one_li8234411390022467901t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_589_properfactor__hom,axiom,
! [H3: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,B: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( domain6553523120543210313t_unit @ R )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( proper8313688649498433056t_unit @ R @ B @ X )
= ( proper8313688649498433056t_unit @ S @ ( H3 @ B ) @ ( H3 @ X ) ) ) ) ) ) ) ) ).
% properfactor_hom
thf(fact_590_properfactor__hom,axiom,
! [H3: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,B: list_a] :
( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( domain6553523120543210313t_unit @ R )
=> ( ( domain_a_b @ S )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( proper8313688649498433056t_unit @ R @ B @ X )
= ( proper19828929941537682xt_a_b @ S @ ( H3 @ B ) @ ( H3 @ X ) ) ) ) ) ) ) ) ).
% properfactor_hom
thf(fact_591_properfactor__hom,axiom,
! [H3: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,B: a] :
( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( domain_a_b @ R )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( proper19828929941537682xt_a_b @ R @ B @ X )
= ( proper8313688649498433056t_unit @ S @ ( H3 @ B ) @ ( H3 @ X ) ) ) ) ) ) ) ) ).
% properfactor_hom
thf(fact_592_properfactor__hom,axiom,
! [H3: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,B: a] :
( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( domain_a_b @ R )
=> ( ( domain_a_b @ S )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( proper19828929941537682xt_a_b @ R @ B @ X )
= ( proper19828929941537682xt_a_b @ S @ ( H3 @ B ) @ ( H3 @ X ) ) ) ) ) ) ) ) ).
% properfactor_hom
thf(fact_593_properfactor__hom,axiom,
! [H3: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X: list_list_a,B: list_list_a] :
( ( member7168557129179038582list_a @ H3 @ ( ring_i4611353245267337884t_unit @ R @ S ) )
=> ( ( domain7810152921033798211t_unit @ R )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( proper6839530779442076320t_unit @ R @ B @ X )
= ( proper8313688649498433056t_unit @ S @ ( H3 @ B ) @ ( H3 @ X ) ) ) ) ) ) ) ) ).
% properfactor_hom
thf(fact_594_properfactor__hom,axiom,
! [H3: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X: list_list_a,B: list_list_a] :
( ( member_list_list_a_a @ H3 @ ( ring_i5684343068699926420it_a_b @ R @ S ) )
=> ( ( domain7810152921033798211t_unit @ R )
=> ( ( domain_a_b @ S )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( proper6839530779442076320t_unit @ R @ B @ X )
= ( proper19828929941537682xt_a_b @ S @ ( H3 @ B ) @ ( H3 @ X ) ) ) ) ) ) ) ) ).
% properfactor_hom
thf(fact_595_domain_Osquare__eq__one,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ X @ X )
= ( one_li8328186300101108157t_unit @ R ) )
=> ( ( X
= ( one_li8328186300101108157t_unit @ R ) )
| ( X
= ( a_inv_8944721093294617173t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_596_domain_Osquare__eq__one,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ X @ X )
= ( one_a_ring_ext_a_b @ R ) )
=> ( ( X
= ( one_a_ring_ext_a_b @ R ) )
| ( X
= ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_597_domain_Osquare__eq__one,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ X @ X )
= ( one_li8234411390022467901t_unit @ R ) )
=> ( ( X
= ( one_li8234411390022467901t_unit @ R ) )
| ( X
= ( a_inv_7033018035630854991t_unit @ R @ ( one_li8234411390022467901t_unit @ R ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_598_ring__hom__memI,axiom,
! [R: partia2670972154091845814t_unit,H3: list_a > list_a,S: partia2670972154091845814t_unit] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H3 @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X2: list_a,Y5: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R @ X2 @ Y5 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X2: list_a,Y5: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y5 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( member_list_a_list_a @ H3 @ ( ring_h7399960747407462284t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_599_ring__hom__memI,axiom,
! [R: partia2670972154091845814t_unit,H3: list_a > a,S: partia2175431115845679010xt_a_b] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H3 @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X2: list_a,Y5: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R @ X2 @ Y5 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X2: list_a,Y5: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y5 ) )
= ( add_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_list_a_a @ H3 @ ( ring_h2895973938487309444it_a_b @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_600_ring__hom__memI,axiom,
! [R: partia2670972154091845814t_unit,H3: list_a > list_list_a,S: partia2956882679547061052t_unit] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( H3 @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) )
=> ( ! [X2: list_a,Y5: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R @ X2 @ Y5 ) )
= ( mult_l4853965630390486993t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X2: list_a,Y5: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y5 ) )
= ( add_li174743652000525320t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_li8234411390022467901t_unit @ S ) )
=> ( member6714375691612171394list_a @ H3 @ ( ring_h8002040739877300486t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_601_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H3: a > list_a,S: partia2670972154091845814t_unit] :
( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H3 @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X2: a,Y5: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y5 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X2: a,Y5: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( add_a_b @ R @ X2 @ Y5 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( member_a_list_a @ H3 @ ( ring_h405018892823518980t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_602_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H3: a > a,S: partia2175431115845679010xt_a_b] :
( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H3 @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X2: a,Y5: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y5 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X2: a,Y5: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( add_a_b @ R @ X2 @ Y5 ) )
= ( add_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_a_a @ H3 @ ( ring_hom_a_b_a_b @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_603_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H3: a > list_list_a,S: partia2956882679547061052t_unit] :
( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_list_a @ ( H3 @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) )
=> ( ! [X2: a,Y5: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y5 ) )
= ( mult_l4853965630390486993t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X2: a,Y5: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( add_a_b @ R @ X2 @ Y5 ) )
= ( add_li174743652000525320t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_li8234411390022467901t_unit @ S ) )
=> ( member_a_list_list_a @ H3 @ ( ring_h6858658657455840382t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_604_ring__hom__memI,axiom,
! [R: partia2956882679547061052t_unit,H3: list_list_a > list_a,S: partia2670972154091845814t_unit] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_a @ ( H3 @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X2: list_list_a,Y5: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( mult_l4853965630390486993t_unit @ R @ X2 @ Y5 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X2: list_list_a,Y5: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( add_li174743652000525320t_unit @ R @ X2 @ Y5 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_li8234411390022467901t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( member7168557129179038582list_a @ H3 @ ( ring_h5031276006722532742t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_605_ring__hom__memI,axiom,
! [R: partia2956882679547061052t_unit,H3: list_list_a > a,S: partia2175431115845679010xt_a_b] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_a @ ( H3 @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X2: list_list_a,Y5: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( mult_l4853965630390486993t_unit @ R @ X2 @ Y5 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X2: list_list_a,Y5: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( add_li174743652000525320t_unit @ R @ X2 @ Y5 ) )
= ( add_a_b @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_li8234411390022467901t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_list_list_a_a @ H3 @ ( ring_h8078271382950527358it_a_b @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_606_ring__hom__memI,axiom,
! [R: partia2956882679547061052t_unit,H3: list_list_a > list_list_a,S: partia2956882679547061052t_unit] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( H3 @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) )
=> ( ! [X2: list_list_a,Y5: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( mult_l4853965630390486993t_unit @ R @ X2 @ Y5 ) )
= ( mult_l4853965630390486993t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X2: list_list_a,Y5: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( add_li174743652000525320t_unit @ R @ X2 @ Y5 ) )
= ( add_li174743652000525320t_unit @ S @ ( H3 @ X2 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_li8234411390022467901t_unit @ R ) )
= ( one_li8234411390022467901t_unit @ S ) )
=> ( member8231385768148312316list_a @ H3 @ ( ring_h8129544334414776832t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_607_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_608_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( 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 ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_609_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( ( partia2464479390973590831t_unit @ R )
!= ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) )
= ( ( one_li8234411390022467901t_unit @ R )
!= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_610_semiring_Ocarrier__one__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_611_semiring_Ocarrier__one__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( 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 ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_612_semiring_Ocarrier__one__zero,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( ( partia2464479390973590831t_unit @ R )
= ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) )
= ( ( one_li8234411390022467901t_unit @ R )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_613_ring__primeE_I1_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( P2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_614_ring__irreducibleE_I1_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( R3
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_615_p_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K: set_list_a,E: set_list_a,V: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E )
=> ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ k ) )
=> ( ( ord_le8861187494160871172list_a @ V @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ V ) ) ) ) ) ).
% p.subalbegra_incl_imp_finite_dimension
thf(fact_616_p_Osubfield__m__inv_I2_J,axiom,
! [K: set_list_a,K2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ K2 @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ K2 ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.subfield_m_inv(2)
thf(fact_617_p_Osubfield__m__inv_I3_J,axiom,
! [K: set_list_a,K2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ K2 ) @ K2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.subfield_m_inv(3)
thf(fact_618_p_Ospace__subgroup__props_I6_J,axiom,
! [K: set_list_a,N: nat,E: set_list_a,K2: list_a,A: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N @ K @ E )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ K2 @ A ) @ E )
=> ( member_list_a @ A @ E ) ) ) ) ) ) ).
% p.space_subgroup_props(6)
thf(fact_619_minus__eq,axiom,
! [X: a,Y: a] :
( ( a_minus_a_b @ r @ X @ Y )
= ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).
% minus_eq
thf(fact_620_pow__mult__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_621_nat__pow__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ).
% nat_pow_distrib
thf(fact_622_nat__pow__comm,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_623_group__commutes__pow,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_624_r__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% r_minus
thf(fact_625_l__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% l_minus
thf(fact_626_r__neg2,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_627_r__neg1,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_628_local_Ominus__add,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_629_a__transpose__inv,axiom,
! [X: a,Y: a,Z: a] :
( ( ( add_a_b @ r @ X @ Y )
= Z )
=> ( ( member_a @ X @ ( 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 @ X ) @ Z )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_630_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_631_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_632_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_633_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_634_add_Oinv__mult__group,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_635_properfactor__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( proper19828929941537682xt_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 ) )
=> ( proper19828929941537682xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% properfactor_prod_r
thf(fact_636_properfactor__prod__l,axiom,
! [A: a,B: a,C: a] :
( ( proper19828929941537682xt_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 ) )
=> ( proper19828929941537682xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ) ).
% properfactor_prod_l
thf(fact_637_r__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( 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 @ X @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_638_l__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_639_m__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( 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 @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_640_m__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% m_comm
thf(fact_641_m__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_642_a__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_643_a__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) ) ) ) ).
% a_comm
thf(fact_644_a__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_645_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_646_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_647_sum__zero__eq__neg,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( X
= ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% sum_zero_eq_neg
thf(fact_648_r__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_649_minus__equality,axiom,
! [Y: a,X: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X )
= Y ) ) ) ) ).
% minus_equality
thf(fact_650_l__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_651_pow__non__zero,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
!= ( zero_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ N )
!= ( zero_a_b @ r ) ) ) ) ).
% pow_non_zero
thf(fact_652_properfactor__of__zero_I2_J,axiom,
! [B: a] :
( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( proper19828929941537682xt_a_b @ r @ B @ ( zero_a_b @ r ) )
= ( B
!= ( zero_a_b @ r ) ) ) ) ).
% properfactor_of_zero(2)
thf(fact_653_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_654_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_655_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_656_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_657_local_Ominus__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_658_add_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X2 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_659_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_660_add_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X2 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_661_add_Oinv__comm,axiom,
! [X: a,Y: a] :
( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_662_nat__pow__mult,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M ) )
= ( pow_a_1026414303147256608_b_nat @ r @ X @ ( plus_plus_nat @ N @ M ) ) ) ) ).
% nat_pow_mult
thf(fact_663_nat__pow__pow,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ M )
= ( pow_a_1026414303147256608_b_nat @ r @ X @ ( times_times_nat @ N @ M ) ) ) ) ).
% nat_pow_pow
thf(fact_664_square__eq__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( X
= ( one_a_ring_ext_a_b @ r ) )
| ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% square_eq_one
thf(fact_665_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_666_inv__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% inv_unique
thf(fact_667_add_Oone__in__subset,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H )
=> ( member_a @ ( a_inv_a_b @ r @ X2 ) @ H ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H )
=> ! [Xa: a] :
( ( member_a @ Xa @ H )
=> ( member_a @ ( add_a_b @ r @ X2 @ Xa ) @ H ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H ) ) ) ) ) ).
% add.one_in_subset
thf(fact_668_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 ) )
=> ( ! [H1: a,H2: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H2 ) @ H ) ) )
=> ( ! [H1: a,H2: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( add_a_b @ r @ H1 @ H2 ) @ H ) ) )
=> ( subring_a_b @ H @ r ) ) ) ) ) ) ).
% subringI
thf(fact_669_p_Odimension__is__inj,axiom,
! [K: set_list_a,N: nat,E: set_list_a,M: nat] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N @ K @ E )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ M @ K @ E )
=> ( N = M ) ) ) ) ).
% p.dimension_is_inj
thf(fact_670_p_Otelescopic__base__dim_I1_J,axiom,
! [K: set_list_a,F2: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( subfie1779122896746047282t_unit @ F2 @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ F2 )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ F2 @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E ) ) ) ) ) ).
% p.telescopic_base_dim(1)
thf(fact_671_p_Ofinite__dimension__def,axiom,
! [K: set_list_a,E: set_list_a] :
( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E )
= ( ? [N2: nat] : ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N2 @ K @ E ) ) ) ).
% p.finite_dimension_def
thf(fact_672_p_Ofinite__dimensionI,axiom,
! [N: nat,K: set_list_a,E: set_list_a] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N @ K @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E ) ) ).
% p.finite_dimensionI
thf(fact_673_p_Ofinite__dimensionE_H,axiom,
! [K: set_list_a,E: set_list_a] :
( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E )
=> ~ ! [N3: nat] :
~ ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N3 @ K @ E ) ) ).
% p.finite_dimensionE'
thf(fact_674_coeff__add,axiom,
! [K: set_a,F: list_a,G: list_a,I: nat] :
( ( 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 ) ) )
=> ( ( coeff_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ F @ G ) @ I )
= ( add_a_b @ r @ ( coeff_a_b @ r @ F @ I ) @ ( coeff_a_b @ r @ G @ I ) ) ) ) ) ) ).
% coeff_add
thf(fact_675_p_Ospace__subgroup__props_I2_J,axiom,
! [K: set_list_a,N: nat,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N @ K @ E )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ E ) ) ) ).
% p.space_subgroup_props(2)
thf(fact_676_p_Ospace__subgroup__props_I3_J,axiom,
! [K: set_list_a,N: nat,E: set_list_a,V1: list_a,V22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N @ K @ E )
=> ( ( member_list_a @ V1 @ E )
=> ( ( member_list_a @ V22 @ E )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ V1 @ V22 ) @ E ) ) ) ) ) ).
% p.space_subgroup_props(3)
thf(fact_677_p_Otelescopic__base,axiom,
! [K: set_list_a,F2: set_list_a,N: nat,M: nat,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( subfie1779122896746047282t_unit @ F2 @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N @ K @ F2 )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ M @ F2 @ E )
=> ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ ( times_times_nat @ N @ M ) @ K @ E ) ) ) ) ) ).
% p.telescopic_base
thf(fact_678_p_Ospace__subgroup__props_I5_J,axiom,
! [K: set_list_a,N: nat,E: set_list_a,K2: list_a,V3: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N @ K @ E )
=> ( ( member_list_a @ K2 @ K )
=> ( ( member_list_a @ V3 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ K2 @ V3 ) @ E ) ) ) ) ) ).
% p.space_subgroup_props(5)
thf(fact_679_p_Ospace__subgroup__props_I4_J,axiom,
! [K: set_list_a,N: nat,E: set_list_a,V3: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N @ K @ E )
=> ( ( member_list_a @ V3 @ E )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ V3 ) @ E ) ) ) ) ).
% p.space_subgroup_props(4)
thf(fact_680_p_Ounique__dimension,axiom,
! [K: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E )
=> ? [X2: nat] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ K @ E )
& ! [Y6: nat] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ Y6 @ K @ E )
=> ( Y6 = X2 ) ) ) ) ) ).
% p.unique_dimension
thf(fact_681_p_Osum__space__dim_I1_J,axiom,
! [K: set_list_a,E: set_list_a,F2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ F2 )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ E @ F2 ) ) ) ) ) ).
% p.sum_space_dim(1)
thf(fact_682_p_Ofinite__dimension__imp__subalgebra,axiom,
! [K: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E )
=> ( embedd1768981623711841426t_unit @ K @ E @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.finite_dimension_imp_subalgebra
thf(fact_683_p_Ospace__subgroup__props_I1_J,axiom,
! [K: set_list_a,N: nat,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N @ K @ E )
=> ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.space_subgroup_props(1)
thf(fact_684_p_Oinv__unique_H,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( Y
= ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) ) ) ) ) ) ).
% p.inv_unique'
thf(fact_685_p_Oinv__char,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ X )
= Y ) ) ) ) ) ).
% p.inv_char
thf(fact_686_p_Ocomm__inv__char,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ X )
= Y ) ) ) ) ).
% p.comm_inv_char
thf(fact_687_nat__pow__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% nat_pow_closed
thf(fact_688_local_Ominus__minus,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
= X ) ) ).
% local.minus_minus
thf(fact_689_a__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_690_m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_691_a__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_692_local_Oadd_Oright__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X )
= ( add_a_b @ r @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_693_minus__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_694_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_695_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_696_p_Osubfield__m__inv_I1_J,axiom,
! [K: set_list_a,K2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ K2 ) @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) ) ) ) ).
% p.subfield_m_inv(1)
thf(fact_697_add_Oinv__eq__1__iff,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X )
= ( zero_a_b @ r ) )
= ( X
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_698_r__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_699_l__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_700_r__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
= X ) ) ).
% r_zero
thf(fact_701_l__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
= X ) ) ).
% l_zero
thf(fact_702_add_Or__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ A @ X ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_703_add_Or__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_704_add_Ol__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ X @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_705_add_Ol__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ A )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_706_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_707_r__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
= X ) ) ).
% r_one
thf(fact_708_l__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
= X ) ) ).
% l_one
thf(fact_709_p_Oinv__one,axiom,
( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.inv_one
thf(fact_710_p_Oinv__neg__one,axiom,
( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.inv_neg_one
thf(fact_711_p_Odimension__zero,axiom,
! [K: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ zero_zero_nat @ K @ E )
=> ( E
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) ) ) ).
% p.dimension_zero
thf(fact_712_p_Odimension__direct__sum__space,axiom,
! [K: set_list_a,N: nat,E: set_list_a,M: nat,F2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N @ K @ E )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ M @ K @ F2 )
=> ( ( ( inf_inf_set_list_a @ E @ F2 )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) )
=> ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ ( plus_plus_nat @ N @ M ) @ K @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ E @ F2 ) ) ) ) ) ) ).
% p.dimension_direct_sum_space
thf(fact_713_ring__primeI,axiom,
! [P2: a] :
( ( P2
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P2 )
=> ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% ring_primeI
thf(fact_714_ring__primeE_I3_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( prime_a_ring_ext_a_b @ r @ P2 ) ) ) ).
% ring_primeE(3)
thf(fact_715_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
thf(fact_716_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_717_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_718_zeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroprimeideal
thf(fact_719_p_Osubring__inter,axiom,
! [I3: set_list_a,J2: set_list_a] :
( ( subrin6918843898125473962t_unit @ I3 @ ( univ_poly_a_b @ r @ k ) )
=> ( ( subrin6918843898125473962t_unit @ J2 @ ( univ_poly_a_b @ r @ k ) )
=> ( subrin6918843898125473962t_unit @ ( inf_inf_set_list_a @ I3 @ J2 ) @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.subring_inter
thf(fact_720_p_Osubalgebra__inter,axiom,
! [K: set_list_a,V: set_list_a,V4: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd1768981623711841426t_unit @ K @ V4 @ ( univ_poly_a_b @ r @ k ) )
=> ( embedd1768981623711841426t_unit @ K @ ( inf_inf_set_list_a @ V @ V4 ) @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.subalgebra_inter
thf(fact_721_p_Osubcring__inter,axiom,
! [I3: set_list_a,J2: set_list_a] :
( ( subcri7763218559781929323t_unit @ I3 @ ( univ_poly_a_b @ r @ k ) )
=> ( ( subcri7763218559781929323t_unit @ J2 @ ( univ_poly_a_b @ r @ k ) )
=> ( subcri7763218559781929323t_unit @ ( inf_inf_set_list_a @ I3 @ J2 ) @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.subcring_inter
thf(fact_722_p_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ N )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.nat_pow_zero
thf(fact_723_zeroprimeideal__domainI,axiom,
( ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
=> ( domain_a_b @ r ) ) ).
% zeroprimeideal_domainI
thf(fact_724_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_725_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_726_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_727_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_728_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_729_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_730_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_731_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_732_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_733_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_734_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_735_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_736_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_737_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_738_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_739_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_740_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_741_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_742_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_743_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_744_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_745_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_746_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_747_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_748_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_749_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_750_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_751_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_752_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_753_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_754_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_755_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_756_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_757_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_758_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_759_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_760_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_761_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_762_Int__subset__iff,axiom,
! [C4: set_list_a,A2: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ C4 @ ( inf_inf_set_list_a @ A2 @ B4 ) )
= ( ( ord_le8861187494160871172list_a @ C4 @ A2 )
& ( ord_le8861187494160871172list_a @ C4 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_763_Int__subset__iff,axiom,
! [C4: set_a,A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C4 @ ( inf_inf_set_a @ A2 @ B4 ) )
= ( ( ord_less_eq_set_a @ C4 @ A2 )
& ( ord_less_eq_set_a @ C4 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_764_p_Ozero__dim,axiom,
! [K: set_list_a] : ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ zero_zero_nat @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) ).
% p.zero_dim
thf(fact_765_p_Odimension__sum__space,axiom,
! [K: set_list_a,N: nat,E: set_list_a,M: nat,F2: set_list_a,K2: nat] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N @ K @ E )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ M @ K @ F2 )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ K2 @ K @ ( inf_inf_set_list_a @ E @ F2 ) )
=> ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K2 ) @ K @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ E @ F2 ) ) ) ) ) ) ).
% p.dimension_sum_space
thf(fact_766_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_767_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_768_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_769_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_770_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_771_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_772_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_773_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_774_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_775_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_776_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_777_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_778_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_779_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_780_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_781_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_782_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_783_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_784_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_785_disjoint__insert_I2_J,axiom,
! [A2: set_list_list_a,B: list_list_a,B4: set_list_list_a] :
( ( bot_bo1875519244922727510list_a
= ( inf_in7423150557312423384list_a @ A2 @ ( insert_list_list_a @ B @ B4 ) ) )
= ( ~ ( member_list_list_a @ B @ A2 )
& ( bot_bo1875519244922727510list_a
= ( inf_in7423150557312423384list_a @ A2 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_786_disjoint__insert_I2_J,axiom,
! [A2: set_list_a,B: list_a,B4: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ B @ B4 ) ) )
= ( ~ ( member_list_a @ B @ A2 )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A2 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_787_disjoint__insert_I2_J,axiom,
! [A2: set_a,B: a,B4: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B @ B4 ) ) )
= ( ~ ( member_a @ B @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_788_disjoint__insert_I1_J,axiom,
! [B4: set_list_list_a,A: list_list_a,A2: set_list_list_a] :
( ( ( inf_in7423150557312423384list_a @ B4 @ ( insert_list_list_a @ A @ A2 ) )
= bot_bo1875519244922727510list_a )
= ( ~ ( member_list_list_a @ A @ B4 )
& ( ( inf_in7423150557312423384list_a @ B4 @ A2 )
= bot_bo1875519244922727510list_a ) ) ) ).
% disjoint_insert(1)
thf(fact_789_disjoint__insert_I1_J,axiom,
! [B4: set_list_a,A: list_a,A2: set_list_a] :
( ( ( inf_inf_set_list_a @ B4 @ ( insert_list_a @ A @ A2 ) )
= bot_bot_set_list_a )
= ( ~ ( member_list_a @ A @ B4 )
& ( ( inf_inf_set_list_a @ B4 @ A2 )
= bot_bot_set_list_a ) ) ) ).
% disjoint_insert(1)
thf(fact_790_disjoint__insert_I1_J,axiom,
! [B4: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B4 @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B4 )
& ( ( inf_inf_set_a @ B4 @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_791_insert__disjoint_I2_J,axiom,
! [A: list_list_a,A2: set_list_list_a,B4: set_list_list_a] :
( ( bot_bo1875519244922727510list_a
= ( inf_in7423150557312423384list_a @ ( insert_list_list_a @ A @ A2 ) @ B4 ) )
= ( ~ ( member_list_list_a @ A @ B4 )
& ( bot_bo1875519244922727510list_a
= ( inf_in7423150557312423384list_a @ A2 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_792_insert__disjoint_I2_J,axiom,
! [A: list_a,A2: set_list_a,B4: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ ( insert_list_a @ A @ A2 ) @ B4 ) )
= ( ~ ( member_list_a @ A @ B4 )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A2 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_793_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B4: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B4 ) )
= ( ~ ( member_a @ A @ B4 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_794_insert__disjoint_I1_J,axiom,
! [A: list_list_a,A2: set_list_list_a,B4: set_list_list_a] :
( ( ( inf_in7423150557312423384list_a @ ( insert_list_list_a @ A @ A2 ) @ B4 )
= bot_bo1875519244922727510list_a )
= ( ~ ( member_list_list_a @ A @ B4 )
& ( ( inf_in7423150557312423384list_a @ A2 @ B4 )
= bot_bo1875519244922727510list_a ) ) ) ).
% insert_disjoint(1)
thf(fact_795_insert__disjoint_I1_J,axiom,
! [A: list_a,A2: set_list_a,B4: set_list_a] :
( ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ A2 ) @ B4 )
= bot_bot_set_list_a )
= ( ~ ( member_list_a @ A @ B4 )
& ( ( inf_inf_set_list_a @ A2 @ B4 )
= bot_bot_set_list_a ) ) ) ).
% insert_disjoint(1)
thf(fact_796_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B4 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B4 )
& ( ( inf_inf_set_a @ A2 @ B4 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_797_Diff__disjoint,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ ( minus_646659088055828811list_a @ B4 @ A2 ) )
= bot_bot_set_list_a ) ).
% Diff_disjoint
thf(fact_798_Diff__disjoint,axiom,
! [A2: set_a,B4: set_a] :
( ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B4 @ A2 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_799_Compl__disjoint,axiom,
! [A2: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ ( uminus7925729386456332763list_a @ A2 ) )
= bot_bot_set_list_a ) ).
% Compl_disjoint
thf(fact_800_Compl__disjoint,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ ( uminus_uminus_set_a @ A2 ) )
= bot_bot_set_a ) ).
% Compl_disjoint
thf(fact_801_Compl__disjoint2,axiom,
! [A2: set_list_a] :
( ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ A2 ) @ A2 )
= bot_bot_set_list_a ) ).
% Compl_disjoint2
thf(fact_802_Compl__disjoint2,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ A2 ) @ A2 )
= bot_bot_set_a ) ).
% Compl_disjoint2
thf(fact_803_local_Onat__pow__0,axiom,
! [X: a] :
( ( pow_a_1026414303147256608_b_nat @ r @ X @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ r ) ) ).
% local.nat_pow_0
thf(fact_804_p_Onat__pow__0,axiom,
! [X: list_a] :
( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ zero_zero_nat )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.nat_pow_0
thf(fact_805_Int__Collect__mono,axiom,
! [A2: set_list_list_a,B4: set_list_list_a,P: list_list_a > $o,Q2: list_list_a > $o] :
( ( ord_le8488217952732425610list_a @ A2 @ B4 )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_le8488217952732425610list_a @ ( inf_in7423150557312423384list_a @ A2 @ ( collect_list_list_a @ P ) ) @ ( inf_in7423150557312423384list_a @ B4 @ ( collect_list_list_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_806_Int__Collect__mono,axiom,
! [A2: set_list_a,B4: set_list_a,P: list_a > $o,Q2: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A2 @ B4 )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ ( collect_list_a @ P ) ) @ ( inf_inf_set_list_a @ B4 @ ( collect_list_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_807_Int__Collect__mono,axiom,
! [A2: set_a,B4: set_a,P: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B4 @ ( collect_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_808_Int__greatest,axiom,
! [C4: set_list_a,A2: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ C4 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ C4 @ B4 )
=> ( ord_le8861187494160871172list_a @ C4 @ ( inf_inf_set_list_a @ A2 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_809_Int__greatest,axiom,
! [C4: set_a,A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
=> ( ( ord_less_eq_set_a @ C4 @ B4 )
=> ( ord_less_eq_set_a @ C4 @ ( inf_inf_set_a @ A2 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_810_Int__absorb2,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B4 )
=> ( ( inf_inf_set_list_a @ A2 @ B4 )
= A2 ) ) ).
% Int_absorb2
thf(fact_811_Int__absorb2,axiom,
! [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( inf_inf_set_a @ A2 @ B4 )
= A2 ) ) ).
% Int_absorb2
thf(fact_812_Int__absorb1,axiom,
! [B4: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B4 @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_813_Int__absorb1,axiom,
! [B4: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_814_Int__lower2,axiom,
! [A2: set_list_a,B4: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_815_Int__lower2,axiom,
! [A2: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_816_Int__lower1,axiom,
! [A2: set_list_a,B4: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B4 ) @ A2 ) ).
% Int_lower1
thf(fact_817_Int__lower1,axiom,
! [A2: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B4 ) @ A2 ) ).
% Int_lower1
thf(fact_818_Int__mono,axiom,
! [A2: set_list_a,C4: set_list_a,B4: set_list_a,D2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ C4 )
=> ( ( ord_le8861187494160871172list_a @ B4 @ D2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B4 ) @ ( inf_inf_set_list_a @ C4 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_819_Int__mono,axiom,
! [A2: set_a,C4: set_a,B4: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C4 )
=> ( ( ord_less_eq_set_a @ B4 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B4 ) @ ( inf_inf_set_a @ C4 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_820_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_821_Int__emptyI,axiom,
! [A2: set_list_list_a,B4: set_list_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A2 )
=> ~ ( member_list_list_a @ X2 @ B4 ) )
=> ( ( inf_in7423150557312423384list_a @ A2 @ B4 )
= bot_bo1875519244922727510list_a ) ) ).
% Int_emptyI
thf(fact_822_Int__emptyI,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ~ ( member_list_a @ X2 @ B4 ) )
=> ( ( inf_inf_set_list_a @ A2 @ B4 )
= bot_bot_set_list_a ) ) ).
% Int_emptyI
thf(fact_823_Int__emptyI,axiom,
! [A2: set_a,B4: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B4 ) )
=> ( ( inf_inf_set_a @ A2 @ B4 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_824_disjoint__iff,axiom,
! [A2: set_list_list_a,B4: set_list_list_a] :
( ( ( inf_in7423150557312423384list_a @ A2 @ B4 )
= bot_bo1875519244922727510list_a )
= ( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A2 )
=> ~ ( member_list_list_a @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_825_disjoint__iff,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B4 )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ~ ( member_list_a @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_826_disjoint__iff,axiom,
! [A2: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ A2 @ B4 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_827_Int__empty__left,axiom,
! [B4: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ B4 )
= bot_bot_set_list_a ) ).
% Int_empty_left
thf(fact_828_Int__empty__left,axiom,
! [B4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B4 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_829_Int__empty__right,axiom,
! [A2: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% Int_empty_right
thf(fact_830_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_831_disjoint__iff__not__equal,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B4 )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ! [Y3: list_a] :
( ( member_list_a @ Y3 @ B4 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_832_disjoint__iff__not__equal,axiom,
! [A2: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ A2 @ B4 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ B4 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_833_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_834_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_835_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_836_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_837_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_838_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_839_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_840_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_841_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_842_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_843_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_844_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_845_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_846_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_847_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_848_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_849_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A4: int,B3: int] :
( ( minus_minus_int @ A4 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_850_Diff__triv,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B4 )
= bot_bot_set_list_a )
=> ( ( minus_646659088055828811list_a @ A2 @ B4 )
= A2 ) ) ).
% Diff_triv
thf(fact_851_Diff__triv,axiom,
! [A2: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ A2 @ B4 )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ B4 )
= A2 ) ) ).
% Diff_triv
thf(fact_852_Int__Diff__disjoint,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A2 @ B4 ) @ ( minus_646659088055828811list_a @ A2 @ B4 ) )
= bot_bot_set_list_a ) ).
% Int_Diff_disjoint
thf(fact_853_Int__Diff__disjoint,axiom,
! [A2: set_a,B4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B4 ) @ ( minus_minus_set_a @ A2 @ B4 ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_854_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_855_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_856_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_857_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_858_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_859_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_860_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_861_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_862_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_863_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_864_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_865_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_866_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_867_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_868_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_869_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_870_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_871_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_872_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_873_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_874_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_875_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_876_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_877_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_878_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_879_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_880_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_881_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_882_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_883_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_884_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_885_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_886_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_887_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_888_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_889_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_890_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_891_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_892_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_893_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_894_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_895_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_896_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_897_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_898_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_899_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_900_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_901_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_902_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_903_disjoint__eq__subset__Compl,axiom,
! [A2: set_list_a,B4: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B4 )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A2 @ ( uminus7925729386456332763list_a @ B4 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_904_disjoint__eq__subset__Compl,axiom,
! [A2: set_a,B4: set_a] :
( ( ( inf_inf_set_a @ A2 @ B4 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ ( uminus_uminus_set_a @ B4 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_905_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_906_semiring_Onat__pow__zero,axiom,
! [R: partia2670972154091845814t_unit,N: nat] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( N != zero_zero_nat )
=> ( ( pow_li1142815632869257134it_nat @ R @ ( zero_l4142658623432671053t_unit @ R ) @ N )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.nat_pow_zero
thf(fact_907_semiring_Onat__pow__zero,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat] :
( ( semiring_a_b @ R )
=> ( ( N != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ R @ ( zero_a_b @ R ) @ N )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.nat_pow_zero
thf(fact_908_maximalideal__prime,axiom,
! [I3: set_a] :
( ( maximalideal_a_b @ I3 @ r )
=> ( primeideal_a_b @ I3 @ r ) ) ).
% maximalideal_prime
thf(fact_909_p_Odimension_Ocases,axiom,
! [A1: nat,A22: set_list_a,A32: set_list_a] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) )
=> ~ ! [V2: list_a,E3: set_list_a,N3: nat] :
( ( A1
= ( suc @ N3 ) )
=> ( ( A32
= ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ A22 @ V2 @ E3 ) )
=> ( ( member_list_a @ V2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ~ ( member_list_a @ V2 @ E3 )
=> ~ ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N3 @ A22 @ E3 ) ) ) ) ) ) ) ).
% p.dimension.cases
thf(fact_910_p_Odimension_Osimps,axiom,
! [A1: nat,A22: set_list_a,A32: set_list_a] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ A1 @ A22 @ A32 )
= ( ? [K4: set_list_a] :
( ( A1 = zero_zero_nat )
& ( A22 = K4 )
& ( A32
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) )
| ? [V5: list_a,E4: set_list_a,N2: nat,K4: set_list_a] :
( ( A1
= ( suc @ N2 ) )
& ( A22 = K4 )
& ( A32
= ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ K4 @ V5 @ E4 ) )
& ( member_list_a @ V5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
& ~ ( member_list_a @ V5 @ E4 )
& ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N2 @ K4 @ E4 ) ) ) ) ).
% p.dimension.simps
thf(fact_911_subring__inter,axiom,
! [I3: set_a,J2: set_a] :
( ( subring_a_b @ I3 @ r )
=> ( ( subring_a_b @ J2 @ r )
=> ( subring_a_b @ ( inf_inf_set_a @ I3 @ J2 ) @ r ) ) ) ).
% subring_inter
thf(fact_912_nat__pow__Suc2,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ ( suc @ N ) )
= ( mult_a_ring_ext_a_b @ r @ X @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ).
% nat_pow_Suc2
thf(fact_913_p_Onat__pow__Suc2,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ ( suc @ N ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) ) ) ) ).
% p.nat_pow_Suc2
thf(fact_914_p_OSuc__dim,axiom,
! [V3: list_a,E: set_list_a,N: nat,K: set_list_a] :
( ( member_list_a @ V3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ~ ( member_list_a @ V3 @ E )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N @ K @ E )
=> ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ ( suc @ N ) @ K @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ V3 @ E ) ) ) ) ) ).
% p.Suc_dim
thf(fact_915_p_Odimension__backwards,axiom,
! [K: set_list_a,N: nat,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ ( suc @ N ) @ K @ E )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
& ? [E5: set_list_a] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N @ K @ E5 )
& ~ ( member_list_a @ X2 @ E5 )
& ( E
= ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ X2 @ E5 ) ) ) ) ) ) ).
% p.dimension_backwards
thf(fact_916_local_Onat__pow__Suc,axiom,
! [X: a,N: nat] :
( ( pow_a_1026414303147256608_b_nat @ r @ X @ ( suc @ N ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ X ) ) ).
% local.nat_pow_Suc
thf(fact_917_p_Onat__pow__Suc,axiom,
! [X: list_a,N: nat] :
( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ ( suc @ N ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) @ X ) ) ).
% p.nat_pow_Suc
thf(fact_918_primeideal__iff__prime,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( 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 @ P2 ) @ r )
= ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% primeideal_iff_prime
thf(fact_919_Group_Onat__pow__0,axiom,
! [G2: partia4785451669175976129t_unit,X: list_a] :
( ( pow_li8657086744513738943it_nat @ G2 @ X @ zero_zero_nat )
= ( one_li6878281577851457998t_unit @ G2 ) ) ).
% Group.nat_pow_0
thf(fact_920_Group_Onat__pow__0,axiom,
! [G2: partia8223610829204095565t_unit,X: a] :
( ( pow_a_1875594501834816709it_nat @ G2 @ X @ zero_zero_nat )
= ( one_a_Product_unit @ G2 ) ) ).
% Group.nat_pow_0
thf(fact_921_Group_Onat__pow__0,axiom,
! [G2: partia2670972154091845814t_unit,X: list_a] :
( ( pow_li1142815632869257134it_nat @ G2 @ X @ zero_zero_nat )
= ( one_li8328186300101108157t_unit @ G2 ) ) ).
% Group.nat_pow_0
thf(fact_922_Group_Onat__pow__0,axiom,
! [G2: partia2175431115845679010xt_a_b,X: a] :
( ( pow_a_1026414303147256608_b_nat @ G2 @ X @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ G2 ) ) ).
% Group.nat_pow_0
thf(fact_923_Group_Onat__pow__Suc,axiom,
! [G2: partia4785451669175976129t_unit,X: list_a,N: nat] :
( ( pow_li8657086744513738943it_nat @ G2 @ X @ ( suc @ N ) )
= ( mult_l6995149843440949818t_unit @ G2 @ ( pow_li8657086744513738943it_nat @ G2 @ X @ N ) @ X ) ) ).
% Group.nat_pow_Suc
thf(fact_924_Group_Onat__pow__Suc,axiom,
! [G2: partia8223610829204095565t_unit,X: a,N: nat] :
( ( pow_a_1875594501834816709it_nat @ G2 @ X @ ( suc @ N ) )
= ( mult_a_Product_unit @ G2 @ ( pow_a_1875594501834816709it_nat @ G2 @ X @ N ) @ X ) ) ).
% Group.nat_pow_Suc
thf(fact_925_Group_Onat__pow__Suc,axiom,
! [G2: partia2670972154091845814t_unit,X: list_a,N: nat] :
( ( pow_li1142815632869257134it_nat @ G2 @ X @ ( suc @ N ) )
= ( mult_l7073676228092353617t_unit @ G2 @ ( pow_li1142815632869257134it_nat @ G2 @ X @ N ) @ X ) ) ).
% Group.nat_pow_Suc
thf(fact_926_Group_Onat__pow__Suc,axiom,
! [G2: partia2175431115845679010xt_a_b,X: a,N: nat] :
( ( pow_a_1026414303147256608_b_nat @ G2 @ X @ ( suc @ N ) )
= ( mult_a_ring_ext_a_b @ G2 @ ( pow_a_1026414303147256608_b_nat @ G2 @ X @ N ) @ X ) ) ).
% Group.nat_pow_Suc
thf(fact_927_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_928_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_929_domain_Oprimeideal__iff__prime,axiom,
! [R: partia2670972154091845814t_unit,P2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P2 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( primei6309817859076077608t_unit @ ( cgenid9131348535277946915t_unit @ R @ P2 ) @ R )
= ( ring_r6430282645014804837t_unit @ R @ P2 ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_930_domain_Oprimeideal__iff__prime,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P2 @ ( 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 @ P2 ) @ R )
= ( ring_ring_prime_a_b @ R @ P2 ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_931_domain_Oprimeideal__iff__prime,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( primei2288432046033540002t_unit @ ( cgenid24865672677839267t_unit @ R @ P2 ) @ R )
= ( ring_r5437400583859147359t_unit @ R @ P2 ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_932_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_933_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_934_p_Ocgenideal__self,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ I @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ I ) ) ) ).
% p.cgenideal_self
thf(fact_935_p_Ocgenideal__is__principalideal,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ I ) @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.cgenideal_is_principalideal
thf(fact_936_p_Ocgenideal__eq__genideal,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ I )
= ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ I @ bot_bot_set_list_a ) ) ) ) ).
% p.cgenideal_eq_genideal
thf(fact_937_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_938_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_939_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_940_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_941_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_942_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_943_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_944_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_945_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_946_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_947_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_948_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_949_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_950_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_951_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_952_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_953_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_954_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_955_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_956_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_957_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_958_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_959_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_960_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_961_Suc__le__D,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
=> ? [M4: nat] :
( M3
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_962_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_963_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_964_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_965_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_966_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_967_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y5: nat,Z3: nat] :
( ( R @ X2 @ Y5 )
=> ( ( R @ Y5 @ Z3 )
=> ( R @ X2 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_968_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_969_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_970_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_971_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_972_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_973_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_974_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_975_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_976_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_977_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_978_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_979_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_980_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_981_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_982_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_983_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_984_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_985_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N2: nat] :
? [K5: nat] :
( N2
= ( plus_plus_nat @ M6 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_986_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_987_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_988_mult__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_989_mult__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_990_mult__le__mono2,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_991_lift__Suc__mono__le,axiom,
! [F: nat > set_list_a,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_le8861187494160871172list_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le8861187494160871172list_a @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_992_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_993_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_994_lift__Suc__antimono__le,axiom,
! [F: nat > set_list_a,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_le8861187494160871172list_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le8861187494160871172list_a @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_995_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_a @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_996_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_997_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_998_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_999_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1000_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1001_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1002_le__diff__conv,axiom,
! [J: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1003_Suc__mult__le__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1004_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( prime_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_1005_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( prime_2011924034616061926t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_1006_ring__prime__def,axiom,
( ring_ring_prime_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,A4: a] :
( ( A4
!= ( zero_a_b @ R2 ) )
& ( prime_a_ring_ext_a_b @ R2 @ A4 ) ) ) ) ).
% ring_prime_def
thf(fact_1007_ring__prime__def,axiom,
( ring_r6430282645014804837t_unit
= ( ^ [R2: partia2670972154091845814t_unit,A4: list_a] :
( ( A4
!= ( zero_l4142658623432671053t_unit @ R2 ) )
& ( prime_2011924034616061926t_unit @ R2 @ A4 ) ) ) ) ).
% ring_prime_def
thf(fact_1008_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2956882679547061052t_unit,R3: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ R3 )
=> ( R3
!= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_1009_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,R3: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R3 )
=> ( R3
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_1010_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,R3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R3 )
=> ( R3
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_1011_domain_Oproperfactor__of__zero_I2_J,axiom,
! [R: partia2670972154091845814t_unit,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( proper8313688649498433056t_unit @ R @ B @ ( zero_l4142658623432671053t_unit @ R ) )
= ( B
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.properfactor_of_zero(2)
thf(fact_1012_domain_Oproperfactor__of__zero_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( proper19828929941537682xt_a_b @ R @ B @ ( zero_a_b @ R ) )
= ( B
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.properfactor_of_zero(2)
thf(fact_1013_domain_Oproperfactor__of__zero_I2_J,axiom,
! [R: partia2956882679547061052t_unit,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( proper6839530779442076320t_unit @ R @ B @ ( zero_l347298301471573063t_unit @ R ) )
= ( B
!= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ).
% domain.properfactor_of_zero(2)
thf(fact_1014_domain_Oring__primeE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r6430282645014804837t_unit @ R @ P2 )
=> ( P2
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_1015_domain_Oring__primeE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P2 )
=> ( P2
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_1016_domain_Oring__primeE_I1_J,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r5437400583859147359t_unit @ R @ P2 )
=> ( P2
!= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_1017_domain_Oring__primeE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,P2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r6430282645014804837t_unit @ R @ P2 )
=> ( prime_2011924034616061926t_unit @ R @ P2 ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_1018_domain_Oring__primeE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P2 )
=> ( prime_a_ring_ext_a_b @ R @ P2 ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_1019_domain_Oring__primeE_I3_J,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r5437400583859147359t_unit @ R @ P2 )
=> ( prime_1232919612140715622t_unit @ R @ P2 ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_1020_p_Oring__primeI,axiom,
! [P2: list_a] :
( ( P2
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) ) ) ).
% p.ring_primeI
thf(fact_1021_inf__compl__bot__left1,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ X ) @ ( inf_inf_set_list_a @ X @ Y ) )
= bot_bot_set_list_a ) ).
% inf_compl_bot_left1
thf(fact_1022_inf__compl__bot__left1,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ ( inf_inf_set_a @ X @ Y ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left1
thf(fact_1023_inf__compl__bot__left2,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ X ) @ Y ) )
= bot_bot_set_list_a ) ).
% inf_compl_bot_left2
thf(fact_1024_inf__compl__bot__left2,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left2
thf(fact_1025_compl__le__compl__iff,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( uminus7925729386456332763list_a @ X ) @ ( uminus7925729386456332763list_a @ Y ) )
= ( ord_le8861187494160871172list_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_1026_compl__le__compl__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_1027_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_1028_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_1029_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_list_a] :
( ( inf_inf_set_list_a @ X @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_1030_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_1031_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( uminus7925729386456332763list_a @ X ) )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1032_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ ( uminus_uminus_set_a @ X ) )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1033_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_list_a] :
( ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ X ) @ X )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1034_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1035_inf__compl__bot__right,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ ( uminus7925729386456332763list_a @ X ) ) )
= bot_bot_set_list_a ) ).
% inf_compl_bot_right
thf(fact_1036_inf__compl__bot__right,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ ( uminus_uminus_set_a @ X ) ) )
= bot_bot_set_a ) ).
% inf_compl_bot_right
thf(fact_1037_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1038_bot__set__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a @ bot_bot_list_a_o ) ) ).
% bot_set_def
thf(fact_1039_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_1040_domain_Opprime__iff__pirreducible,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_1041_domain_Opprime__iff__pirreducible,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_1042_compl__le__swap2,axiom,
! [Y: set_list_a,X: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( uminus7925729386456332763list_a @ Y ) @ X )
=> ( ord_le8861187494160871172list_a @ ( uminus7925729386456332763list_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1043_compl__le__swap2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ X )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1044_compl__le__swap1,axiom,
! [Y: set_list_a,X: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ ( uminus7925729386456332763list_a @ X ) )
=> ( ord_le8861187494160871172list_a @ X @ ( uminus7925729386456332763list_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1045_compl__le__swap1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ ( uminus_uminus_set_a @ X ) )
=> ( ord_less_eq_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1046_compl__mono,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ord_le8861187494160871172list_a @ ( uminus7925729386456332763list_a @ Y ) @ ( uminus7925729386456332763list_a @ X ) ) ) ).
% compl_mono
thf(fact_1047_compl__mono,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ ( uminus_uminus_set_a @ X ) ) ) ).
% compl_mono
thf(fact_1048_diff__shunt__var,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ( minus_646659088055828811list_a @ X @ Y )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1049_diff__shunt__var,axiom,
! [X: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1050_diff__eq,axiom,
( minus_646659088055828811list_a
= ( ^ [X3: set_list_a,Y3: set_list_a] : ( inf_inf_set_list_a @ X3 @ ( uminus7925729386456332763list_a @ Y3 ) ) ) ) ).
% diff_eq
thf(fact_1051_diff__eq,axiom,
( minus_minus_set_a
= ( ^ [X3: set_a,Y3: set_a] : ( inf_inf_set_a @ X3 @ ( uminus_uminus_set_a @ Y3 ) ) ) ) ).
% diff_eq
thf(fact_1052_inf__cancel__left2,axiom,
! [X: set_list_a,A: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ X ) @ A ) @ ( inf_inf_set_list_a @ X @ B ) )
= bot_bot_set_list_a ) ).
% inf_cancel_left2
thf(fact_1053_inf__cancel__left2,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ A ) @ ( inf_inf_set_a @ X @ B ) )
= bot_bot_set_a ) ).
% inf_cancel_left2
thf(fact_1054_inf__cancel__left1,axiom,
! [X: set_list_a,A: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X @ A ) @ ( inf_inf_set_list_a @ ( uminus7925729386456332763list_a @ X ) @ B ) )
= bot_bot_set_list_a ) ).
% inf_cancel_left1
thf(fact_1055_inf__cancel__left1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ A ) @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X ) @ B ) )
= bot_bot_set_a ) ).
% inf_cancel_left1
thf(fact_1056_inf__shunt,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ( inf_inf_set_list_a @ X @ Y )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ X @ ( uminus7925729386456332763list_a @ Y ) ) ) ).
% inf_shunt
thf(fact_1057_inf__shunt,axiom,
! [X: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ ( uminus_uminus_set_a @ Y ) ) ) ).
% inf_shunt
thf(fact_1058_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_1059_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1060_cgenideal__eq__genideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( cgenid547466209912283029xt_a_b @ r @ I )
= ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% cgenideal_eq_genideal
thf(fact_1061_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_1062_subring__props_I7_J,axiom,
! [K: set_a,H12: a,H22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H12 @ K )
=> ( ( member_a @ H22 @ K )
=> ( member_a @ ( add_a_b @ r @ H12 @ H22 ) @ K ) ) ) ) ).
% subring_props(7)
thf(fact_1063_subring__props_I6_J,axiom,
! [K: set_a,H12: a,H22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H12 @ K )
=> ( ( member_a @ H22 @ K )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H22 ) @ K ) ) ) ) ).
% subring_props(6)
thf(fact_1064_subring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( K != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_1065_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_1066_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_1067_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_1068_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_1069_subset__Idl__subset,axiom,
! [I3: set_a,H: set_a] :
( ( ord_less_eq_set_a @ I3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H @ I3 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H ) @ ( genideal_a_b @ r @ I3 ) ) ) ) ).
% subset_Idl_subset
thf(fact_1070_genideal__self,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ r @ S ) ) ) ).
% genideal_self
thf(fact_1071_pprime__iff__pirreducible,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_1072_genideal__self_H,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_1073_genideal__zero,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% genideal_zero
thf(fact_1074_Idl__subset__ideal_H,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_1075_genideal__one,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) )
= ( partia707051561876973205xt_a_b @ r ) ) ).
% genideal_one
thf(fact_1076_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_1077_verit__comp__simplify1_I2_J,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1078_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1079_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1080_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1081_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_1082_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_1083_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1084_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1085_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_1086_long__division__a__inv_I1_J,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( 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 ) @ P2 ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ P2 @ Q ) ) ) ) ) ) ).
% long_division_a_inv(1)
thf(fact_1087_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_1088_long__division__closed_I1_J,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( 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 @ P2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_1089_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_1090_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_1091_ring_Opdiv_Ocong,axiom,
polynomial_pdiv_a_b = polynomial_pdiv_a_b ).
% ring.pdiv.cong
thf(fact_1092_domain_Ouniv__poly__is__principal,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ring_p715737262848045090t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_1093_domain_Ouniv__poly__is__principal,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_1094_principal__domain_Oprimeness__condition,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_a] :
( ( ring_p715737262848045090t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ P2 )
= ( ring_r5437400583859147359t_unit @ R @ P2 ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_1095_principal__domain_Oprimeness__condition,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ P2 )
= ( ring_ring_prime_a_b @ R @ P2 ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_1096_principal__domain_Oprimeness__condition,axiom,
! [R: partia2670972154091845814t_unit,P2: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ P2 )
= ( ring_r6430282645014804837t_unit @ R @ P2 ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_1097_domain_Olong__division__closed_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member_list_list_a @ ( polyno5893782122288709345t_unit @ R @ P2 @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_1098_domain_Olong__division__closed_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( 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 @ P2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_1099_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_a] :
( ( ring_p715737262848045090t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ P2 )
=> ( maxima7552488817642790894t_unit @ ( cgenid24865672677839267t_unit @ R @ P2 ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_1100_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ P2 )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ R @ P2 ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_1101_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia2670972154091845814t_unit,P2: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ P2 )
=> ( maxima6585700282301356660t_unit @ ( cgenid9131348535277946915t_unit @ R @ P2 ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_1102_domain_Olong__division__add_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ A @ B ) @ Q )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno5893782122288709345t_unit @ R @ A @ Q ) @ ( polyno5893782122288709345t_unit @ R @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_1103_domain_Olong__division__add_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_1104_domain_Olong__division__a__inv_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ R @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) @ Q )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno5893782122288709345t_unit @ R @ P2 @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_1105_domain_Olong__division__a__inv_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( 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 ) @ P2 ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ ( polynomial_pdiv_a_b @ R @ P2 @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_1106_pdiv__pmod,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ ( polynomial_pdiv_a_b @ r @ P2 @ Q ) ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q ) ) ) ) ) ) ).
% pdiv_pmod
thf(fact_1107_p_Ozeromaximalideal__eq__field,axiom,
( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ k ) )
= ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.zeromaximalideal_eq_field
thf(fact_1108_p_Ozeromaximalideal__fieldI,axiom,
( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ k ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.zeromaximalideal_fieldI
thf(fact_1109_univ__poly__not__field,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ~ ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_not_field
thf(fact_1110_long__division__closed_I2_J,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( 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 @ P2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(2)
thf(fact_1111_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_1112_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_1113_long__division__a__inv_I2_J,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( 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 ) @ P2 ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q ) ) ) ) ) ) ).
% long_division_a_inv(2)
thf(fact_1114_p_Ocring__fieldI2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( A3
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ? [X5: list_a] :
( ( member_list_a @ X5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A3 @ X5 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.cring_fieldI2
thf(fact_1115_Ring_Oone__not__zero,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_1116_Ring_Oone__not__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_1117_Ring_Oone__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_1118_field_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% field.axioms(1)
thf(fact_1119_p_Ofield__intro2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( 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 ) ) )
=> ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.field_intro2
thf(fact_1120_comm__inv__char,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ X )
= Y ) ) ) ) ).
% comm_inv_char
thf(fact_1121_inv__char,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ X )
= Y ) ) ) ) ) ).
% inv_char
thf(fact_1122_inv__unique_H,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( Y
= ( m_inv_a_ring_ext_a_b @ r @ X ) ) ) ) ) ) ).
% inv_unique'
thf(fact_1123_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A3
!= ( zero_a_b @ r ) )
=> ? [X5: a] :
( ( member_a @ X5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A3 @ X5 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_1124_p_OUnits__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.Units_closed
thf(fact_1125_p_OUnits__pow__closed,axiom,
! [X: list_a,D: nat] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ D ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.Units_pow_closed
thf(fact_1126_p_Oinv__eq__imp__eq,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ X )
= ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ Y ) )
=> ( X = Y ) ) ) ) ).
% p.inv_eq_imp_eq
thf(fact_1127_zeromaximalideal__eq__field,axiom,
( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
= ( field_a_b @ r ) ) ).
% zeromaximalideal_eq_field
thf(fact_1128_zeromaximalideal__fieldI,axiom,
( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
=> ( field_a_b @ r ) ) ).
% zeromaximalideal_fieldI
thf(fact_1129_subfield__m__inv_I1_J,axiom,
! [K: set_a,K2: 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 @ ( m_inv_a_ring_ext_a_b @ r @ K2 ) @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ).
% subfield_m_inv(1)
thf(fact_1130_pirreducibleE_I2_J,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pirreducibleE(2)
thf(fact_1131_p_Oprod__unit__l,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( 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 @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ) ).
% p.prod_unit_l
thf(fact_1132_p_Oprod__unit__r,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( 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 @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ) ).
% p.prod_unit_r
thf(fact_1133_p_Ounit__factor,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( 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 @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).
% p.unit_factor
thf(fact_1134_p_Oideal__eq__carrier__iff,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ A ) )
= ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.ideal_eq_carrier_iff
thf(fact_1135_pprimeE_I2_J,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pprimeE(2)
thf(fact_1136_p_OUnits__inv__comm,axiom,
! [X: list_a,Y: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).
% p.Units_inv_comm
thf(fact_1137_p_Oproperfactor__unitE,axiom,
! [U: list_a,A: list_a] :
( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( proper8313688649498433056t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ U )
=> ~ ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.properfactor_unitE
thf(fact_1138_p_Oinv__eq__one__eq,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= ( X
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.inv_eq_one_eq
thf(fact_1139_pirreducibleE_I3_J,axiom,
! [K: set_a,P2: list_a,Q: list_a,R3: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R3 ) )
=> ( ( member_list_a @ Q @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ) ) ) ).
% pirreducibleE(3)
thf(fact_1140_p_OUnits__l__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.Units_l_inv_ex
thf(fact_1141_p_OUnits__r__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ X2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.Units_r_inv_ex
thf(fact_1142_p_Oinv__eq__neg__one__eq,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ X )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) )
= ( X
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).
% p.inv_eq_neg_one_eq
thf(fact_1143_subfield__m__inv_I2_J,axiom,
! [K: set_a,K2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ K2 @ ( m_inv_a_ring_ext_a_b @ r @ K2 ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% subfield_m_inv(2)
thf(fact_1144_subfield__m__inv_I3_J,axiom,
! [K: set_a,K2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ K2 ) @ K2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% subfield_m_inv(3)
thf(fact_1145_inv__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) )
= ( one_a_ring_ext_a_b @ r ) ) ).
% inv_one
thf(fact_1146_p_Ocring__fieldI,axiom,
( ( ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) )
= ( 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 ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.cring_fieldI
thf(fact_1147_inv__neg__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) )
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ).
% inv_neg_one
thf(fact_1148_p_OUnits__m__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.Units_m_closed
thf(fact_1149_p_OUnits__one__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) ).
% p.Units_one_closed
thf(fact_1150_p_OUnits__inv__Units,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.Units_inv_Units
thf(fact_1151_p_OUnits__inv__inv,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) )
= X ) ) ).
% p.Units_inv_inv
thf(fact_1152_p_OUnits__l__cancel,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% p.Units_l_cancel
thf(fact_1153_p_OUnits__inv__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.Units_inv_closed
thf(fact_1154_p_OUnits__minus__one__closed,axiom,
member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) ).
% p.Units_minus_one_closed
thf(fact_1155_p_OUnits__l__inv,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.Units_l_inv
thf(fact_1156_p_OUnits__r__inv,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.Units_r_inv
thf(fact_1157_p_Ounits__of__pow,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( pow_li8657086744513738943it_nat @ ( units_6477118173342999439t_unit @ ( univ_poly_a_b @ r @ k ) ) @ X @ N )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) ) ) ).
% p.units_of_pow
thf(fact_1158_rupture__is__field__iff__pirreducible,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ).
% rupture_is_field_iff_pirreducible
thf(fact_1159_Units__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_1160_carrier__is__subcring,axiom,
subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subcring
thf(fact_1161_subcringI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subcring_a_b @ H @ r ) ) ).
% subcringI'
thf(fact_1162_subdomainI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subdomain_a_b @ H @ r ) ) ).
% subdomainI'
thf(fact_1163_Units__pow__closed,axiom,
! [X: a,D: nat] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X @ D ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_pow_closed
thf(fact_1164_inv__eq__imp__eq,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( m_inv_a_ring_ext_a_b @ r @ Y ) )
=> ( X = Y ) ) ) ) ).
% inv_eq_imp_eq
thf(fact_1165_subcring__inter,axiom,
! [I3: set_a,J2: set_a] :
( ( subcring_a_b @ I3 @ r )
=> ( ( subcring_a_b @ J2 @ r )
=> ( subcring_a_b @ ( inf_inf_set_a @ I3 @ J2 ) @ r ) ) ) ).
% subcring_inter
thf(fact_1166_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_1167_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_1168_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_1169_Units__inv__comm,axiom,
! [X: a,Y: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( 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 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_inv_comm
thf(fact_1170_subcringI,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( ! [H1: a,H2: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( ( mult_a_ring_ext_a_b @ r @ H1 @ H2 )
= ( mult_a_ring_ext_a_b @ r @ H2 @ H1 ) ) ) )
=> ( subcring_a_b @ H @ r ) ) ) ).
% subcringI
thf(fact_1171_properfactor__unitE,axiom,
! [U: a,A: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( proper19828929941537682xt_a_b @ r @ A @ U )
=> ~ ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% properfactor_unitE
thf(fact_1172_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_1173_inv__eq__one__eq,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( one_a_ring_ext_a_b @ r ) )
= ( X
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% inv_eq_one_eq
thf(fact_1174_ring__irreducibleE_I4_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_1175_Units__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_1176_Units__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X2 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_1177_subdomainI,axiom,
! [H: set_a] :
( ( subcring_a_b @ H @ r )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ! [H1: a,H2: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ r @ H1 @ H2 )
= ( zero_a_b @ r ) )
=> ( ( H1
= ( zero_a_b @ r ) )
| ( H2
= ( zero_a_b @ r ) ) ) ) ) )
=> ( subdomain_a_b @ H @ r ) ) ) ) ).
% subdomainI
thf(fact_1178_inv__eq__self,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( X
= ( m_inv_a_ring_ext_a_b @ r @ X ) )
=> ( ( X
= ( one_a_ring_ext_a_b @ r ) )
| ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% inv_eq_self
thf(fact_1179_inv__eq__neg__one__eq,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) )
= ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% inv_eq_neg_one_eq
thf(fact_1180_ring__irreducibleE_I5_J,axiom,
! [R3: a,A: a,B: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R3
= ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_1181_cring__fieldI,axiom,
( ( ( 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 ) ) )
=> ( field_a_b @ r ) ) ).
% cring_fieldI
thf(fact_1182_field__intro2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) ) )
=> ( field_a_b @ r ) ) ) ).
% field_intro2
thf(fact_1183_ring__irreducibleI,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R3
= ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ r @ R3 ) ) ) ) ).
% ring_irreducibleI
thf(fact_1184_Units__m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( 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 @ X @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_1185_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_1186_Units__inv__Units,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_inv_Units
thf(fact_1187_Units__inv__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ X ) )
= X ) ) ).
% Units_inv_inv
thf(fact_1188_Units__l__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( 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 @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% Units_l_cancel
thf(fact_1189_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_1190_Units__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_inv_closed
thf(fact_1191_Units__l__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% Units_l_inv
thf(fact_1192_Units__r__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( m_inv_a_ring_ext_a_b @ r @ X ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% Units_r_inv
thf(fact_1193_units__of__pow,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( units_8174867845824275201xt_a_b @ r ) @ X @ N )
= ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ).
% units_of_pow
thf(fact_1194_pirreducibleI,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2 != nil_a )
=> ( ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q3: list_a,R4: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q3 @ R4 ) )
=> ( ( member_list_a @ Q3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( member_list_a @ R4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ) ) ).
% pirreducibleI
thf(fact_1195_coeff_Osimps_I1_J,axiom,
( ( coeff_a_b @ r @ nil_a )
= ( ^ [Uu: nat] : ( zero_a_b @ r ) ) ) ).
% coeff.simps(1)
thf(fact_1196_polynomial__pow__not__zero,axiom,
! [P2: list_a,N: nat] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ N )
!= nil_a ) ) ) ).
% polynomial_pow_not_zero
thf(fact_1197_subring__polynomial__pow__not__zero,axiom,
! [K: set_a,P2: list_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2 != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P2 @ N )
!= nil_a ) ) ) ) ).
% subring_polynomial_pow_not_zero
thf(fact_1198_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_1199_pirreducibleE_I1_J,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( P2 != nil_a ) ) ) ) ).
% pirreducibleE(1)
thf(fact_1200_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_1201_pprimeE_I1_J,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( P2 != nil_a ) ) ) ) ).
% pprimeE(1)
thf(fact_1202_exists__unique__long__division,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ? [X2: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q @ X2 )
& ! [Y6: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q @ Y6 )
=> ( Y6 = X2 ) ) ) ) ) ) ) ).
% exists_unique_long_division
thf(fact_1203_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_1204_normalize_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [V2: a,Va: list_a] :
( X
!= ( cons_a @ V2 @ Va ) ) ) ).
% normalize.cases
thf(fact_1205_p_Ocoeff_Osimps_I1_J,axiom,
( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ k ) @ nil_list_a )
= ( ^ [Uu: nat] : ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.coeff.simps(1)
thf(fact_1206_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_1207_associated__polynomials__iff,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( 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 ) @ P2 @ Q )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
& ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ X3 @ nil_a ) @ Q ) ) ) ) ) ) ) ) ).
% associated_polynomials_iff
thf(fact_1208_is__root__imp__pdivides,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
=> ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P2 ) ) ) ).
% is_root_imp_pdivides
thf(fact_1209_p_Onormalize_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ~ ! [V2: list_a,Va: list_list_a] :
( X
!= ( cons_list_a @ V2 @ Va ) ) ) ).
% p.normalize.cases
thf(fact_1210_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_1211_zero__pdivides,axiom,
! [P2: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P2 )
= ( P2 = nil_a ) ) ).
% zero_pdivides
thf(fact_1212_p_Oassociated__sym,axiom,
! [A: list_a,B: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ A ) ) ).
% p.associated_sym
thf(fact_1213_p_Oassoc__subst,axiom,
! [A: list_a,B: list_a,F: list_a > list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B )
=> ( ! [A3: list_a,B2: list_a] :
( ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
& ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
& ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ A3 @ B2 ) )
=> ( ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
& ( member_list_a @ ( F @ B2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
& ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ ( F @ A3 ) @ ( F @ B2 ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% p.assoc_subst
thf(fact_1214_p_Oassociated__trans,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ C )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ C ) ) ) ) ) ).
% p.associated_trans
thf(fact_1215_p_OUnits__assoc,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B ) ) ) ).
% p.Units_assoc
thf(fact_1216_pdivides__zero,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P2 @ nil_a ) ) ) ).
% pdivides_zero
thf(fact_1217_associated__polynomials__imp__same__is__root,axiom,
! [P2: list_a,Q: list_a,X: a] :
( ( member_list_a @ P2 @ ( 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 ) ) @ P2 @ Q )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
= ( polyno4133073214067823460ot_a_b @ r @ Q @ X ) ) ) ) ) ).
% associated_polynomials_imp_same_is_root
thf(fact_1218_p_Omult__cong__l,axiom,
! [A: list_a,A6: list_a,B: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ A6 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ A6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A6 @ B ) ) ) ) ) ) ).
% p.mult_cong_l
thf(fact_1219_p_Omult__cong__r,axiom,
! [B: list_a,B6: list_a,A: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ B6 )
=> ( ( 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 @ B6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B6 ) ) ) ) ) ) ).
% p.mult_cong_r
thf(fact_1220_p_OUnits__cong,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).
% p.Units_cong
thf(fact_1221_p_Oassociated__iff__same__ideal,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B )
= ( ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ A )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ B ) ) ) ) ) ).
% p.associated_iff_same_ideal
thf(fact_1222_p_Oproperfactor__cong__l,axiom,
! [X6: list_a,X: list_a,Y: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ X6 @ X )
=> ( ( proper8313688649498433056t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ X6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( proper8313688649498433056t_unit @ ( univ_poly_a_b @ r @ k ) @ X6 @ Y ) ) ) ) ) ) ).
% p.properfactor_cong_l
thf(fact_1223_p_Oproperfactor__cong__r,axiom,
! [X: list_a,Y: list_a,Y2: list_a] :
( ( proper8313688649498433056t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ Y2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( proper8313688649498433056t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y2 ) ) ) ) ) ) ).
% p.properfactor_cong_r
thf(fact_1224_polynomial__pow__division,axiom,
! [P2: list_a,N: nat,M: nat] :
( ( member_list_a @ P2 @ ( 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 ) ) @ P2 @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ M ) ) ) ) ).
% polynomial_pow_division
thf(fact_1225_pmod__zero__iff__pdivides,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( 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 @ P2 @ Q )
= nil_a )
= ( polyno5814909790663948098es_a_b @ r @ Q @ P2 ) ) ) ) ) ).
% pmod_zero_iff_pdivides
thf(fact_1226_p_OassociatedI2,axiom,
! [U: list_a,A: list_a,B: list_a] :
( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( A
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ U ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B ) ) ) ) ).
% p.associatedI2
thf(fact_1227_p_OassociatedI2_H,axiom,
! [A: list_a,B: list_a,U: list_a] :
( ( A
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ U ) )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B ) ) ) ) ).
% p.associatedI2'
thf(fact_1228_cgenideal__pirreducible,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ Q )
=> ( ( member_list_a @ Q @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ Q ) ) ) ) ) ) ).
% cgenideal_pirreducible
thf(fact_1229_p_Omonic__degree__one__root__condition,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ A ) @ nil_list_a ) ) @ B )
= ( A = B ) ) ) ).
% p.monic_degree_one_root_condition
thf(fact_1230_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_1231_pprimeE_I3_J,axiom,
! [K: set_a,P2: list_a,Q: list_a,R3: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R3 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q )
| ( polyno5814909790663948098es_a_b @ r @ P2 @ R3 ) ) ) ) ) ) ) ) ).
% pprimeE(3)
thf(fact_1232_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_1233_pirreducible__pow__pdivides__iff,axiom,
! [K: set_a,P2: list_a,Q: list_a,R3: list_a,N: nat] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( ~ ( polyno5814909790663948098es_a_b @ r @ P2 @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P2 @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R3 ) )
= ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P2 @ N ) @ R3 ) ) ) ) ) ) ) ) ).
% pirreducible_pow_pdivides_iff
thf(fact_1234_pdivides__imp__is__root,axiom,
! [P2: list_a,X: a] :
( ( P2 != nil_a )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P2 )
=> ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ) ).
% pdivides_imp_is_root
thf(fact_1235_pprimeI,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2 != nil_a )
=> ( ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q3: list_a,R4: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q3 @ R4 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q3 )
| ( polyno5814909790663948098es_a_b @ r @ P2 @ R4 ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ) ) ).
% pprimeI
thf(fact_1236_p_Oassociated__refl,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ A ) ) ).
% p.associated_refl
thf(fact_1237_alg__multE_I2_J,axiom,
! [X: a,P2: list_a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != 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 @ X ) @ nil_a ) ) @ N ) @ P2 )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) ) ) ) ) ) ).
% alg_multE(2)
thf(fact_1238_le__alg__mult__imp__pdivides,axiom,
! [X: a,P2: list_a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) )
=> ( 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 @ X ) @ nil_a ) ) @ N ) @ P2 ) ) ) ) ).
% le_alg_mult_imp_pdivides
thf(fact_1239_associated__sym,axiom,
! [A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( associ5860276527279195403xt_a_b @ r @ B @ A ) ) ).
% associated_sym
thf(fact_1240_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_1241_assoc__subst,axiom,
! [A: a,B: a,F: a > a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ! [A3: a,B2: a] :
( ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ A3 @ B2 ) )
=> ( ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ ( F @ B2 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ ( F @ A3 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% assoc_subst
thf(fact_1242_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_1243_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_1244_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_1245_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_1246_properfactor__cong__r,axiom,
! [X: a,Y: a,Y2: a] :
( ( proper19828929941537682xt_a_b @ r @ X @ Y )
=> ( ( associ5860276527279195403xt_a_b @ r @ Y @ Y2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ X @ Y2 ) ) ) ) ) ) ).
% properfactor_cong_r
thf(fact_1247_properfactor__cong__l,axiom,
! [X6: a,X: a,Y: a] :
( ( associ5860276527279195403xt_a_b @ r @ X6 @ X )
=> ( ( proper19828929941537682xt_a_b @ r @ X @ Y )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ X6 @ Y ) ) ) ) ) ) ).
% properfactor_cong_l
thf(fact_1248_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_1249_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_1250_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_1251_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 )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
& ( A
= ( mult_a_ring_ext_a_b @ r @ X3 @ B ) ) ) ) ) ) ) ).
% ring_associated_iff
thf(fact_1252_associated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ A ) ) ).
% associated_refl
thf(fact_1253_alg__multE_I1_J,axiom,
! [X: a,P2: list_a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != 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 @ X ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) ) @ P2 ) ) ) ) ).
% alg_multE(1)
thf(fact_1254_long__divisionE,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( 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 @ P2 @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q ) ) ) ) ) ) ) ).
% long_divisionE
thf(fact_1255_long__divisionI,axiom,
! [K: set_a,P2: list_a,Q: list_a,B: list_a,R3: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( 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 @ P2 @ Q @ ( produc6837034575241423639list_a @ B @ R3 ) )
=> ( ( produc6837034575241423639list_a @ B @ R3 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q ) ) ) ) ) ) ) ) ).
% long_divisionI
thf(fact_1256_poly__add_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
~ ! [P1: list_a,P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).
% poly_add.cases
thf(fact_1257_combine_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [K3: a,Ks: list_a,U2: a,Us: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ K3 @ Ks ) @ ( cons_a @ U2 @ Us ) ) )
=> ( ! [Us: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Us ) )
=> ~ ! [Ks: list_a] :
( X
!= ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).
% combine.cases
thf(fact_1258_poly__mult_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ P22 ) )
=> ~ ! [V2: a,Va: list_a,P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ V2 @ Va ) @ P22 ) ) ) ).
% poly_mult.cases
thf(fact_1259_exists__long__division,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ~ ! [B2: list_a] :
( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ! [R4: list_a] :
( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ r @ P2 @ Q @ ( produc6837034575241423639list_a @ B2 @ R4 ) ) ) ) ) ) ) ) ).
% exists_long_division
thf(fact_1260_poly__mult__var,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( P2 = nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ ( var_a_b @ r ) )
= nil_a ) )
& ( ( P2 != nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ ( var_a_b @ r ) )
= ( append_a @ P2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ).
% poly_mult_var
thf(fact_1261_p_Olead__coeff__in__carrier,axiom,
! [K: set_list_a,A: list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ ( cons_list_a @ A @ P2 ) )
=> ( member_list_a @ A @ ( 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 ) ) ) ) ) ).
% p.lead_coeff_in_carrier
thf(fact_1262_p_Ocombine_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [K3: list_a,Ks: list_list_a,U2: list_a,Us: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ K3 @ Ks ) @ ( cons_list_a @ U2 @ Us ) ) )
=> ( ! [Us: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ Us ) )
=> ~ ! [Ks: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ Ks @ nil_list_a ) ) ) ) ).
% p.combine.cases
thf(fact_1263_p_Opoly__mult_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [P22: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ P22 ) )
=> ~ ! [V2: list_a,Va: list_list_a,P22: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ V2 @ Va ) @ P22 ) ) ) ).
% p.poly_mult.cases
thf(fact_1264_p_Ocoeff__iff__polynomial__cond,axiom,
! [K: set_list_a,P12: list_list_a,P23: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P12 )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P23 )
=> ( ( P12 = P23 )
= ( ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ k ) @ P12 )
= ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ k ) @ P23 ) ) ) ) ) ).
% p.coeff_iff_polynomial_cond
thf(fact_1265_p_Opoly__coeff__in__carrier,axiom,
! [K: set_list_a,P2: list_list_a,I: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P2 )
=> ( member_list_a @ ( coeff_6360649920519955023t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ I ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.poly_coeff_in_carrier
thf(fact_1266_p_Olead__coeff__not__zero,axiom,
! [K: set_list_a,A: list_a,P2: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ ( cons_list_a @ A @ P2 ) )
=> ( member_list_a @ A @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) ) ) ).
% p.lead_coeff_not_zero
thf(fact_1267_p_Ozero__is__polynomial,axiom,
! [K: set_list_a] : ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ nil_list_a ) ).
% p.zero_is_polynomial
thf(fact_1268_p_Ocarrier__polynomial,axiom,
! [K: set_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P2 )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ P2 ) ) ) ).
% p.carrier_polynomial
thf(fact_1269_p_Oconst__is__polynomial,axiom,
! [A: list_a,K: set_list_a] :
( ( member_list_a @ A @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ ( cons_list_a @ A @ nil_list_a ) ) ) ).
% p.const_is_polynomial
thf(fact_1270_p_Omonom__is__polynomial,axiom,
! [K: set_list_a,A: list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( member_list_a @ A @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ N ) ) ) ) ).
% p.monom_is_polynomial
thf(fact_1271_alg__mult__gt__zero__iff__is__root,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) )
= ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_1272_p_Oconst__term__not__zero,axiom,
! [P2: list_list_a] :
( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( P2 != nil_list_a ) ) ).
% p.const_term_not_zero
thf(fact_1273_p_Opoly__add_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
~ ! [P1: list_list_a,P22: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ P1 @ P22 ) ) ).
% p.poly_add.cases
thf(fact_1274_coeff__iff__polynomial__cond,axiom,
! [K: set_a,P12: list_a,P23: list_a] :
( ( polynomial_a_b @ r @ K @ P12 )
=> ( ( polynomial_a_b @ r @ K @ P23 )
=> ( ( P12 = P23 )
= ( ( coeff_a_b @ r @ P12 )
= ( coeff_a_b @ r @ P23 ) ) ) ) ) ).
% coeff_iff_polynomial_cond
thf(fact_1275_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_1276_poly__coeff__in__carrier,axiom,
! [K: set_a,P2: list_a,I: nat] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P2 )
=> ( member_a @ ( coeff_a_b @ r @ P2 @ I ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_coeff_in_carrier
thf(fact_1277_lead__coeff__not__zero,axiom,
! [K: set_a,A: a,P2: list_a] :
( ( polynomial_a_b @ r @ K @ ( cons_a @ A @ P2 ) )
=> ( member_a @ A @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% lead_coeff_not_zero
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( formal4452980811800949548iv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ ( formal4452980811800949548iv_a_b @ r @ f ) @ ( formal4452980811800949548iv_a_b @ r @ f ) ) )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( formal4452980811800949548iv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( formal4452980811800949548iv_a_b @ r @ f ) ) @ ( formal4452980811800949548iv_a_b @ r @ f ) ) ) ).
%------------------------------------------------------------------------------