TPTP Problem File: SLH0045^1.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 : Interpolation_Polynomials_HOL_Algebra/0001_Lagrange_Interpolation/prob_00092_003233__17147014_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1648 ( 182 unt; 370 typ; 0 def)
% Number of atoms : 4834 (1417 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 21197 ( 282 ~; 66 |; 86 &;17593 @)
% ( 0 <=>;3170 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 9 avg)
% Number of types : 48 ( 47 usr)
% Number of type conns : 866 ( 866 >; 0 *; 0 +; 0 <<)
% Number of symbols : 324 ( 323 usr; 17 con; 0-4 aty)
% Number of variables : 3569 ( 21 ^;3478 !; 70 ?;3569 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:36:45.379
%------------------------------------------------------------------------------
% Could-be-implicit typings (47)
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member_set_list_a_a: ( set_list_a > a ) > set_set_list_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member_a_list_list_a: ( a > list_list_a ) > set_a_list_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
member_a_list_a: ( a > list_a ) > set_a_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member_a_set_list_a: ( a > set_list_a ) > set_a_set_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
member6842060177613954879list_a: list_l7815035709764258326list_a > set_li3407770045201608054list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member5342144027231129785list_a: list_list_list_a > set_list_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
member352051402189872281list_a: list_list_set_list_a > set_li664707282716828624list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member5524387281408368019list_a: list_set_list_a > set_list_set_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__c,type,
member_c: c > set_c > $o ).
thf(sy_v_A,type,
a2: set_c ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_f,type,
f: c > list_a ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1277)
thf(fact_0_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_1_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_2_domain_Ouniv__poly__is__cring,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( cring_3148771470849435808t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_cring
thf(fact_3_domain_Ouniv__poly__is__cring,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( cring_5991999922451032090t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_cring
thf(fact_4_univ__poly__is__cring,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( cring_3148771470849435808t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_cring
thf(fact_5_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_6_carrier__is__subcring,axiom,
subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subcring
thf(fact_7_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_8_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_9_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_10_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_11_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_12_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_13_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_14_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_15_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_16_subcringI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subcring_a_b @ H @ r ) ) ).
% subcringI'
thf(fact_17_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_18_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_19_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_20_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_21_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_22_subdomainI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subdomain_a_b @ H @ r ) ) ).
% subdomainI'
thf(fact_23_cring_Ocgenideal__is__principalideal,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ R @ I ) @ R ) ) ) ).
% cring.cgenideal_is_principalideal
thf(fact_24_cring_Ocgenideal__is__principalideal,axiom,
! [R: partia2670972154091845814t_unit,I: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ R ) )
=> ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ R @ I ) @ R ) ) ) ).
% cring.cgenideal_is_principalideal
thf(fact_25_cring_Ocgenideal__is__principalideal,axiom,
! [R: partia2956882679547061052t_unit,I: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ I @ ( partia2464479390973590831t_unit @ R ) )
=> ( princi2534607884127416211t_unit @ ( cgenid24865672677839267t_unit @ R @ I ) @ R ) ) ) ).
% cring.cgenideal_is_principalideal
thf(fact_26_univ__poly__is__abelian__monoid,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( abelia226231641709521465t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_abelian_monoid
thf(fact_27_cring_OsubcringI_H,axiom,
! [R: partia2956882679547061052t_unit,H: set_list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ H @ R )
=> ( subcri8676831449680469861t_unit @ H @ R ) ) ) ).
% cring.subcringI'
thf(fact_28_cring_OsubcringI_H,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ H @ R )
=> ( subcri7763218559781929323t_unit @ H @ R ) ) ) ).
% cring.subcringI'
thf(fact_29_cring_OsubcringI_H,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( cring_a_b @ R )
=> ( ( subring_a_b @ H @ R )
=> ( subcring_a_b @ H @ R ) ) ) ).
% cring.subcringI'
thf(fact_30_cring_Ocarrier__is__subcring,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( subcring_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% cring.carrier_is_subcring
thf(fact_31_cring_Ocarrier__is__subcring,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( subcri7763218559781929323t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).
% cring.carrier_is_subcring
thf(fact_32_cring_Ocarrier__is__subcring,axiom,
! [R: partia2956882679547061052t_unit] :
( ( cring_5991999922451032090t_unit @ R )
=> ( subcri8676831449680469861t_unit @ ( partia2464479390973590831t_unit @ R ) @ R ) ) ).
% cring.carrier_is_subcring
thf(fact_33_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_34_add_Oint__pow__mult__distrib,axiom,
! [X: a,Y: a,I: int] :
( ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ) ).
% add.int_pow_mult_distrib
thf(fact_35_add_Oint__pow__distrib,axiom,
! [X: a,Y: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ).
% add.int_pow_distrib
thf(fact_36_ring__iso__memE_I3_J,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_37_ring__iso__memE_I3_J,axiom,
! [H2: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_38_ring__iso__memE_I3_J,axiom,
! [H2: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H2 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_39_ring__iso__memE_I3_J,axiom,
! [H2: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_40_ring__iso__memE_I3_J,axiom,
! [H2: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H2 @ ( ring_i5684343068699926420it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_41_ring__iso__memE_I3_J,axiom,
! [H2: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H2 @ ( ring_i4611353245267337884t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_42_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_43_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_44_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_45_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_46_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_47_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_48_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_49_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_50_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_51_is__cring,axiom,
cring_a_b @ r ).
% is_cring
thf(fact_52_add_Oint__pow__closed,axiom,
! [X: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_pow_a_b_int @ r @ I @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.int_pow_closed
thf(fact_53_carrier__polynomial__shell,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% carrier_polynomial_shell
thf(fact_54_field_Oaxioms_I1_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( domain1617769409708967785t_unit @ R ) ) ).
% field.axioms(1)
thf(fact_55_field_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% field.axioms(1)
thf(fact_56_field_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% field.axioms(1)
thf(fact_57_fieldE_I1_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( field_1861437471013600865t_unit @ R )
=> ( cring_5991999922451032090t_unit @ R ) ) ).
% fieldE(1)
thf(fact_58_fieldE_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( cring_3148771470849435808t_unit @ R ) ) ).
% fieldE(1)
thf(fact_59_fieldE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( cring_a_b @ R ) ) ).
% fieldE(1)
thf(fact_60_fieldE_I1_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( cring_3470013030684506304t_unit @ R ) ) ).
% fieldE(1)
thf(fact_61_subdomainE_I7_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subdomainE(7)
thf(fact_62_subdomainE_I7_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( add_a_b @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subdomainE(7)
thf(fact_63_subdomainE_I4_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( H != bot_bot_set_a ) ) ).
% subdomainE(4)
thf(fact_64_semiring_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( abelia226231641709521465t_unit @ R ) ) ).
% semiring.axioms(1)
thf(fact_65_semiring_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( abelian_monoid_a_b @ R ) ) ).
% semiring.axioms(1)
thf(fact_66_subdomain_Oaxioms_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( subcring_a_b @ H @ R ) ) ).
% subdomain.axioms(1)
thf(fact_67_abelian__monoid_Oa__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( add_a_b @ G @ X @ Y ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_68_abelian__monoid_Oa__closed,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ G @ X @ Y ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_69_abelian__monoid_Oa__closed,axiom,
! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ G @ X @ Y ) @ ( partia2464479390973590831t_unit @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_70_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X @ ( add_a_b @ G @ Y @ Z ) )
= ( add_a_b @ G @ Y @ ( add_a_b @ G @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_71_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X @ ( add_li7652885771158616974t_unit @ G @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ G @ Y @ ( add_li7652885771158616974t_unit @ G @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_72_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ X @ ( add_li174743652000525320t_unit @ G @ Y @ Z ) )
= ( add_li174743652000525320t_unit @ G @ Y @ ( add_li174743652000525320t_unit @ G @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_73_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
! [A: list_a,P2: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_75_mem__Collect__eq,axiom,
! [A: c,P2: c > $o] :
( ( member_c @ A @ ( collect_c @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_76_mem__Collect__eq,axiom,
! [A: list_list_a,P2: list_list_a > $o] :
( ( member_list_list_a @ A @ ( collect_list_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X2: list_a] : ( member_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_79_Collect__mem__eq,axiom,
! [A2: set_c] :
( ( collect_c
@ ^ [X2: c] : ( member_c @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A2: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X2: list_list_a] : ( member_list_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_81_abelian__monoid_Oa__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( add_a_b @ G @ X @ Y ) @ Z )
= ( add_a_b @ G @ X @ ( add_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_82_abelian__monoid_Oa__assoc,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( add_li7652885771158616974t_unit @ G @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ G @ X @ ( add_li7652885771158616974t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_83_abelian__monoid_Oa__assoc,axiom,
! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ ( add_li174743652000525320t_unit @ G @ X @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ G @ X @ ( add_li174743652000525320t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_84_abelian__monoid_Oa__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X @ Y )
= ( add_a_b @ G @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_85_abelian__monoid_Oa__comm,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X @ Y )
= ( add_li7652885771158616974t_unit @ G @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_86_abelian__monoid_Oa__comm,axiom,
! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ X @ Y )
= ( add_li174743652000525320t_unit @ G @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_87_abelian__monoidE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_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 ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_88_abelian__monoidE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_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 ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_89_abelian__monoidE_I1_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( abelia3641329199688042803t_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 ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_90_abelian__monoidE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( abelian_monoid_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 ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_91_abelian__monoidE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( abelia226231641709521465t_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 ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_92_abelian__monoidE_I3_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( abelia3641329199688042803t_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 ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_93_abelian__monoidE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_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 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_94_abelian__monoidE_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_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 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_95_abelian__monoidE_I5_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( abelia3641329199688042803t_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 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_96_domain_OsubdomainI_H,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ H @ R )
=> ( subdom7821232466298058046t_unit @ H @ R ) ) ) ).
% domain.subdomainI'
thf(fact_97_domain_OsubdomainI_H,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ H @ R )
=> ( subdomain_a_b @ H @ R ) ) ) ).
% domain.subdomainI'
thf(fact_98_domain_Ouniv__poly__is__abelian__monoid,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( abelia226231641709521465t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_abelian_monoid
thf(fact_99_domain_Ouniv__poly__is__abelian__monoid,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( abelia3641329199688042803t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_abelian_monoid
thf(fact_100_cring_Ois__cring,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( cring_3148771470849435808t_unit @ R ) ) ).
% cring.is_cring
thf(fact_101_cring_Ois__cring,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( cring_a_b @ R ) ) ).
% cring.is_cring
thf(fact_102_cring_Ois__cring,axiom,
! [R: partia2956882679547061052t_unit] :
( ( cring_5991999922451032090t_unit @ R )
=> ( cring_5991999922451032090t_unit @ R ) ) ).
% cring.is_cring
thf(fact_103_principalideal_Ois__principalideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ( principalideal_a_b @ I2 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_104_subringE_I7_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subring_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( add_a_b @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subringE(7)
thf(fact_105_subringE_I7_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subringE(7)
thf(fact_106_subringE_I4_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( H != bot_bot_set_list_a ) ) ).
% subringE(4)
thf(fact_107_subringE_I4_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( H != bot_bot_set_a ) ) ).
% subringE(4)
thf(fact_108_ring__hom__closed,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_109_ring__hom__closed,axiom,
! [H2: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a] :
( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H2 @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_110_ring__hom__closed,axiom,
! [H2: a > list_list_a,R: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,X: a] :
( ( member_a_list_list_a @ H2 @ ( ring_h6858658657455840382t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_list_a @ ( H2 @ X ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_111_ring__hom__closed,axiom,
! [H2: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a] :
( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_112_ring__hom__closed,axiom,
! [H2: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H2 @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_113_ring__hom__closed,axiom,
! [H2: list_a > list_list_a,R: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,X: list_a] :
( ( member6714375691612171394list_a @ H2 @ ( ring_h8002040739877300486t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( H2 @ X ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_114_ring__hom__closed,axiom,
! [H2: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X: list_list_a] :
( ( member_list_list_a_a @ H2 @ ( ring_h8078271382950527358it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_115_ring__hom__closed,axiom,
! [H2: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X: list_list_a] :
( ( member7168557129179038582list_a @ H2 @ ( ring_h5031276006722532742t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_a @ ( H2 @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_116_ring__hom__closed,axiom,
! [H2: list_list_a > list_list_a,R: partia2956882679547061052t_unit,S: partia2956882679547061052t_unit,X: list_list_a] :
( ( member8231385768148312316list_a @ H2 @ ( ring_h8129544334414776832t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( H2 @ X ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_117_subcringE_I7_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subcringE(7)
thf(fact_118_subcringE_I7_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( add_a_b @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subcringE(7)
thf(fact_119_subcringE_I4_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( H != bot_bot_set_a ) ) ).
% subcringE(4)
thf(fact_120_domain_Oaxioms_I1_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( domain7810152921033798211t_unit @ R )
=> ( cring_5991999922451032090t_unit @ R ) ) ).
% domain.axioms(1)
thf(fact_121_domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( cring_a_b @ R ) ) ).
% domain.axioms(1)
thf(fact_122_domain_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( cring_3148771470849435808t_unit @ R ) ) ).
% domain.axioms(1)
thf(fact_123_ring__iso__memE_I1_J,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_124_ring__iso__memE_I1_J,axiom,
! [H2: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a] :
( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H2 @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_125_ring__iso__memE_I1_J,axiom,
! [H2: a > list_list_a,R: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,X: a] :
( ( member_a_list_list_a @ H2 @ ( ring_i4464730343205239444t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_list_a @ ( H2 @ X ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_126_ring__iso__memE_I1_J,axiom,
! [H2: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a] :
( ( member_list_a_a @ H2 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_127_ring__iso__memE_I1_J,axiom,
! [H2: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H2 @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_128_ring__iso__memE_I1_J,axiom,
! [H2: list_a > list_list_a,R: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,X: list_a] :
( ( member6714375691612171394list_a @ H2 @ ( ring_i7582117978422105628t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( H2 @ X ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_129_ring__iso__memE_I1_J,axiom,
! [H2: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X: list_list_a] :
( ( member_list_list_a_a @ H2 @ ( ring_i5684343068699926420it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_130_ring__iso__memE_I1_J,axiom,
! [H2: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X: list_list_a] :
( ( member7168557129179038582list_a @ H2 @ ( ring_i4611353245267337884t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_a @ ( H2 @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_131_ring__iso__memE_I1_J,axiom,
! [H2: list_list_a > list_list_a,R: partia2956882679547061052t_unit,S: partia2956882679547061052t_unit,X: list_list_a] :
( ( member8231385768148312316list_a @ H2 @ ( ring_i6186174840089424918t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( H2 @ X ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_132_subcring_Oaxioms_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( subrin6918843898125473962t_unit @ H @ R ) ) ).
% subcring.axioms(1)
thf(fact_133_subcring_Oaxioms_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( subring_a_b @ H @ R ) ) ).
% subcring.axioms(1)
thf(fact_134_cring_Ocring__simprules_I23_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( cring_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 ) ) ) ) ) ) ) ).
% cring.cring_simprules(23)
thf(fact_135_cring_Ocring__simprules_I23_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( cring_3148771470849435808t_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 ) ) ) ) ) ) ) ).
% cring.cring_simprules(23)
thf(fact_136_cring_Ocring__simprules_I23_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( cring_5991999922451032090t_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 ) ) ) ) ) ) ) ).
% cring.cring_simprules(23)
thf(fact_137_cring_Ocring__simprules_I10_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_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 ) ) ) ) ) ).
% cring.cring_simprules(10)
thf(fact_138_cring_Ocring__simprules_I10_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_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 ) ) ) ) ) ).
% cring.cring_simprules(10)
thf(fact_139_cring_Ocring__simprules_I10_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( cring_5991999922451032090t_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 ) ) ) ) ) ).
% cring.cring_simprules(10)
thf(fact_140_cring_Ocring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( cring_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 ) ) ) ) ) ) ) ).
% cring.cring_simprules(7)
thf(fact_141_cring_Ocring__simprules_I7_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( cring_3148771470849435808t_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 ) ) ) ) ) ) ) ).
% cring.cring_simprules(7)
thf(fact_142_cring_Ocring__simprules_I7_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( cring_5991999922451032090t_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 ) ) ) ) ) ) ) ).
% cring.cring_simprules(7)
thf(fact_143_cring_Ocring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_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 ) ) ) ) ) ).
% cring.cring_simprules(1)
thf(fact_144_cring_Ocring__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_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 ) ) ) ) ) ).
% cring.cring_simprules(1)
thf(fact_145_cring_Ocring__simprules_I1_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( cring_5991999922451032090t_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 ) ) ) ) ) ).
% cring.cring_simprules(1)
thf(fact_146_ring__hom__add,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_147_ring__hom__add,axiom,
! [H2: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_148_ring__hom__add,axiom,
! [H2: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_149_ring__hom__add,axiom,
! [H2: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_150_ring__hom__add,axiom,
! [H2: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H2 @ ( ring_h8078271382950527358it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_151_ring__hom__add,axiom,
! [H2: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H2 @ ( ring_h5031276006722532742t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_152_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_153_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_154_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_155_univ__poly__a__minus__consistent,axiom,
! [K: set_a,Q: list_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) ) ) ) ).
% univ_poly_a_minus_consistent
thf(fact_156_domain_Ouniv__poly__not__field,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ~ ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_not_field
thf(fact_157_domain_Ouniv__poly__not__field,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ~ ( field_1861437471013600865t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_not_field
thf(fact_158_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_159_const__term__simprules__shell_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) )
= ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ) ).
% const_term_simprules_shell(3)
thf(fact_160_poly__of__const__in__carrier,axiom,
! [S2: a] :
( ( member_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( poly_of_const_a_b @ r @ S2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% poly_of_const_in_carrier
thf(fact_161_add_Oint__pow__mult,axiom,
! [X: a,I: int,J: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ ( plus_plus_int @ I @ J ) @ X )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ J @ X ) ) ) ) ).
% add.int_pow_mult
thf(fact_162_const__term__simprules__shell_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_a @ ( const_term_a_b @ r @ P ) @ K ) ) ) ).
% const_term_simprules_shell(1)
thf(fact_163_subcringI,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( ( mult_a_ring_ext_a_b @ r @ H12 @ H23 )
= ( mult_a_ring_ext_a_b @ r @ H23 @ H12 ) ) ) )
=> ( subcring_a_b @ H @ r ) ) ) ).
% subcringI
thf(fact_164_add_Oint__pow__pow,axiom,
! [X: a,M: int,N: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ M @ ( add_pow_a_b_int @ r @ N @ X ) )
= ( add_pow_a_b_int @ r @ ( times_times_int @ N @ M ) @ X ) ) ) ).
% add.int_pow_pow
thf(fact_165_add__pow__ldistr__int,axiom,
! [A: a,B: a,K2: int] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ K2 @ A ) @ B )
= ( add_pow_a_b_int @ r @ K2 @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_ldistr_int
thf(fact_166_add__pow__rdistr__int,axiom,
! [A: a,B: a,K2: int] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ K2 @ B ) )
= ( add_pow_a_b_int @ r @ K2 @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_rdistr_int
thf(fact_167_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_168_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_169_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_170_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_171_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_172_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_173_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_174_empty_Oprems_I2_J,axiom,
! [X: c] :
( ( member_c @ X @ bot_bot_set_c )
=> ( member_list_a @ ( f @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% empty.prems(2)
thf(fact_175_ring_Opoly__of__const_Ocong,axiom,
poly_of_const_a_b = poly_of_const_a_b ).
% ring.poly_of_const.cong
thf(fact_176_ring_Oconst__term_Ocong,axiom,
const_term_a_b = const_term_a_b ).
% ring.const_term.cong
thf(fact_177_domain_Oconst__term__simprules__shell_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( const_term_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ R @ ( const_term_a_b @ R @ P ) @ ( const_term_a_b @ R @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(2)
thf(fact_178_domain_Oconst__term__simprules__shell_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ Q ) )
= ( mult_l7073676228092353617t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P ) @ ( const_6738166269504826821t_unit @ R @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(2)
thf(fact_179_subringE_I6_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subring_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subringE(6)
thf(fact_180_subringE_I6_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subringE(6)
thf(fact_181_subcringE_I6_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subcringE(6)
thf(fact_182_subcringE_I6_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subcringE(6)
thf(fact_183_subcring_Osub__m__comm,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 )
= ( mult_l7073676228092353617t_unit @ R @ H22 @ H1 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_184_subcring_Osub__m__comm,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( mult_a_ring_ext_a_b @ R @ H1 @ H22 )
= ( mult_a_ring_ext_a_b @ R @ H22 @ H1 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_185_subdomainE_I8_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 )
= ( mult_l7073676228092353617t_unit @ R @ H22 @ H1 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_186_subdomainE_I8_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( mult_a_ring_ext_a_b @ R @ H1 @ H22 )
= ( mult_a_ring_ext_a_b @ R @ H22 @ H1 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_187_subdomainE_I6_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subdomainE(6)
thf(fact_188_subdomainE_I6_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subdomainE(6)
thf(fact_189_cring_Ocring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_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 ) ) ) ) ) ).
% cring.cring_simprules(5)
thf(fact_190_cring_Ocring__simprules_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_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 ) ) ) ) ) ).
% cring.cring_simprules(5)
thf(fact_191_cring_Ocring__simprules_I5_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( cring_5991999922451032090t_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 ) ) ) ) ) ).
% cring.cring_simprules(5)
thf(fact_192_cring_Ocring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( cring_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 ) ) ) ) ) ) ) ).
% cring.cring_simprules(11)
thf(fact_193_cring_Ocring__simprules_I11_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( cring_3148771470849435808t_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 ) ) ) ) ) ) ) ).
% cring.cring_simprules(11)
thf(fact_194_cring_Ocring__simprules_I11_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( cring_5991999922451032090t_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 ) ) ) ) ) ) ) ).
% cring.cring_simprules(11)
thf(fact_195_cring_Ocring__simprules_I14_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% cring.cring_simprules(14)
thf(fact_196_cring_Ocring__simprules_I14_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ Y )
= ( mult_l7073676228092353617t_unit @ R @ Y @ X ) ) ) ) ) ).
% cring.cring_simprules(14)
thf(fact_197_cring_Ocring__simprules_I14_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ X @ Y )
= ( mult_l4853965630390486993t_unit @ R @ Y @ X ) ) ) ) ) ).
% cring.cring_simprules(14)
thf(fact_198_cring_Ocring__simprules_I24_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( cring_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 @ 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 ) ) ) ) ) ) ) ).
% cring.cring_simprules(24)
thf(fact_199_cring_Ocring__simprules_I24_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( cring_3148771470849435808t_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 @ X @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) )
= ( mult_l7073676228092353617t_unit @ R @ Y @ ( mult_l7073676228092353617t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(24)
thf(fact_200_cring_Ocring__simprules_I24_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( cring_5991999922451032090t_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 @ X @ ( mult_l4853965630390486993t_unit @ R @ Y @ Z ) )
= ( mult_l4853965630390486993t_unit @ R @ Y @ ( mult_l4853965630390486993t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(24)
thf(fact_201_ring__hom__mult,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_202_ring__hom__mult,axiom,
! [H2: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_203_ring__hom__mult,axiom,
! [H2: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_204_ring__hom__mult,axiom,
! [H2: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_205_ring__hom__mult,axiom,
! [H2: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H2 @ ( ring_h8078271382950527358it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_206_ring__hom__mult,axiom,
! [H2: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H2 @ ( ring_h5031276006722532742t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_207_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_208_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_209_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_210_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_211_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_212_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_213_ring__iso__memE_I2_J,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_214_ring__iso__memE_I2_J,axiom,
! [H2: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_215_ring__iso__memE_I2_J,axiom,
! [H2: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H2 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_216_ring__iso__memE_I2_J,axiom,
! [H2: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_217_ring__iso__memE_I2_J,axiom,
! [H2: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H2 @ ( ring_i5684343068699926420it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_218_ring__iso__memE_I2_J,axiom,
! [H2: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H2 @ ( ring_i4611353245267337884t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_219_cring_Ocring__simprules_I13_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( cring_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 ) ) ) ) ) ) ) ).
% cring.cring_simprules(13)
thf(fact_220_cring_Ocring__simprules_I13_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( cring_3148771470849435808t_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 ) ) ) ) ) ) ) ).
% cring.cring_simprules(13)
thf(fact_221_cring_Ocring__simprules_I13_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( cring_5991999922451032090t_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 ) ) ) ) ) ) ) ).
% cring.cring_simprules(13)
thf(fact_222_cring_Ocring__simprules_I25_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( cring_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 ) ) ) ) ) ) ) ).
% cring.cring_simprules(25)
thf(fact_223_cring_Ocring__simprules_I25_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( cring_3148771470849435808t_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 ) ) ) ) ) ) ) ).
% cring.cring_simprules(25)
thf(fact_224_cring_Ocring__simprules_I25_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( cring_5991999922451032090t_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 ) ) ) ) ) ) ) ).
% cring.cring_simprules(25)
thf(fact_225_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_226_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_227_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_228_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_229_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_230_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_231_cring_Ocring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% cring.cring_simprules(4)
thf(fact_232_cring_Ocring__simprules_I4_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% cring.cring_simprules(4)
thf(fact_233_cring_Ocring__simprules_I4_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( a_minu2241224857956505934t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% cring.cring_simprules(4)
thf(fact_234_domain_Oconst__term__simprules__shell_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_a @ ( const_term_a_b @ R @ P ) @ K ) ) ) ) ).
% domain.const_term_simprules_shell(1)
thf(fact_235_domain_Oconst__term__simprules__shell_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ R @ P ) @ K ) ) ) ) ).
% domain.const_term_simprules_shell(1)
thf(fact_236_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_237_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_238_domain_Oconst__term__simprules__shell_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( const_term_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ Q ) )
= ( add_a_b @ R @ ( const_term_a_b @ R @ P ) @ ( const_term_a_b @ R @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(3)
thf(fact_239_domain_Oconst__term__simprules__shell_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ Q ) )
= ( add_li7652885771158616974t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P ) @ ( const_6738166269504826821t_unit @ R @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(3)
thf(fact_240_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a,Q: list_list_list_a,P: 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 ) @ P @ Q )
= ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_241_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Q: list_a,P: 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 ) @ P @ Q )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_242_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a,P: 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 ) @ P @ Q )
= ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_243_monoid__cancelI,axiom,
( ! [A3: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A3 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B2 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A3 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B2 @ C2 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_244_const__term__simprules__shell_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ) ).
% const_term_simprules_shell(2)
thf(fact_245_line__extension__mem__iff,axiom,
! [U: a,K: set_a,A: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ K )
& ? [Y2: a] :
( ( member_a @ Y2 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ A ) @ Y2 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_246_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_247_map__norm__in__poly__ring__carrier,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_list_a @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ).
% map_norm_in_poly_ring_carrier
thf(fact_248_cgenideal__prod,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_mu8047982887099575916xt_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ ( cgenid547466209912283029xt_a_b @ r @ B ) )
= ( cgenid547466209912283029xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% cgenideal_prod
thf(fact_249_univ__poly__a__inv__consistent,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ) ).
% univ_poly_a_inv_consistent
thf(fact_250_assms_I3_J,axiom,
! [X: c] :
( ( member_c @ X @ a2 )
=> ( member_list_a @ ( f @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% assms(3)
thf(fact_251_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_252_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_253_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_254_add_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X3 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_255_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X3 )
= X3 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_256_add_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X3 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_257_local_Ominus__unique,axiom,
! [Y: a,X: a,Y3: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y3 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y3 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_258_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_259_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_260_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_261_assms_I1_J,axiom,
finite_finite_c @ a2 ).
% assms(1)
thf(fact_262_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_263_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_264_add_Oint__pow__one,axiom,
! [Z: int] :
( ( add_pow_a_b_int @ r @ Z @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% add.int_pow_one
thf(fact_265_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_266_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_267_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_268_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_269_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_270_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_271_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_272_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_273_cring_Ocring__simprules_I22_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( a_inv_7033018035630854991t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) )
= ( zero_l347298301471573063t_unit @ R ) ) ) ).
% cring.cring_simprules(22)
thf(fact_274_cring_Ocring__simprules_I22_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( a_inv_a_b @ R @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ).
% cring.cring_simprules(22)
thf(fact_275_cring_Ocring__simprules_I22_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( a_inv_5715216516650856053t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% cring.cring_simprules(22)
thf(fact_276_cring_Ocring__simprules_I22_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% cring.cring_simprules(22)
thf(fact_277_subringE_I5_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H2: a] :
( ( subring_a_b @ H @ R )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( a_inv_a_b @ R @ H2 ) @ H ) ) ) ).
% subringE(5)
thf(fact_278_subringE_I5_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H2: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H2 ) @ H ) ) ) ).
% subringE(5)
thf(fact_279_subcringE_I5_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H2: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H2 ) @ H ) ) ) ).
% subcringE(5)
thf(fact_280_subcringE_I5_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H2: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( a_inv_a_b @ R @ H2 ) @ H ) ) ) ).
% subcringE(5)
thf(fact_281_subringE_I2_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subrin5643252653130547402t_unit @ H @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ H ) ) ).
% subringE(2)
thf(fact_282_subringE_I2_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H ) ) ).
% subringE(2)
thf(fact_283_subringE_I2_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).
% subringE(2)
thf(fact_284_subdomainE_I5_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H2: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H2 ) @ H ) ) ) ).
% subdomainE(5)
thf(fact_285_subdomainE_I5_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H2: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( a_inv_a_b @ R @ H2 ) @ H ) ) ) ).
% subdomainE(5)
thf(fact_286_subcringE_I2_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subcri7783154434480317835t_unit @ H @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ H ) ) ).
% subcringE(2)
thf(fact_287_subcringE_I2_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).
% subcringE(2)
thf(fact_288_subcringE_I2_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H ) ) ).
% subcringE(2)
thf(fact_289_subdomainE_I2_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subdom3220114454046903646t_unit @ H @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ H ) ) ).
% subdomainE(2)
thf(fact_290_subdomainE_I2_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).
% subdomainE(2)
thf(fact_291_subdomainE_I2_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H ) ) ).
% subdomainE(2)
thf(fact_292_cring_Ocring__simprules_I17_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X @ ( a_inv_5715216516650856053t_unit @ R @ X ) )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% cring.cring_simprules(17)
thf(fact_293_cring_Ocring__simprules_I17_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( a_inv_a_b @ R @ X ) )
= ( zero_a_b @ R ) ) ) ) ).
% cring.cring_simprules(17)
thf(fact_294_cring_Ocring__simprules_I17_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ X ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% cring.cring_simprules(17)
thf(fact_295_cring_Ocring__simprules_I17_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X @ ( a_inv_7033018035630854991t_unit @ R @ X ) )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% cring.cring_simprules(17)
thf(fact_296_cring_Ocring__simprules_I9_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( a_inv_5715216516650856053t_unit @ R @ X ) @ X )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% cring.cring_simprules(9)
thf(fact_297_cring_Ocring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% cring.cring_simprules(9)
thf(fact_298_cring_Ocring__simprules_I9_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ X )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% cring.cring_simprules(9)
thf(fact_299_cring_Ocring__simprules_I9_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ X )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% cring.cring_simprules(9)
thf(fact_300_cring_Osum__zero__eq__neg,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( add_se2486902527185523630t_unit @ R @ X @ Y )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( X
= ( a_inv_5715216516650856053t_unit @ R @ Y ) ) ) ) ) ) ).
% cring.sum_zero_eq_neg
thf(fact_301_cring_Osum__zero__eq__neg,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ).
% cring.sum_zero_eq_neg
thf(fact_302_cring_Osum__zero__eq__neg,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( X
= ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ) ) ).
% cring.sum_zero_eq_neg
thf(fact_303_cring_Osum__zero__eq__neg,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( add_li174743652000525320t_unit @ R @ X @ Y )
= ( zero_l347298301471573063t_unit @ R ) )
=> ( X
= ( a_inv_7033018035630854991t_unit @ R @ Y ) ) ) ) ) ) ).
% cring.sum_zero_eq_neg
thf(fact_304_cring_Ocring__simprules_I21_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( a_inv_a_b @ R @ X ) )
= X ) ) ) ).
% cring.cring_simprules(21)
thf(fact_305_cring_Ocring__simprules_I21_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) )
= X ) ) ) ).
% cring.cring_simprules(21)
thf(fact_306_cring_Ocring__simprules_I21_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( a_inv_7033018035630854991t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) )
= X ) ) ) ).
% cring.cring_simprules(21)
thf(fact_307_cring_Ocring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_inv_a_b @ R @ X ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% cring.cring_simprules(3)
thf(fact_308_cring_Ocring__simprules_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% cring.cring_simprules(3)
thf(fact_309_cring_Ocring__simprules_I3_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).
% cring.cring_simprules(3)
thf(fact_310_a__minus__def,axiom,
( a_minus_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,X2: a,Y2: a] : ( add_a_b @ R2 @ X2 @ ( a_inv_a_b @ R2 @ Y2 ) ) ) ) ).
% a_minus_def
thf(fact_311_a__minus__def,axiom,
( a_minu3984020753470702548t_unit
= ( ^ [R2: partia2670972154091845814t_unit,X2: list_a,Y2: list_a] : ( add_li7652885771158616974t_unit @ R2 @ X2 @ ( a_inv_8944721093294617173t_unit @ R2 @ Y2 ) ) ) ) ).
% a_minus_def
thf(fact_312_cring_Ocring__simprules_I2_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% cring.cring_simprules(2)
thf(fact_313_cring_Ocring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% cring.cring_simprules(2)
thf(fact_314_cring_Ocring__simprules_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% cring.cring_simprules(2)
thf(fact_315_cring_Ocring__simprules_I2_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( cring_5991999922451032090t_unit @ R )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% cring.cring_simprules(2)
thf(fact_316_abelian__monoid_Ozero__closed,axiom,
! [G: partia7496981018696276118t_unit] :
( ( abelia3322010900105369177t_unit @ G )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ G ) @ ( partia141011252114345353t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_317_abelian__monoid_Ozero__closed,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ G )
=> ( member_a @ ( zero_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_318_abelian__monoid_Ozero__closed,axiom,
! [G: partia2670972154091845814t_unit] :
( ( abelia226231641709521465t_unit @ G )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_319_abelian__monoid_Ozero__closed,axiom,
! [G: partia2956882679547061052t_unit] :
( ( abelia3641329199688042803t_unit @ G )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ G ) @ ( partia2464479390973590831t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_320_abelian__monoidE_I2_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( abelia3322010900105369177t_unit @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_321_abelian__monoidE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_322_abelian__monoidE_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( abelia226231641709521465t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_323_abelian__monoidE_I2_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( abelia3641329199688042803t_unit @ R )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_324_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_325_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_326_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_327_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_328_subdomain_Osubintegral,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit,H1: set_list_a,H22: set_list_a] :
( ( subdom3220114454046903646t_unit @ H @ R )
=> ( ( member_set_list_a @ H1 @ H )
=> ( ( member_set_list_a @ H22 @ H )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ H1 @ H22 )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( H1
= ( zero_s2910681146719230829t_unit @ R ) )
| ( H22
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_329_subdomain_Osubintegral,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( H1
= ( zero_l4142658623432671053t_unit @ R ) )
| ( H22
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_330_subdomain_Osubintegral,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ R @ H1 @ H22 )
= ( zero_a_b @ R ) )
=> ( ( H1
= ( zero_a_b @ R ) )
| ( H22
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_331_domain_Oconst__term__simprules__shell_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( const_term_a_b @ R @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) )
= ( a_inv_a_b @ R @ ( const_term_a_b @ R @ P ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(4)
thf(fact_332_domain_Oconst__term__simprules__shell_I4_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) )
= ( a_inv_8944721093294617173t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(4)
thf(fact_333_cring_Ocring__simprules_I20_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ).
% cring.cring_simprules(20)
thf(fact_334_cring_Ocring__simprules_I20_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(20)
thf(fact_335_cring_Ocring__simprules_I20_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( a_inv_7033018035630854991t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
= ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ ( a_inv_7033018035630854991t_unit @ R @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(20)
thf(fact_336_cring_Ocring__simprules_I19_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% cring.cring_simprules(19)
thf(fact_337_cring_Ocring__simprules_I19_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= Y ) ) ) ) ).
% cring.cring_simprules(19)
thf(fact_338_cring_Ocring__simprules_I19_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
= Y ) ) ) ) ).
% cring.cring_simprules(19)
thf(fact_339_cring_Ocring__simprules_I18_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% cring.cring_simprules(18)
thf(fact_340_cring_Ocring__simprules_I18_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ Y ) )
= Y ) ) ) ) ).
% cring.cring_simprules(18)
thf(fact_341_cring_Ocring__simprules_I18_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ Y ) )
= Y ) ) ) ) ).
% cring.cring_simprules(18)
thf(fact_342_cring_Ocring__simprules_I29_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ).
% cring.cring_simprules(29)
thf(fact_343_cring_Ocring__simprules_I29_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ Y ) )
= ( a_inv_8944721093294617173t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(29)
thf(fact_344_cring_Ocring__simprules_I29_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ X @ ( a_inv_7033018035630854991t_unit @ R @ Y ) )
= ( a_inv_7033018035630854991t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(29)
thf(fact_345_cring_Ocring__simprules_I28_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ).
% cring.cring_simprules(28)
thf(fact_346_cring_Ocring__simprules_I28_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ Y )
= ( a_inv_8944721093294617173t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(28)
thf(fact_347_cring_Ocring__simprules_I28_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ Y )
= ( a_inv_7033018035630854991t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(28)
thf(fact_348_cring_Ocring__simprules_I15_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( a_minu2241224857956505934t_unit @ R @ X @ Y )
= ( add_li174743652000525320t_unit @ R @ X @ ( a_inv_7033018035630854991t_unit @ R @ Y ) ) ) ) ).
% cring.cring_simprules(15)
thf(fact_349_cring_Ocring__simprules_I15_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( a_minus_a_b @ R @ X @ Y )
= ( add_a_b @ R @ X @ ( a_inv_a_b @ R @ Y ) ) ) ) ).
% cring.cring_simprules(15)
thf(fact_350_cring_Ocring__simprules_I15_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( a_minu3984020753470702548t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ).
% cring.cring_simprules(15)
thf(fact_351_domain_Ointegral__iff,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B )
= ( zero_s2910681146719230829t_unit @ R ) )
= ( ( A
= ( zero_s2910681146719230829t_unit @ R ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_352_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_353_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_354_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_355_domain_Om__rcancel,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a,C: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( A
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ B @ A )
= ( mult_s7802724872828879953t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_356_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_357_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_358_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_359_domain_Om__lcancel,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a,C: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( A
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B )
= ( mult_s7802724872828879953t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_360_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_361_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_362_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_363_domain_Ointegral,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( A
= ( zero_s2910681146719230829t_unit @ R ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_364_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_365_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_366_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_367_cring_Ocring__simprules_I16_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X @ ( zero_s2910681146719230829t_unit @ R ) )
= X ) ) ) ).
% cring.cring_simprules(16)
thf(fact_368_cring_Ocring__simprules_I16_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
= X ) ) ) ).
% cring.cring_simprules(16)
thf(fact_369_cring_Ocring__simprules_I16_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= X ) ) ) ).
% cring.cring_simprules(16)
thf(fact_370_cring_Ocring__simprules_I16_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X @ ( zero_l347298301471573063t_unit @ R ) )
= X ) ) ) ).
% cring.cring_simprules(16)
thf(fact_371_cring_Ocring__simprules_I8_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X )
= X ) ) ) ).
% cring.cring_simprules(8)
thf(fact_372_cring_Ocring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% cring.cring_simprules(8)
thf(fact_373_cring_Ocring__simprules_I8_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% cring.cring_simprules(8)
thf(fact_374_cring_Ocring__simprules_I8_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
= X ) ) ) ).
% cring.cring_simprules(8)
thf(fact_375_cring_Ocring__simprules_I27_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ X @ ( zero_s2910681146719230829t_unit @ R ) )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% cring.cring_simprules(27)
thf(fact_376_cring_Ocring__simprules_I27_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_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 ) ) ) ) ).
% cring.cring_simprules(27)
thf(fact_377_cring_Ocring__simprules_I27_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% cring.cring_simprules(27)
thf(fact_378_cring_Ocring__simprules_I27_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ X @ ( zero_l347298301471573063t_unit @ R ) )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% cring.cring_simprules(27)
thf(fact_379_cring_Ocring__simprules_I26_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% cring.cring_simprules(26)
thf(fact_380_cring_Ocring__simprules_I26_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_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 ) ) ) ) ).
% cring.cring_simprules(26)
thf(fact_381_cring_Ocring__simprules_I26_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% cring.cring_simprules(26)
thf(fact_382_cring_Ocring__simprules_I26_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% cring.cring_simprules(26)
thf(fact_383_Ring_Ointegral,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( A
= ( zero_s2910681146719230829t_unit @ R ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_384_Ring_Ointegral,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( field_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 ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_385_Ring_Ointegral,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( field_6388047844668329575t_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 ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_386_Ring_Ointegral,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( field_1861437471013600865t_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 ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_387_abelian__monoidI,axiom,
! [R: partia7496981018696276118t_unit] :
( ! [X3: set_list_a,Y4: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ R ) )
=> ( member_set_list_a @ ( add_se2486902527185523630t_unit @ R @ X3 @ Y4 ) @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ! [X3: set_list_a,Y4: set_list_a,Z2: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Z2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( add_se2486902527185523630t_unit @ R @ X3 @ Y4 ) @ Z2 )
= ( add_se2486902527185523630t_unit @ R @ X3 @ ( add_se2486902527185523630t_unit @ R @ Y4 @ Z2 ) ) ) ) ) )
=> ( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X3 )
= X3 ) )
=> ( ! [X3: set_list_a,Y4: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X3 @ Y4 )
= ( add_se2486902527185523630t_unit @ R @ Y4 @ X3 ) ) ) )
=> ( abelia3322010900105369177t_unit @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_388_abelian__monoidI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X3 @ Y4 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ! [X3: a,Y4: a,Z2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X3 @ Y4 ) @ Z2 )
= ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y4 @ Z2 ) ) ) ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= X3 ) )
=> ( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ Y4 )
= ( add_a_b @ R @ Y4 @ X3 ) ) ) )
=> ( abelian_monoid_a_b @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_389_abelian__monoidI,axiom,
! [R: partia2670972154091845814t_unit] :
( ! [X3: list_a,Y4: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y4 ) @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ! [X3: list_a,Y4: list_a,Z2: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y4 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ X3 @ ( add_li7652885771158616974t_unit @ R @ Y4 @ Z2 ) ) ) ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X3 )
= X3 ) )
=> ( ! [X3: list_a,Y4: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X3 @ Y4 )
= ( add_li7652885771158616974t_unit @ R @ Y4 @ X3 ) ) ) )
=> ( abelia226231641709521465t_unit @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_390_abelian__monoidI,axiom,
! [R: partia2956882679547061052t_unit] :
( ! [X3: list_list_a,Y4: list_list_a] :
( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X3 @ Y4 ) @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ! [X3: list_list_a,Y4: list_list_a,Z2: list_list_a] :
( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X3 @ Y4 ) @ Z2 )
= ( add_li174743652000525320t_unit @ R @ X3 @ ( add_li174743652000525320t_unit @ R @ Y4 @ Z2 ) ) ) ) ) )
=> ( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X3 )
= X3 ) )
=> ( ! [X3: list_list_a,Y4: list_list_a] :
( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X3 @ Y4 )
= ( add_li174743652000525320t_unit @ R @ Y4 @ X3 ) ) ) )
=> ( abelia3641329199688042803t_unit @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_391_abelian__monoid_Ominus__unique,axiom,
! [G: partia7496981018696276118t_unit,Y: set_list_a,X: set_list_a,Y3: set_list_a] :
( ( abelia3322010900105369177t_unit @ G )
=> ( ( ( add_se2486902527185523630t_unit @ G @ Y @ X )
= ( zero_s2910681146719230829t_unit @ G ) )
=> ( ( ( add_se2486902527185523630t_unit @ G @ X @ Y3 )
= ( zero_s2910681146719230829t_unit @ G ) )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ G ) )
=> ( ( member_set_list_a @ Y3 @ ( partia141011252114345353t_unit @ G ) )
=> ( Y = Y3 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_392_abelian__monoid_Ominus__unique,axiom,
! [G: partia2175431115845679010xt_a_b,Y: a,X: a,Y3: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( ( add_a_b @ G @ Y @ X )
= ( zero_a_b @ G ) )
=> ( ( ( add_a_b @ G @ X @ Y3 )
= ( zero_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( Y = Y3 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_393_abelian__monoid_Ominus__unique,axiom,
! [G: partia2670972154091845814t_unit,Y: list_a,X: list_a,Y3: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( ( add_li7652885771158616974t_unit @ G @ Y @ X )
= ( zero_l4142658623432671053t_unit @ G ) )
=> ( ( ( add_li7652885771158616974t_unit @ G @ X @ Y3 )
= ( zero_l4142658623432671053t_unit @ G ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( Y = Y3 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_394_abelian__monoid_Ominus__unique,axiom,
! [G: partia2956882679547061052t_unit,Y: list_list_a,X: list_list_a,Y3: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( ( add_li174743652000525320t_unit @ G @ Y @ X )
= ( zero_l347298301471573063t_unit @ G ) )
=> ( ( ( add_li174743652000525320t_unit @ G @ X @ Y3 )
= ( zero_l347298301471573063t_unit @ G ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( Y = Y3 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_395_abelian__monoid_Or__zero,axiom,
! [G: partia7496981018696276118t_unit,X: set_list_a] :
( ( abelia3322010900105369177t_unit @ G )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G ) )
=> ( ( add_se2486902527185523630t_unit @ G @ X @ ( zero_s2910681146719230829t_unit @ G ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_396_abelian__monoid_Or__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X @ ( zero_a_b @ G ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_397_abelian__monoid_Or__zero,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X @ ( zero_l4142658623432671053t_unit @ G ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_398_abelian__monoid_Or__zero,axiom,
! [G: partia2956882679547061052t_unit,X: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ X @ ( zero_l347298301471573063t_unit @ G ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_399_abelian__monoid_Ol__zero,axiom,
! [G: partia7496981018696276118t_unit,X: set_list_a] :
( ( abelia3322010900105369177t_unit @ G )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G ) )
=> ( ( add_se2486902527185523630t_unit @ G @ ( zero_s2910681146719230829t_unit @ G ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_400_abelian__monoid_Ol__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( zero_a_b @ G ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_401_abelian__monoid_Ol__zero,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( zero_l4142658623432671053t_unit @ G ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_402_abelian__monoid_Ol__zero,axiom,
! [G: partia2956882679547061052t_unit,X: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ ( zero_l347298301471573063t_unit @ G ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_403_abelian__monoidE_I4_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( abelia3322010900105369177t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_404_abelian__monoidE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_405_abelian__monoidE_I4_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_406_abelian__monoidE_I4_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( abelia3641329199688042803t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_407_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_408_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_409_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_410_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_411_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X @ ( zero_s2910681146719230829t_unit @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_412_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_413_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_414_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_415_semiring_Or__null,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ X @ ( zero_s2910681146719230829t_unit @ R ) )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% semiring.r_null
thf(fact_416_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_417_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_418_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_419_semiring_Ol__null,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% semiring.l_null
thf(fact_420_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_421_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_422_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_423_cring_Ocgenideal__prod,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( set_mu8047982887099575916xt_a_b @ R @ ( cgenid547466209912283029xt_a_b @ R @ A ) @ ( cgenid547466209912283029xt_a_b @ R @ B ) )
= ( cgenid547466209912283029xt_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ A @ B ) ) ) ) ) ) ).
% cring.cgenideal_prod
thf(fact_424_cring_Ocgenideal__prod,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( set_mu3586181839180059898t_unit @ R @ ( cgenid9131348535277946915t_unit @ R @ A ) @ ( cgenid9131348535277946915t_unit @ R @ B ) )
= ( cgenid9131348535277946915t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ A @ B ) ) ) ) ) ) ).
% cring.cgenideal_prod
thf(fact_425_cring_Ocgenideal__prod,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( set_mu4265036031142965114t_unit @ R @ ( cgenid24865672677839267t_unit @ R @ A ) @ ( cgenid24865672677839267t_unit @ R @ B ) )
= ( cgenid24865672677839267t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ A @ B ) ) ) ) ) ) ).
% cring.cgenideal_prod
thf(fact_426_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
=> ( ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P )
= ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_427_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_428_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_429_domain_Omap__norm__in__poly__ring__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_list_list_a @ ( map_a_list_a @ ( poly_of_const_a_b @ R ) @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ R @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ) ).
% domain.map_norm_in_poly_ring_carrier
thf(fact_430_domain_Omap__norm__in__poly__ring__carrier,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member5342144027231129785list_a @ ( map_li5729356230488778442list_a @ ( poly_o8716471131768098070t_unit @ R ) @ P ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ) ).
% domain.map_norm_in_poly_ring_carrier
thf(fact_431_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_432_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_433_ring__primeE_I1_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( P
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_434_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_435_map__in__poly__ring__carrier,axiom,
! [P: list_a,F: a > list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [A3: a] :
( ( A3
!= ( zero_a_b @ r ) )
=> ( ( F @ A3 )
!= nil_a ) )
=> ( member_list_list_a @ ( map_a_list_a @ F @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).
% map_in_poly_ring_carrier
thf(fact_436_is__root__poly__mult__imp__is__root,axiom,
! [P: list_a,Q: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_437_norm__map__in__poly__ring__carrier,axiom,
! [P: list_a,F: a > list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( map_a_list_a @ F @ P ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% norm_map_in_poly_ring_carrier
thf(fact_438_eval__rewrite,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( P
= ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ K ) @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ P ) @ ( var_a_b @ r ) ) ) ) ) ).
% eval_rewrite
thf(fact_439_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_440_subring__polynomial__pow__not__zero,axiom,
! [K: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N )
!= nil_a ) ) ) ) ).
% subring_polynomial_pow_not_zero
thf(fact_441_polynomial__pow__not__zero,axiom,
! [P: list_a,N: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N )
!= nil_a ) ) ) ).
% polynomial_pow_not_zero
thf(fact_442_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_443_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_444_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_445_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_446_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_447_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_448_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_449_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_450_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_451_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_452_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_453_add_Oint__pow__inv,axiom,
! [X: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( a_inv_a_b @ r @ X ) )
= ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% add.int_pow_inv
thf(fact_454_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_455_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_456_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_457_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_458_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_459_const__term__not__zero,axiom,
! [P: list_a] :
( ( ( const_term_a_b @ r @ P )
!= ( zero_a_b @ r ) )
=> ( P != nil_a ) ) ).
% const_term_not_zero
thf(fact_460_ring__primeE_I3_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( prime_a_ring_ext_a_b @ r @ P ) ) ) ).
% ring_primeE(3)
thf(fact_461_ring__primeI,axiom,
! [P: a] :
( ( P
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P )
=> ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% ring_primeI
thf(fact_462_const__term__simprules__shell_I4_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
= ( a_inv_a_b @ r @ ( const_term_a_b @ r @ P ) ) ) ) ) ).
% const_term_simprules_shell(4)
thf(fact_463_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_464_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_465_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_466_univ__poly__zero__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ).
% univ_poly_zero_closed
thf(fact_467_univ__poly__zero__closed,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] : ( member_list_list_a @ nil_list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ).
% univ_poly_zero_closed
thf(fact_468_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_469_univ__poly__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K ) )
= nil_a ) ).
% univ_poly_zero
thf(fact_470_univ__poly__zero,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) )
= nil_list_a ) ).
% univ_poly_zero
thf(fact_471_ring_Onormalize_Ocong,axiom,
normal637505603836502915t_unit = normal637505603836502915t_unit ).
% ring.normalize.cong
thf(fact_472_ring_Onormalize_Ocong,axiom,
normalize_a_b = normalize_a_b ).
% ring.normalize.cong
thf(fact_473_ring_Oeval_Ocong,axiom,
eval_l34571156754992824t_unit = eval_l34571156754992824t_unit ).
% ring.eval.cong
thf(fact_474_ring_Oeval_Ocong,axiom,
eval_a_b = eval_a_b ).
% ring.eval.cong
thf(fact_475_ring_Ois__root_Ocong,axiom,
polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).
% ring.is_root.cong
thf(fact_476_domain_Opolynomial__pow__not__zero,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( P != nil_list_list_a )
=> ( ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ N )
!= nil_list_list_a ) ) ) ) ).
% domain.polynomial_pow_not_zero
thf(fact_477_domain_Opolynomial__pow__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ N )
!= nil_a ) ) ) ) ).
% domain.polynomial_pow_not_zero
thf(fact_478_domain_Opolynomial__pow__not__zero,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ N )
!= nil_list_a ) ) ) ) ).
% domain.polynomial_pow_not_zero
thf(fact_479_domain_Osubring__polynomial__pow__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K ) @ P @ N )
!= nil_a ) ) ) ) ) ).
% domain.subring_polynomial_pow_not_zero
thf(fact_480_domain_Osubring__polynomial__pow__not__zero,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( P != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ N )
!= nil_list_a ) ) ) ) ) ).
% domain.subring_polynomial_pow_not_zero
thf(fact_481_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a,X: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( 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 ) ) @ P @ Q ) @ X )
=> ( ( polyno5142720416380192742t_unit @ R @ P @ X )
| ( polyno5142720416380192742t_unit @ R @ Q @ X ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_482_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( 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 ) ) @ P @ Q ) @ X )
=> ( ( polyno6951661231331188332t_unit @ R @ P @ X )
| ( polyno6951661231331188332t_unit @ R @ Q @ X ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_483_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,X: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ Q ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P @ X )
| ( polyno4133073214067823460ot_a_b @ R @ Q @ X ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_484_domain_Oeval__rewrite,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( P
= ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ R @ K ) @ ( map_a_list_a @ ( poly_of_const_a_b @ R ) @ P ) @ ( var_a_b @ R ) ) ) ) ) ) ).
% domain.eval_rewrite
thf(fact_485_domain_Oeval__rewrite,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( P
= ( eval_l1088911609197519410t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( map_li5729356230488778442list_a @ ( poly_o8716471131768098070t_unit @ R ) @ P ) @ ( var_li8453953174693405341t_unit @ R ) ) ) ) ) ) ).
% domain.eval_rewrite
thf(fact_486_domain_Onorm__map__in__poly__ring__carrier,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,F: list_list_a > list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ! [A3: list_list_a] :
( ( member_list_list_a @ A3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member5342144027231129785list_a @ ( F @ A3 ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) )
=> ( member6842060177613954879list_a @ ( normal5368706450718127095t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( map_li5227692475714150986list_a @ F @ P ) ) @ ( partia2412307164297199803t_unit @ ( univ_p3766428211910075458t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.norm_map_in_poly_ring_carrier
thf(fact_487_domain_Onorm__map__in__poly__ring__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,F: a > list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) )
=> ( member_list_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( map_a_list_a @ F @ P ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ) ) ).
% domain.norm_map_in_poly_ring_carrier
thf(fact_488_domain_Onorm__map__in__poly__ring__carrier,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,F: list_a > list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( F @ A3 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) )
=> ( member5342144027231129785list_a @ ( normal1297324897130370429t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( map_li5729356230488778442list_a @ F @ P ) ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.norm_map_in_poly_ring_carrier
thf(fact_489_domain_Omap__in__poly__ring__carrier,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,F: set_list_a > list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ! [A3: set_list_a] :
( ( member_set_list_a @ A3 @ ( partia141011252114345353t_unit @ R ) )
=> ( member5524387281408368019list_a @ ( F @ A3 ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) ) )
=> ( ! [A3: set_list_a] :
( ( A3
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( F @ A3 )
!= nil_set_list_a ) )
=> ( member352051402189872281list_a @ ( map_se1776605471917444810list_a @ F @ P ) @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ) ) ) ) ) ).
% domain.map_in_poly_ring_carrier
thf(fact_490_domain_Omap__in__poly__ring__carrier,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,F: list_list_a > list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ! [A3: list_list_a] :
( ( member_list_list_a @ A3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member5342144027231129785list_a @ ( F @ A3 ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) )
=> ( ! [A3: list_list_a] :
( ( A3
!= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( F @ A3 )
!= nil_list_list_a ) )
=> ( member6842060177613954879list_a @ ( map_li5227692475714150986list_a @ F @ P ) @ ( partia2412307164297199803t_unit @ ( univ_p3766428211910075458t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ) ) ) ) ).
% domain.map_in_poly_ring_carrier
thf(fact_491_domain_Omap__in__poly__ring__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,F: a > list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) )
=> ( ! [A3: a] :
( ( A3
!= ( zero_a_b @ R ) )
=> ( ( F @ A3 )
!= nil_a ) )
=> ( member_list_list_a @ ( map_a_list_a @ F @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ) ) ) ).
% domain.map_in_poly_ring_carrier
thf(fact_492_domain_Omap__in__poly__ring__carrier,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,F: list_a > list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( F @ A3 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) )
=> ( ! [A3: list_a] :
( ( A3
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( F @ A3 )
!= nil_list_a ) )
=> ( member5342144027231129785list_a @ ( map_li5729356230488778442list_a @ F @ P ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ) ) ) ).
% domain.map_in_poly_ring_carrier
thf(fact_493_pirreducibleE_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( P != nil_a ) ) ) ) ).
% pirreducibleE(1)
thf(fact_494_list_Omap__disc__iff,axiom,
! [F: a > list_a,A: list_a] :
( ( ( map_a_list_a @ F @ A )
= nil_list_a )
= ( A = nil_a ) ) ).
% list.map_disc_iff
thf(fact_495_list_Omap__disc__iff,axiom,
! [F: a > a,A: list_a] :
( ( ( map_a_a @ F @ A )
= nil_a )
= ( A = nil_a ) ) ).
% list.map_disc_iff
thf(fact_496_Nil__is__map__conv,axiom,
! [F: a > list_a,Xs: list_a] :
( ( nil_list_a
= ( map_a_list_a @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_497_Nil__is__map__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( nil_a
= ( map_a_a @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_498_map__is__Nil__conv,axiom,
! [F: a > list_a,Xs: list_a] :
( ( ( map_a_list_a @ F @ Xs )
= nil_list_a )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_499_map__is__Nil__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( ( map_a_a @ F @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_500_subdomainI,axiom,
! [H: set_a] :
( ( subcring_a_b @ H @ r )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ r @ H12 @ H23 )
= ( zero_a_b @ r ) )
=> ( ( H12
= ( zero_a_b @ r ) )
| ( H23
= ( zero_a_b @ r ) ) ) ) ) )
=> ( subdomain_a_b @ H @ r ) ) ) ) ).
% subdomainI
thf(fact_501_add_Oone__in__subset,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H != bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ H )
=> ( member_a @ ( a_inv_a_b @ r @ X3 ) @ H ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ H )
=> ! [Xa: a] :
( ( member_a @ Xa @ H )
=> ( member_a @ ( add_a_b @ r @ X3 @ Xa ) @ H ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H ) ) ) ) ) ).
% add.one_in_subset
thf(fact_502_domain_Oring__primeE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P )
=> ( prime_a_ring_ext_a_b @ R @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_503_domain_Oring__primeE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r6430282645014804837t_unit @ R @ P )
=> ( prime_2011924034616061926t_unit @ R @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_504_domain_Oring__primeE_I3_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r5437400583859147359t_unit @ R @ P )
=> ( prime_1232919612140715622t_unit @ R @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_505_univ__poly__a__inv__length,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
= ( size_size_list_a @ P ) ) ) ) ).
% univ_poly_a_inv_length
thf(fact_506_normalize_Osimps_I1_J,axiom,
( ( normalize_a_b @ r @ nil_a )
= nil_a ) ).
% normalize.simps(1)
thf(fact_507_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_508_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_509_const__term__def,axiom,
! [P: list_a] :
( ( const_term_a_b @ r @ P )
= ( eval_a_b @ r @ P @ ( zero_a_b @ r ) ) ) ).
% const_term_def
thf(fact_510_eval__var,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X )
= X ) ) ).
% eval_var
thf(fact_511_eval__poly__of__const,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_of_const_a_b @ r @ X ) @ Y )
= X ) ) ).
% eval_poly_of_const
thf(fact_512_eval__in__carrier__2,axiom,
! [X: list_a,Y: a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier_2
thf(fact_513_inv__unique,axiom,
! [Y: a,X: a,Y3: 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 @ Y3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y3 ) ) ) ) ) ) ).
% inv_unique
thf(fact_514_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X3 )
= X3 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_515_line__extension__in__carrier,axiom,
! [K: set_a,A: a,E: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_516_set__mult__closed,axiom,
! [H: set_a,K: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_mu8047982887099575916xt_a_b @ r @ H @ K ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_mult_closed
thf(fact_517_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 ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A3 @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_518_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_519_is__root__def,axiom,
! [P: list_a,X: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
= ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P @ X )
= ( zero_a_b @ r ) )
& ( P != nil_a ) ) ) ).
% is_root_def
thf(fact_520_length__map,axiom,
! [F: a > list_a,Xs: list_a] :
( ( size_s349497388124573686list_a @ ( map_a_list_a @ F @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_map
thf(fact_521_length__map,axiom,
! [F: a > a,Xs: list_a] :
( ( size_size_list_a @ ( map_a_a @ F @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_map
thf(fact_522_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 )
=> ( ! [H3: a] :
( ( member_a @ H3 @ H )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ H ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H23 ) @ H ) ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( add_a_b @ r @ H12 @ H23 ) @ H ) ) )
=> ( subring_a_b @ H @ r ) ) ) ) ) ) ).
% subringI
thf(fact_523_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_524_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_525_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_526_map__eq__imp__length__eq,axiom,
! [F: a > list_a,Xs: list_a,G2: a > list_a,Ys: list_a] :
( ( ( map_a_list_a @ F @ Xs )
= ( map_a_list_a @ G2 @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_527_map__eq__imp__length__eq,axiom,
! [F: a > a,Xs: list_a,G2: a > a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( map_a_a @ G2 @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_528_subringE_I3_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subrin5643252653130547402t_unit @ H @ R )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ H ) ) ).
% subringE(3)
thf(fact_529_subringE_I3_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H ) ) ).
% subringE(3)
thf(fact_530_subringE_I3_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ H ) ) ).
% subringE(3)
thf(fact_531_ring__hom__one,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( H2 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_hom_one
thf(fact_532_ring__hom__one,axiom,
! [H2: a > set_list_a,R: partia2175431115845679010xt_a_b,S: partia7496981018696276118t_unit] :
( ( member_a_set_list_a @ H2 @ ( ring_h6109298854714515236t_unit @ R @ S ) )
=> ( ( H2 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_se1127990129394575805t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_533_ring__hom__one,axiom,
! [H2: set_list_a > a,R: partia7496981018696276118t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_set_list_a_a @ H2 @ ( ring_h8906680420194085028it_a_b @ R @ S ) )
=> ( ( H2 @ ( one_se1127990129394575805t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_hom_one
thf(fact_534_ring__hom__one,axiom,
! [H2: set_list_a > set_list_a,R: partia7496981018696276118t_unit,S: partia7496981018696276118t_unit] :
( ( member5068272912271824380list_a @ H2 @ ( ring_h6076331213207892940t_unit @ R @ S ) )
=> ( ( H2 @ ( one_se1127990129394575805t_unit @ R ) )
= ( one_se1127990129394575805t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_535_subcringE_I3_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subcri7783154434480317835t_unit @ H @ R )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ H ) ) ).
% subcringE(3)
thf(fact_536_subcringE_I3_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H ) ) ).
% subcringE(3)
thf(fact_537_subdomainE_I3_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subdom3220114454046903646t_unit @ H @ R )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ H ) ) ).
% subdomainE(3)
thf(fact_538_subdomainE_I3_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H ) ) ).
% subdomainE(3)
thf(fact_539_ring__iso__memE_I4_J,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( H2 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_540_ring__iso__memE_I4_J,axiom,
! [H2: a > set_list_a,R: partia2175431115845679010xt_a_b,S: partia7496981018696276118t_unit] :
( ( member_a_set_list_a @ H2 @ ( ring_i5325512697602418746t_unit @ R @ S ) )
=> ( ( H2 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_se1127990129394575805t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_541_ring__iso__memE_I4_J,axiom,
! [H2: set_list_a > a,R: partia7496981018696276118t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_set_list_a_a @ H2 @ ( ring_i8122894263081988538it_a_b @ R @ S ) )
=> ( ( H2 @ ( one_se1127990129394575805t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_542_ring__iso__memE_I4_J,axiom,
! [H2: set_list_a > set_list_a,R: partia7496981018696276118t_unit,S: partia7496981018696276118t_unit] :
( ( member5068272912271824380list_a @ H2 @ ( ring_i6425008796027319266t_unit @ R @ S ) )
=> ( ( H2 @ ( one_se1127990129394575805t_unit @ R ) )
= ( one_se1127990129394575805t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_543_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_544_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_545_domain_Ozero__not__one,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( zero_s2910681146719230829t_unit @ R )
!= ( one_se1127990129394575805t_unit @ R ) ) ) ).
% domain.zero_not_one
thf(fact_546_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_547_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_548_domain_Oone__not__zero,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% domain.one_not_zero
thf(fact_549_cring_Ocring__simprules_I6_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% cring.cring_simprules(6)
thf(fact_550_cring_Ocring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% cring.cring_simprules(6)
thf(fact_551_cring_Ocring__simprules_I6_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% cring.cring_simprules(6)
thf(fact_552_cring_Ocring__simprules_I6_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( cring_5991999922451032090t_unit @ R )
=> ( member_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% cring.cring_simprules(6)
thf(fact_553_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_554_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_555_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_556_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_557_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_558_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_559_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_560_subringE_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subringE(1)
thf(fact_561_subringE_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subringE(1)
thf(fact_562_subringE_I1_J,axiom,
! [H: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subrin3541368690557094692t_unit @ H @ R )
=> ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% subringE(1)
thf(fact_563_subdomain_Osub__one__not__zero,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_564_subdomain_Osub__one__not__zero,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subdom3220114454046903646t_unit @ H @ R )
=> ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_565_subdomain_Osub__one__not__zero,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_566_subcringE_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subcringE(1)
thf(fact_567_subcringE_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subcringE(1)
thf(fact_568_subcringE_I1_J,axiom,
! [H: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subcri8676831449680469861t_unit @ H @ R )
=> ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% subcringE(1)
thf(fact_569_subdomainE_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subdomainE(1)
thf(fact_570_subdomainE_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subdomainE(1)
thf(fact_571_subdomainE_I1_J,axiom,
! [H: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subdom561091866123308472t_unit @ H @ R )
=> ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% subdomainE(1)
thf(fact_572_cring_Ocring__simprules_I12_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ ( one_se1127990129394575805t_unit @ R ) @ X )
= X ) ) ) ).
% cring.cring_simprules(12)
thf(fact_573_cring_Ocring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_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 ) ) ) ).
% cring.cring_simprules(12)
thf(fact_574_cring_Ocring__simprules_I12_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ X )
= X ) ) ) ).
% cring.cring_simprules(12)
thf(fact_575_cring_Ocring__simprules_I12_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( one_li8234411390022467901t_unit @ R ) @ X )
= X ) ) ) ).
% cring.cring_simprules(12)
thf(fact_576_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ ( one_se1127990129394575805t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_577_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_578_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_579_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_580_domain_Osquare__eq__one,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ X @ X )
= ( one_se1127990129394575805t_unit @ R ) )
=> ( ( X
= ( one_se1127990129394575805t_unit @ R ) )
| ( X
= ( a_inv_5715216516650856053t_unit @ R @ ( one_se1127990129394575805t_unit @ R ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_581_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_582_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_583_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_584_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > a,S: partia2175431115845679010xt_a_b] :
( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H2 @ X3 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y4 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X3 @ Y4 ) )
= ( add_a_b @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ( ( H2 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_585_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > list_a,S: partia2670972154091845814t_unit] :
( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H2 @ X3 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y4 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X3 @ Y4 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ( ( H2 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_586_ring__hom__memI,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > a,S: partia2175431115845679010xt_a_b] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H2 @ X3 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X3: list_a,Y4: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R @ X3 @ Y4 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ! [X3: list_a,Y4: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y4 ) )
= ( add_a_b @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ( ( H2 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_587_ring__hom__memI,axiom,
! [R: partia7496981018696276118t_unit,H2: set_list_a > a,S: partia2175431115845679010xt_a_b] :
( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( member_a @ ( H2 @ X3 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X3: set_list_a,Y4: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( H2 @ ( mult_s7802724872828879953t_unit @ R @ X3 @ Y4 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ! [X3: set_list_a,Y4: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( H2 @ ( add_se2486902527185523630t_unit @ R @ X3 @ Y4 ) )
= ( add_a_b @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ( ( H2 @ ( one_se1127990129394575805t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_set_list_a_a @ H2 @ ( ring_h8906680420194085028it_a_b @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_588_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > set_list_a,S: partia7496981018696276118t_unit] :
( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_set_list_a @ ( H2 @ X3 ) @ ( partia141011252114345353t_unit @ S ) ) )
=> ( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y4 ) )
= ( mult_s7802724872828879953t_unit @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X3 @ Y4 ) )
= ( add_se2486902527185523630t_unit @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ( ( H2 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_se1127990129394575805t_unit @ S ) )
=> ( member_a_set_list_a @ H2 @ ( ring_h6109298854714515236t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_589_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > list_list_a,S: partia2956882679547061052t_unit] :
( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_list_a @ ( H2 @ X3 ) @ ( partia2464479390973590831t_unit @ S ) ) )
=> ( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y4 ) )
= ( mult_l4853965630390486993t_unit @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X3 @ Y4 ) )
= ( add_li174743652000525320t_unit @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ( ( H2 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_li8234411390022467901t_unit @ S ) )
=> ( member_a_list_list_a @ H2 @ ( ring_h6858658657455840382t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_590_ring__hom__memI,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > list_a,S: partia2670972154091845814t_unit] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H2 @ X3 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X3: list_a,Y4: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R @ X3 @ Y4 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ! [X3: list_a,Y4: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y4 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ( ( H2 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_591_ring__hom__memI,axiom,
! [R: partia2956882679547061052t_unit,H2: list_list_a > a,S: partia2175431115845679010xt_a_b] :
( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_a @ ( H2 @ X3 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X3: list_list_a,Y4: list_list_a] :
( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( mult_l4853965630390486993t_unit @ R @ X3 @ Y4 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ! [X3: list_list_a,Y4: list_list_a] :
( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( add_li174743652000525320t_unit @ R @ X3 @ Y4 ) )
= ( add_a_b @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ( ( H2 @ ( one_li8234411390022467901t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_list_list_a_a @ H2 @ ( ring_h8078271382950527358it_a_b @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_592_ring__hom__memI,axiom,
! [R: partia7496981018696276118t_unit,H2: set_list_a > list_a,S: partia2670972154091845814t_unit] :
( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( member_list_a @ ( H2 @ X3 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X3: set_list_a,Y4: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( H2 @ ( mult_s7802724872828879953t_unit @ R @ X3 @ Y4 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ! [X3: set_list_a,Y4: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( H2 @ ( add_se2486902527185523630t_unit @ R @ X3 @ Y4 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ( ( H2 @ ( one_se1127990129394575805t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( member5910328476188217884list_a @ H2 @ ( ring_h8038483918290310060t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_593_ring__hom__memI,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > set_list_a,S: partia7496981018696276118t_unit] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_set_list_a @ ( H2 @ X3 ) @ ( partia141011252114345353t_unit @ S ) ) )
=> ( ! [X3: list_a,Y4: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R @ X3 @ Y4 ) )
= ( mult_s7802724872828879953t_unit @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ! [X3: list_a,Y4: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y4 ) )
= ( add_se2486902527185523630t_unit @ S @ ( H2 @ X3 ) @ ( H2 @ Y4 ) ) ) ) )
=> ( ( ( H2 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_se1127990129394575805t_unit @ S ) )
=> ( member4263473470251683292list_a @ H2 @ ( ring_h6188449271506562988t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_594_cring_Ocring__fieldI2,axiom,
! [R: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( ( zero_s2910681146719230829t_unit @ R )
!= ( one_se1127990129394575805t_unit @ R ) )
=> ( ! [A3: set_list_a] :
( ( member_set_list_a @ A3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( A3
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ? [X4: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
& ( ( mult_s7802724872828879953t_unit @ R @ A3 @ X4 )
= ( one_se1127990129394575805t_unit @ R ) ) ) ) )
=> ( field_26233345952514695t_unit @ R ) ) ) ) ).
% cring.cring_fieldI2
thf(fact_595_cring_Ocring__fieldI2,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( ( 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 ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( mult_a_ring_ext_a_b @ R @ A3 @ X4 )
= ( one_a_ring_ext_a_b @ R ) ) ) ) )
=> ( field_a_b @ R ) ) ) ) ).
% cring.cring_fieldI2
thf(fact_596_cring_Ocring__fieldI2,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( ( zero_l4142658623432671053t_unit @ R )
!= ( one_li8328186300101108157t_unit @ R ) )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( A3
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
& ( ( mult_l7073676228092353617t_unit @ R @ A3 @ X4 )
= ( one_li8328186300101108157t_unit @ R ) ) ) ) )
=> ( field_6388047844668329575t_unit @ R ) ) ) ) ).
% cring.cring_fieldI2
thf(fact_597_cring_Ocring__fieldI2,axiom,
! [R: partia2956882679547061052t_unit] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( ( zero_l347298301471573063t_unit @ R )
!= ( one_li8234411390022467901t_unit @ R ) )
=> ( ! [A3: list_list_a] :
( ( member_list_list_a @ A3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( A3
!= ( zero_l347298301471573063t_unit @ R ) )
=> ? [X4: list_list_a] :
( ( member_list_list_a @ X4 @ ( partia2464479390973590831t_unit @ R ) )
& ( ( mult_l4853965630390486993t_unit @ R @ A3 @ X4 )
= ( one_li8234411390022467901t_unit @ R ) ) ) ) )
=> ( field_1861437471013600865t_unit @ R ) ) ) ) ).
% cring.cring_fieldI2
thf(fact_598_domain_Ouniv__poly__a__inv__length,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) )
= ( size_size_list_a @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_length
thf(fact_599_domain_Ouniv__poly__a__inv__length,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( size_s349497388124573686list_a @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) )
= ( size_s349497388124573686list_a @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_length
thf(fact_600_domain_OpirreducibleE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
=> ( P != nil_list_a ) ) ) ) ) ).
% domain.pirreducibleE(1)
thf(fact_601_domain_OpirreducibleE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
=> ( P != nil_a ) ) ) ) ) ).
% domain.pirreducibleE(1)
thf(fact_602_list_Osimps_I8_J,axiom,
! [F: a > list_a] :
( ( map_a_list_a @ F @ nil_a )
= nil_list_a ) ).
% list.simps(8)
thf(fact_603_list_Osimps_I8_J,axiom,
! [F: a > a] :
( ( map_a_a @ F @ nil_a )
= nil_a ) ).
% list.simps(8)
thf(fact_604_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( prime_5738381090551951334t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_605_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_606_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_607_ring__prime__def,axiom,
( ring_r1091214237498979717t_unit
= ( ^ [R2: partia7496981018696276118t_unit,A4: set_list_a] :
( ( A4
!= ( zero_s2910681146719230829t_unit @ R2 ) )
& ( prime_5738381090551951334t_unit @ R2 @ A4 ) ) ) ) ).
% ring_prime_def
thf(fact_608_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_609_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_610_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia7496981018696276118t_unit,R3: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ R3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r5115406448772830318t_unit @ R @ R3 )
=> ( R3
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_611_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_612_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_613_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_614_domain_Oring__primeE_I1_J,axiom,
! [R: partia7496981018696276118t_unit,P: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ P @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r1091214237498979717t_unit @ R @ P )
=> ( P
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_615_domain_Oring__primeE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P )
=> ( P
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_616_domain_Oring__primeE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r6430282645014804837t_unit @ R @ P )
=> ( P
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_617_domain_Oring__primeE_I1_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r5437400583859147359t_unit @ R @ P )
=> ( P
!= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_618_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_619_domainI,axiom,
! [R: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ! [A3: set_list_a,B2: set_list_a] :
( ( ( mult_s7802724872828879953t_unit @ R @ A3 @ B2 )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( A3
= ( zero_s2910681146719230829t_unit @ R ) )
| ( B2
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) )
=> ( domain1617769409708967785t_unit @ R ) ) ) ) ).
% domainI
thf(fact_620_domainI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) )
=> ( ! [A3: a,B2: a] :
( ( ( mult_a_ring_ext_a_b @ R @ A3 @ B2 )
= ( zero_a_b @ R ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A3
= ( zero_a_b @ R ) )
| ( B2
= ( zero_a_b @ R ) ) ) ) ) )
=> ( domain_a_b @ R ) ) ) ) ).
% domainI
thf(fact_621_domainI,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ! [A3: list_a,B2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ R @ A3 @ B2 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( A3
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B2
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) )
=> ( domain6553523120543210313t_unit @ R ) ) ) ) ).
% domainI
thf(fact_622_domainI,axiom,
! [R: partia2956882679547061052t_unit] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( ( one_li8234411390022467901t_unit @ R )
!= ( zero_l347298301471573063t_unit @ R ) )
=> ( ! [A3: list_list_a,B2: list_list_a] :
( ( ( mult_l4853965630390486993t_unit @ R @ A3 @ B2 )
= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( member_list_list_a @ A3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( A3
= ( zero_l347298301471573063t_unit @ R ) )
| ( B2
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) )
=> ( domain7810152921033798211t_unit @ R ) ) ) ) ).
% domainI
thf(fact_623_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_624_a__lcos__m__assoc,axiom,
! [M2: set_a,G2: a,H2: a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G2 @ ( a_l_coset_a_b @ r @ H2 @ M2 ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G2 @ H2 ) @ M2 ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_625_a__lcos__mult__one,axiom,
! [M2: set_a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M2 )
= M2 ) ) ).
% a_lcos_mult_one
thf(fact_626_pirreducibleI,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ R4 ) )
=> ( ( member_list_a @ Q2 @ ( 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 ) @ P ) ) ) ) ) ) ).
% pirreducibleI
thf(fact_627_subset__empty,axiom,
! [A2: set_c] :
( ( ord_less_eq_set_c @ A2 @ bot_bot_set_c )
= ( A2 = bot_bot_set_c ) ) ).
% subset_empty
thf(fact_628_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_629_normalize_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [V: a,Va: list_a] :
( X
!= ( cons_a @ V @ Va ) ) ) ).
% normalize.cases
thf(fact_630_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_631_empty__iff,axiom,
! [C: list_list_a] :
~ ( member_list_list_a @ C @ bot_bo1875519244922727510list_a ) ).
% empty_iff
thf(fact_632_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_633_empty__iff,axiom,
! [C: c] :
~ ( member_c @ C @ bot_bot_set_c ) ).
% empty_iff
thf(fact_634_all__not__in__conv,axiom,
! [A2: set_list_a] :
( ( ! [X2: list_a] :
~ ( member_list_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_635_all__not__in__conv,axiom,
! [A2: set_list_list_a] :
( ( ! [X2: list_list_a] :
~ ( member_list_list_a @ X2 @ A2 ) )
= ( A2 = bot_bo1875519244922727510list_a ) ) ).
% all_not_in_conv
thf(fact_636_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_637_all__not__in__conv,axiom,
! [A2: set_c] :
( ( ! [X2: c] :
~ ( member_c @ X2 @ A2 ) )
= ( A2 = bot_bot_set_c ) ) ).
% all_not_in_conv
thf(fact_638_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_639_Collect__empty__eq,axiom,
! [P2: c > $o] :
( ( ( collect_c @ P2 )
= bot_bot_set_c )
= ( ! [X2: c] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_640_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X2: a] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_641_empty__Collect__eq,axiom,
! [P2: c > $o] :
( ( bot_bot_set_c
= ( collect_c @ P2 ) )
= ( ! [X2: c] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_642_a__l__coset__subset__G,axiom,
! [H: set_a,X: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_643_poly__of__const__def,axiom,
( ( poly_of_const_a_b @ r )
= ( ^ [K3: a] : ( normalize_a_b @ r @ ( cons_a @ K3 @ nil_a ) ) ) ) ).
% poly_of_const_def
thf(fact_644_pirreducibleE_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pirreducibleE(2)
thf(fact_645_empty__subsetI,axiom,
! [A2: set_c] : ( ord_less_eq_set_c @ bot_bot_set_c @ A2 ) ).
% empty_subsetI
thf(fact_646_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_647_pirreducibleE_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a,R3: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P
= ( 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_648_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_649_ring_Omonom_Ocong,axiom,
monom_a_b = monom_a_b ).
% ring.monom.cong
thf(fact_650_list__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P2 @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_651_list__induct2_H,axiom,
! [P2: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P2 @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a] : ( P2 @ ( cons_a @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y4: a,Ys2: list_a] : ( P2 @ nil_a @ ( cons_a @ Y4 @ Ys2 ) )
=> ( ! [X3: a,Xs2: list_a,Y4: a,Ys2: list_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_652_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y2: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_653_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X3: a] :
( X
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y4: a,Xs2: list_a] :
( X
!= ( cons_a @ X3 @ ( cons_a @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_654_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X21: a,X22: list_a] :
( Y
!= ( cons_a @ X21 @ X22 ) ) ) ).
% list.exhaust
thf(fact_655_list_OdiscI,axiom,
! [List: list_a,X212: a,X222: list_a] :
( ( List
= ( cons_a @ X212 @ X222 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_656_list_Odistinct_I1_J,axiom,
! [X212: a,X222: list_a] :
( nil_a
!= ( cons_a @ X212 @ X222 ) ) ).
% list.distinct(1)
thf(fact_657_map__eq__Cons__conv,axiom,
! [F: a > list_a,Xs: list_a,Y: list_a,Ys: list_list_a] :
( ( ( map_a_list_a @ F @ Xs )
= ( cons_list_a @ Y @ Ys ) )
= ( ? [Z3: a,Zs: list_a] :
( ( Xs
= ( cons_a @ Z3 @ Zs ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_a_list_a @ F @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_658_map__eq__Cons__conv,axiom,
! [F: a > a,Xs: list_a,Y: a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( ? [Z3: a,Zs: list_a] :
( ( Xs
= ( cons_a @ Z3 @ Zs ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_a_a @ F @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_659_Cons__eq__map__conv,axiom,
! [X: list_a,Xs: list_list_a,F: a > list_a,Ys: list_a] :
( ( ( cons_list_a @ X @ Xs )
= ( map_a_list_a @ F @ Ys ) )
= ( ? [Z3: a,Zs: list_a] :
( ( Ys
= ( cons_a @ Z3 @ Zs ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_a_list_a @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_660_Cons__eq__map__conv,axiom,
! [X: a,Xs: list_a,F: a > a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= ( map_a_a @ F @ Ys ) )
= ( ? [Z3: a,Zs: list_a] :
( ( Ys
= ( cons_a @ Z3 @ Zs ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_a_a @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_661_map__eq__Cons__D,axiom,
! [F: a > list_a,Xs: list_a,Y: list_a,Ys: list_list_a] :
( ( ( map_a_list_a @ F @ Xs )
= ( cons_list_a @ Y @ Ys ) )
=> ? [Z2: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_a_list_a @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_662_map__eq__Cons__D,axiom,
! [F: a > a,Xs: list_a,Y: a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ? [Z2: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_a_a @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_663_Cons__eq__map__D,axiom,
! [X: list_a,Xs: list_list_a,F: a > list_a,Ys: list_a] :
( ( ( cons_list_a @ X @ Xs )
= ( map_a_list_a @ F @ Ys ) )
=> ? [Z2: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z2 @ Zs2 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_a_list_a @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_664_Cons__eq__map__D,axiom,
! [X: a,Xs: list_a,F: a > a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= ( map_a_a @ F @ Ys ) )
=> ? [Z2: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z2 @ Zs2 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_a_a @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_665_list_Osimps_I9_J,axiom,
! [F: a > list_a,X212: a,X222: list_a] :
( ( map_a_list_a @ F @ ( cons_a @ X212 @ X222 ) )
= ( cons_list_a @ ( F @ X212 ) @ ( map_a_list_a @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_666_list_Osimps_I9_J,axiom,
! [F: a > a,X212: a,X222: list_a] :
( ( map_a_a @ F @ ( cons_a @ X212 @ X222 ) )
= ( cons_a @ ( F @ X212 ) @ ( map_a_a @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_667_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs3: list_a,Ws: list_a,P2: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y4: a,Ys2: list_a,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_668_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs3: list_a,P2: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y4: a,Ys2: list_a,Z2: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_669_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P2: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y4: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_670_cring_Oideal__eq__carrier__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: a] :
( ( cring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% cring.ideal_eq_carrier_iff
thf(fact_671_cring_Oideal__eq__carrier__iff,axiom,
! [R: partia2670972154091845814t_unit,A: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( partia5361259788508890537t_unit @ R )
= ( cgenid9131348535277946915t_unit @ R @ A ) )
= ( member_list_a @ A @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) ).
% cring.ideal_eq_carrier_iff
thf(fact_672_cring_Oideal__eq__carrier__iff,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( partia2464479390973590831t_unit @ R )
= ( cgenid24865672677839267t_unit @ R @ A ) )
= ( member_list_list_a @ A @ ( units_4903515905731149798t_unit @ R ) ) ) ) ) ).
% cring.ideal_eq_carrier_iff
thf(fact_673_domain_Oring__irreducibleE_I4_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 )
=> ~ ( member_list_list_a @ R3 @ ( units_4903515905731149798t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_674_domain_Oring__irreducibleE_I4_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 )
=> ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_675_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia2670972154091845814t_unit,R3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R3 )
=> ~ ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_676_univ__poly__one,axiom,
! [R: partia7496981018696276118t_unit,K: set_set_list_a] :
( ( one_li1622763072977731901t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) )
= ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ nil_set_list_a ) ) ).
% univ_poly_one
thf(fact_677_univ__poly__one,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ K ) )
= ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) ) ).
% univ_poly_one
thf(fact_678_univ__poly__one,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) )
= ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ nil_list_a ) ) ).
% univ_poly_one
thf(fact_679_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_680_emptyE,axiom,
! [A: list_list_a] :
~ ( member_list_list_a @ A @ bot_bo1875519244922727510list_a ) ).
% emptyE
thf(fact_681_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_682_emptyE,axiom,
! [A: c] :
~ ( member_c @ A @ bot_bot_set_c ) ).
% emptyE
thf(fact_683_equals0D,axiom,
! [A2: set_list_a,A: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_684_equals0D,axiom,
! [A2: set_list_list_a,A: list_list_a] :
( ( A2 = bot_bo1875519244922727510list_a )
=> ~ ( member_list_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_685_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_686_equals0D,axiom,
! [A2: set_c,A: c] :
( ( A2 = bot_bot_set_c )
=> ~ ( member_c @ A @ A2 ) ) ).
% equals0D
thf(fact_687_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y4: list_a] :
~ ( member_list_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_688_equals0I,axiom,
! [A2: set_list_list_a] :
( ! [Y4: list_list_a] :
~ ( member_list_list_a @ Y4 @ A2 )
=> ( A2 = bot_bo1875519244922727510list_a ) ) ).
% equals0I
thf(fact_689_equals0I,axiom,
! [A2: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_690_equals0I,axiom,
! [A2: set_c] :
( ! [Y4: c] :
~ ( member_c @ Y4 @ A2 )
=> ( A2 = bot_bot_set_c ) ) ).
% equals0I
thf(fact_691_ex__in__conv,axiom,
! [A2: set_list_a] :
( ( ? [X2: list_a] : ( member_list_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_692_ex__in__conv,axiom,
! [A2: set_list_list_a] :
( ( ? [X2: list_list_a] : ( member_list_list_a @ X2 @ A2 ) )
= ( A2 != bot_bo1875519244922727510list_a ) ) ).
% ex_in_conv
thf(fact_693_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_694_ex__in__conv,axiom,
! [A2: set_c] :
( ( ? [X2: c] : ( member_c @ X2 @ A2 ) )
= ( A2 != bot_bot_set_c ) ) ).
% ex_in_conv
thf(fact_695_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2956882679547061052t_unit,R3: list_list_a,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ R3 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( R3
= ( mult_l4853965630390486993t_unit @ R @ A @ B ) )
=> ( ( member_list_list_a @ A @ ( units_4903515905731149798t_unit @ R ) )
| ( member_list_list_a @ B @ ( units_4903515905731149798t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_696_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,R3: a,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( 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 ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_697_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2670972154091845814t_unit,R3: list_a,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( R3
= ( mult_l7073676228092353617t_unit @ R @ A @ B ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ R ) )
| ( member_list_a @ B @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_698_var__def,axiom,
( var_li8453953174693405341t_unit
= ( ^ [R2: partia2670972154091845814t_unit] : ( cons_list_a @ ( one_li8328186300101108157t_unit @ R2 ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ nil_list_a ) ) ) ) ).
% var_def
thf(fact_699_var__def,axiom,
( var_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b] : ( cons_a @ ( one_a_ring_ext_a_b @ R2 ) @ ( cons_a @ ( zero_a_b @ R2 ) @ nil_a ) ) ) ) ).
% var_def
thf(fact_700_var__def,axiom,
( var_se6008125447796440765t_unit
= ( ^ [R2: partia7496981018696276118t_unit] : ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R2 ) @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R2 ) @ nil_set_list_a ) ) ) ) ).
% var_def
thf(fact_701_domain_OpirreducibleE_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
=> ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).
% domain.pirreducibleE(2)
thf(fact_702_domain_OpirreducibleE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).
% domain.pirreducibleE(2)
thf(fact_703_domain_OpirreducibleE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a,R3: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( P
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q @ R3 ) )
=> ( ( member_list_list_a @ Q @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
| ( member_list_list_a @ R3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ) ) ) ) ).
% domain.pirreducibleE(3)
thf(fact_704_domain_OpirreducibleE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a,R3: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( P
= ( 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 ) ) ) ) ) ) ) ) ) ) ) ).
% domain.pirreducibleE(3)
thf(fact_705_domain_OpirreducibleI,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( P != nil_list_a )
=> ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ! [Q2: list_list_a,R4: list_list_a] :
( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( P
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q2 @ R4 ) )
=> ( ( member_list_list_a @ Q2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
| ( member_list_list_a @ R4 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) ) ) ) ) ) ) ).
% domain.pirreducibleI
thf(fact_706_domain_OpirreducibleI,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ! [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q2 @ R4 ) )
=> ( ( member_list_a @ Q2 @ ( 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 ) @ P ) ) ) ) ) ) ) ).
% domain.pirreducibleI
thf(fact_707_domain_Ounitary__monom__eq__var__pow,axiom,
! [R: partia7496981018696276118t_unit,K: set_set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K @ R )
=> ( ( monom_317758005976320064t_unit @ R @ ( one_se1127990129394575805t_unit @ R ) @ N )
= ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ K ) @ ( var_se6008125447796440765t_unit @ R ) @ N ) ) ) ) ).
% domain.unitary_monom_eq_var_pow
thf(fact_708_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_709_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_710_is__root__imp__pdivides,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
=> ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P ) ) ) ).
% is_root_imp_pdivides
thf(fact_711_poly__mult__var,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( P = nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ ( var_a_b @ r ) )
= nil_a ) )
& ( ( P != nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ ( var_a_b @ r ) )
= ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ).
% poly_mult_var
thf(fact_712_pdivides__imp__is__root,axiom,
! [P: list_a,X: a] :
( ( P != 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 ) ) @ P )
=> ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ) ).
% pdivides_imp_is_root
thf(fact_713_eval__as__unique__hom,axiom,
! [K: set_a,X: a,H2: list_a > a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ r @ H2 )
=> ( ! [K4: a] :
( ( member_a @ K4 @ K )
=> ( ( H2 @ ( cons_a @ K4 @ nil_a ) )
= K4 ) )
=> ( ( ( H2 @ ( var_a_b @ r ) )
= X )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( H2 @ P )
= ( eval_a_b @ r @ P @ X ) ) ) ) ) ) ) ) ).
% eval_as_unique_hom
thf(fact_714_pdivides__imp__root__sharing,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( eval_a_b @ r @ P @ A )
= ( zero_a_b @ r ) )
=> ( ( eval_a_b @ r @ Q @ A )
= ( zero_a_b @ r ) ) ) ) ) ) ).
% pdivides_imp_root_sharing
thf(fact_715_determination__of__hom,axiom,
! [K: set_a,A2: partia2175431115845679010xt_a_b,H2: list_a > a,G2: list_a > a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ A2 @ H2 )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ A2 @ G2 )
=> ( ! [K4: a] :
( ( member_a @ K4 @ K )
=> ( ( H2 @ ( cons_a @ K4 @ nil_a ) )
= ( G2 @ ( cons_a @ K4 @ nil_a ) ) ) )
=> ( ( ( H2 @ ( var_a_b @ r ) )
= ( G2 @ ( var_a_b @ r ) ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( H2 @ P )
= ( G2 @ P ) ) ) ) ) ) ) ) ).
% determination_of_hom
thf(fact_716_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_717_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_718_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_719_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_720_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_721_zero__pdivides,axiom,
! [P: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P )
= ( P = nil_a ) ) ).
% zero_pdivides
thf(fact_722_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_723_local_Onormalize__idem,axiom,
! [P: list_a,Q: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( normalize_a_b @ r @ P ) @ Q ) )
= ( normalize_a_b @ r @ ( append_a @ P @ Q ) ) ) ).
% local.normalize_idem
thf(fact_724_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_725_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_726_Units__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_727_Units__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_728_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_729_pdivides__zero,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P @ nil_a ) ) ) ).
% pdivides_zero
thf(fact_730_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_731_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_732_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_733_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_734_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_735_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_736_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_737_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_738_map__append,axiom,
! [F: a > list_a,Xs: list_a,Ys: list_a] :
( ( map_a_list_a @ F @ ( append_a @ Xs @ Ys ) )
= ( append_list_a @ ( map_a_list_a @ F @ Xs ) @ ( map_a_list_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_739_map__append,axiom,
! [F: a > a,Xs: list_a,Ys: list_a] :
( ( map_a_a @ F @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( map_a_a @ F @ Xs ) @ ( map_a_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_740_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_741_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_742_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_743_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_744_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_745_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_746_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_747_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_748_append__eq__map__conv,axiom,
! [Ys: list_list_a,Zs3: list_list_a,F: a > list_a,Xs: list_a] :
( ( ( append_list_a @ Ys @ Zs3 )
= ( map_a_list_a @ F @ Xs ) )
= ( ? [Us: list_a,Vs: list_a] :
( ( Xs
= ( append_a @ Us @ Vs ) )
& ( Ys
= ( map_a_list_a @ F @ Us ) )
& ( Zs3
= ( map_a_list_a @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_749_append__eq__map__conv,axiom,
! [Ys: list_a,Zs3: list_a,F: a > a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs3 )
= ( map_a_a @ F @ Xs ) )
= ( ? [Us: list_a,Vs: list_a] :
( ( Xs
= ( append_a @ Us @ Vs ) )
& ( Ys
= ( map_a_a @ F @ Us ) )
& ( Zs3
= ( map_a_a @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_750_map__eq__append__conv,axiom,
! [F: a > list_a,Xs: list_a,Ys: list_list_a,Zs3: list_list_a] :
( ( ( map_a_list_a @ F @ Xs )
= ( append_list_a @ Ys @ Zs3 ) )
= ( ? [Us: list_a,Vs: list_a] :
( ( Xs
= ( append_a @ Us @ Vs ) )
& ( Ys
= ( map_a_list_a @ F @ Us ) )
& ( Zs3
= ( map_a_list_a @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_751_map__eq__append__conv,axiom,
! [F: a > a,Xs: list_a,Ys: list_a,Zs3: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( append_a @ Ys @ Zs3 ) )
= ( ? [Us: list_a,Vs: list_a] :
( ( Xs
= ( append_a @ Us @ Vs ) )
& ( Ys
= ( map_a_a @ F @ Us ) )
& ( Zs3
= ( map_a_a @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_752_rev__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P2 @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_753_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs3: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs3 )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs3
= ( cons_a @ X @ Xs ) ) )
| ? [Ys4: list_a] :
( ( Ys
= ( cons_a @ X @ Ys4 ) )
& ( ( append_a @ Ys4 @ Zs3 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_754_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs3: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs3 ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs3 ) )
| ? [Ys4: list_a] :
( ( ( cons_a @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_a @ Ys4 @ Zs3 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_755_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y4: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y4 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_756_rev__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ( P2 @ nil_a )
=> ( ! [X3: a,Xs2: list_a] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_757_domain_Ozero__pdivides__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( polyno8016796738000020810t_unit @ R @ nil_list_a @ nil_list_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_758_domain_Ozero__pdivides__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( polyno5814909790663948098es_a_b @ R @ nil_a @ nil_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_759_domain_Ozero__pdivides,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( polyno8016796738000020810t_unit @ R @ nil_list_a @ P )
= ( P = nil_list_a ) ) ) ).
% domain.zero_pdivides
thf(fact_760_domain_Ozero__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( domain_a_b @ R )
=> ( ( polyno5814909790663948098es_a_b @ R @ nil_a @ P )
= ( P = nil_a ) ) ) ).
% domain.zero_pdivides
thf(fact_761_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X3: a,Xs3: list_a,Y4: a,Ys5: list_a] :
( ( X3 != Y4 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X3 @ nil_a ) @ Xs3 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y4 @ nil_a ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_762_ring__hom__ring_Oeval__hom,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,K: set_a,A: a,P: list_a] :
( ( ring_h5357930050666032198t_unit @ R @ S @ H2 )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( H2 @ ( eval_a_b @ R @ P @ A ) )
= ( eval_l34571156754992824t_unit @ S @ ( map_a_list_a @ H2 @ P ) @ ( H2 @ A ) ) ) ) ) ) ) ).
% ring_hom_ring.eval_hom
thf(fact_763_ring__hom__ring_Oeval__hom,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a,K: set_a,A: a,P: list_a] :
( ( ring_h4024360765257340990_b_a_b @ R @ S @ H2 )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( H2 @ ( eval_a_b @ R @ P @ A ) )
= ( eval_a_b @ S @ ( map_a_a @ H2 @ P ) @ ( H2 @ A ) ) ) ) ) ) ) ).
% ring_hom_ring.eval_hom
thf(fact_764_ring__hom__ring_Oeval__hom,axiom,
! [R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H2: list_a > list_a,K: set_list_a,A: list_a,P: list_list_a] :
( ( ring_h1334922693953046990t_unit @ R @ S @ H2 )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( H2 @ ( eval_l34571156754992824t_unit @ R @ P @ A ) )
= ( eval_l34571156754992824t_unit @ S @ ( map_list_a_list_a @ H2 @ P ) @ ( H2 @ A ) ) ) ) ) ) ) ).
% ring_hom_ring.eval_hom
thf(fact_765_ring__hom__ring_Oeval__hom,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,K: set_list_a,A: list_a,P: list_list_a] :
( ( ring_h7848885096329822662it_a_b @ R @ S @ H2 )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( H2 @ ( eval_l34571156754992824t_unit @ R @ P @ A ) )
= ( eval_a_b @ S @ ( map_list_a_a @ H2 @ P ) @ ( H2 @ A ) ) ) ) ) ) ) ).
% ring_hom_ring.eval_hom
thf(fact_766_ring__hom__ring_Oeval__hom,axiom,
! [R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,H2: list_list_a > list_a,K: set_list_list_a,A: list_list_a,P: list_list_list_a] :
( ( ring_h4589914651911841480t_unit @ R @ S @ H2 )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
=> ( ( H2 @ ( eval_l1088911609197519410t_unit @ R @ P @ A ) )
= ( eval_l34571156754992824t_unit @ S @ ( map_li1646474281249396926list_a @ H2 @ P ) @ ( H2 @ A ) ) ) ) ) ) ) ).
% ring_hom_ring.eval_hom
thf(fact_767_ring__hom__ring_Oeval__hom,axiom,
! [R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,H2: list_list_a > a,K: set_list_list_a,A: list_list_a,P: list_list_list_a] :
( ( ring_h3841606220870141376it_a_b @ R @ S @ H2 )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
=> ( ( H2 @ ( eval_l1088911609197519410t_unit @ R @ P @ A ) )
= ( eval_a_b @ S @ ( map_list_list_a_a @ H2 @ P ) @ ( H2 @ A ) ) ) ) ) ) ) ).
% ring_hom_ring.eval_hom
thf(fact_768_domain_Opdivides__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( polyno5814909790663948098es_a_b @ R @ P @ nil_a ) ) ) ) ).
% domain.pdivides_zero
thf(fact_769_domain_Opdivides__zero,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( polyno8016796738000020810t_unit @ R @ P @ nil_list_a ) ) ) ) ).
% domain.pdivides_zero
thf(fact_770_domain_Odetermination__of__hom,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,A2: partia2175431115845679010xt_a_b,H2: list_a > a,G2: list_a > a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R @ K ) @ A2 @ H2 )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R @ K ) @ A2 @ G2 )
=> ( ! [K4: a] :
( ( member_a @ K4 @ K )
=> ( ( H2 @ ( cons_a @ K4 @ nil_a ) )
= ( G2 @ ( cons_a @ K4 @ nil_a ) ) ) )
=> ( ( ( H2 @ ( var_a_b @ R ) )
= ( G2 @ ( var_a_b @ R ) ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( H2 @ P )
= ( G2 @ P ) ) ) ) ) ) ) ) ) ).
% domain.determination_of_hom
thf(fact_771_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,Q: list_set_list_a,A: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( polyno9075941895896075626t_unit @ R @ P @ Q )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( eval_s5133945360527818456t_unit @ R @ P @ A )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( eval_s5133945360527818456t_unit @ R @ Q @ A )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_772_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a,A: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( polyno4453881341673752516t_unit @ R @ P @ Q )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( eval_l1088911609197519410t_unit @ R @ P @ A )
= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( eval_l1088911609197519410t_unit @ R @ Q @ A )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_773_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,A: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P @ Q )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( eval_a_b @ R @ P @ A )
= ( zero_a_b @ R ) )
=> ( ( eval_a_b @ R @ Q @ A )
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_774_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,A: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P @ Q )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( eval_l34571156754992824t_unit @ R @ P @ A )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ Q @ A )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_775_domain_Oeval__as__unique__hom,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,X: a,H2: list_a > a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R @ K ) @ R @ H2 )
=> ( ! [K4: a] :
( ( member_a @ K4 @ K )
=> ( ( H2 @ ( cons_a @ K4 @ nil_a ) )
= K4 ) )
=> ( ( ( H2 @ ( var_a_b @ R ) )
= X )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( H2 @ P )
= ( eval_a_b @ R @ P @ X ) ) ) ) ) ) ) ) ) ).
% domain.eval_as_unique_hom
thf(fact_776_domain_Oeval__as__unique__hom,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,X: list_a,H2: list_list_a > list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_h4589914651911841480t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ R @ H2 )
=> ( ! [K4: list_a] :
( ( member_list_a @ K4 @ K )
=> ( ( H2 @ ( cons_list_a @ K4 @ nil_list_a ) )
= K4 ) )
=> ( ( ( H2 @ ( var_li8453953174693405341t_unit @ R ) )
= X )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( H2 @ P )
= ( eval_l34571156754992824t_unit @ R @ P @ X ) ) ) ) ) ) ) ) ) ).
% domain.eval_as_unique_hom
thf(fact_777_domain_Oeval__as__unique__hom,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a,X: list_list_a,H2: list_list_list_a > list_list_a,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_h7694777735462631100t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ R @ H2 )
=> ( ! [K4: list_list_a] :
( ( member_list_list_a @ K4 @ K )
=> ( ( H2 @ ( cons_list_list_a @ K4 @ nil_list_list_a ) )
= K4 ) )
=> ( ( ( H2 @ ( var_li3532061862469730199t_unit @ R ) )
= X )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
=> ( ( H2 @ P )
= ( eval_l1088911609197519410t_unit @ R @ P @ X ) ) ) ) ) ) ) ) ) ).
% domain.eval_as_unique_hom
thf(fact_778_domain_Opdivides__imp__is__root,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,X: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( P != nil_set_list_a )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( polyno9075941895896075626t_unit @ R @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X ) @ nil_set_list_a ) ) @ P )
=> ( polyno4320237611291262604t_unit @ R @ P @ X ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_779_domain_Opdivides__imp__is__root,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( P != nil_list_a )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( polyno8016796738000020810t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ P )
=> ( polyno6951661231331188332t_unit @ R @ P @ X ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_780_domain_Opdivides__imp__is__root,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,X: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( P != nil_list_list_a )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( polyno4453881341673752516t_unit @ R @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ nil_list_list_a ) ) @ P )
=> ( polyno5142720416380192742t_unit @ R @ P @ X ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_781_domain_Opdivides__imp__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,X: a] :
( ( domain_a_b @ R )
=> ( ( P != 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 ) ) @ P )
=> ( polyno4133073214067823460ot_a_b @ R @ P @ X ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_782_domain_Opoly__mult__var,axiom,
! [R: partia7496981018696276118t_unit,K: set_set_list_a,P: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K @ R )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
=> ( ( ( P = nil_set_list_a )
=> ( ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P @ ( var_se6008125447796440765t_unit @ R ) )
= nil_set_list_a ) )
& ( ( P != nil_set_list_a )
=> ( ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P @ ( var_se6008125447796440765t_unit @ R ) )
= ( append_set_list_a @ P @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ nil_set_list_a ) ) ) ) ) ) ) ) ).
% domain.poly_mult_var
thf(fact_783_domain_Opoly__mult__var,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ( P = nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ ( var_a_b @ R ) )
= nil_a ) )
& ( ( P != nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ ( var_a_b @ R ) )
= ( append_a @ P @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) ) ) ) ) ) ) ).
% domain.poly_mult_var
thf(fact_784_domain_Opoly__mult__var,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ( P = nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ ( var_li8453953174693405341t_unit @ R ) )
= nil_list_a ) )
& ( ( P != nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ ( var_li8453953174693405341t_unit @ R ) )
= ( append_list_a @ P @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) ) ) ) ) ) ) ).
% domain.poly_mult_var
thf(fact_785_domain_Ois__root__imp__pdivides,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,X: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( polyno4320237611291262604t_unit @ R @ P @ X )
=> ( polyno9075941895896075626t_unit @ R @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X ) @ nil_set_list_a ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_786_domain_Ois__root__imp__pdivides,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,X: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( polyno5142720416380192742t_unit @ R @ P @ X )
=> ( polyno4453881341673752516t_unit @ R @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ nil_list_list_a ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_787_domain_Ois__root__imp__pdivides,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ R @ P @ X )
=> ( polyno8016796738000020810t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_788_domain_Ois__root__imp__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,X: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P @ X )
=> ( polyno5814909790663948098es_a_b @ R @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ X ) @ nil_a ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_789_alg__multE_I1_J,axiom,
! [X: a,P: list_a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P @ X ) ) @ P ) ) ) ) ).
% alg_multE(1)
thf(fact_790_eval__append__aux,axiom,
! [P: list_a,B: a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P @ ( cons_a @ B @ nil_a ) ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ A ) @ B ) ) ) ) ) ).
% eval_append_aux
thf(fact_791_const__term__eq__last,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( append_a @ P @ ( cons_a @ A @ nil_a ) ) )
= A ) ) ) ).
% const_term_eq_last
thf(fact_792_const__term__explicit,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= A )
=> ~ ! [P3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( P
!= ( append_a @ P3 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ).
% const_term_explicit
thf(fact_793_pirreducible__pow__pdivides__iff,axiom,
! [K: set_a,P: list_a,Q: list_a,R3: list_a,N: nat] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ~ ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R3 ) )
= ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) @ R3 ) ) ) ) ) ) ) ) ).
% pirreducible_pow_pdivides_iff
thf(fact_794_polynomial__pow__division,axiom,
! [P: list_a,N: nat,M: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ M ) ) ) ) ).
% polynomial_pow_division
thf(fact_795_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_796_subring__props_I7_J,axiom,
! [K: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H22 @ K )
=> ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ K ) ) ) ) ).
% subring_props(7)
thf(fact_797_subring__props_I6_J,axiom,
! [K: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H22 @ K )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ K ) ) ) ) ).
% subring_props(6)
thf(fact_798_subring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( K != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_799_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_800_subring__props_I5_J,axiom,
! [K: set_a,H2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H2 @ K )
=> ( member_a @ ( a_inv_a_b @ r @ H2 ) @ K ) ) ) ).
% subring_props(5)
thf(fact_801_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_802_normalize__length__le,axiom,
! [P: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) @ ( size_size_list_a @ P ) ) ).
% normalize_length_le
thf(fact_803_normalize__in__carrier,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( normalize_a_b @ r @ P ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% normalize_in_carrier
thf(fact_804_eval__in__carrier,axiom,
! [P: list_a,X: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier
thf(fact_805_pprimeE_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( P != nil_a ) ) ) ) ).
% pprimeE(1)
thf(fact_806_const__term__simprules_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( const_term_a_b @ r @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% const_term_simprules(1)
thf(fact_807_pprime__iff__pirreducible,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_808_pprimeE_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pprimeE(2)
thf(fact_809_eval__normalize,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( normalize_a_b @ r @ P ) @ A )
= ( eval_a_b @ r @ P @ A ) ) ) ) ).
% eval_normalize
thf(fact_810_line__extension__smult__closed,axiom,
! [K: set_a,E: set_a,A: a,K2: a,U: a] :
( ( subfield_a_b @ K @ r )
=> ( ! [K4: a,V: a] :
( ( member_a @ K4 @ K )
=> ( ( member_a @ V @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K4 @ V ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ U ) @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_811_pprimeE_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a,R3: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R3 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
| ( polyno5814909790663948098es_a_b @ r @ P @ R3 ) ) ) ) ) ) ) ) ).
% pprimeE(3)
thf(fact_812_List_Ofinite__set,axiom,
! [Xs: list_c] : ( finite_finite_c @ ( set_c2 @ Xs ) ) ).
% List.finite_set
thf(fact_813_List_Ofinite__set,axiom,
! [Xs: list_a] : ( finite_finite_a @ ( set_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_814_map__eq__conv,axiom,
! [F: a > list_a,Xs: list_a,G2: a > list_a] :
( ( ( map_a_list_a @ F @ Xs )
= ( map_a_list_a @ G2 @ Xs ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( F @ X2 )
= ( G2 @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_815_map__eq__conv,axiom,
! [F: a > a,Xs: list_a,G2: a > a] :
( ( ( map_a_a @ F @ Xs )
= ( map_a_a @ G2 @ Xs ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( F @ X2 )
= ( G2 @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_816_pprimeI,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ R4 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q2 )
| ( polyno5814909790663948098es_a_b @ r @ P @ R4 ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ) ) ).
% pprimeI
thf(fact_817_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_818_set__empty2,axiom,
! [Xs: list_c] :
( ( bot_bot_set_c
= ( set_c2 @ Xs ) )
= ( Xs = nil_c ) ) ).
% set_empty2
thf(fact_819_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_820_set__empty,axiom,
! [Xs: list_c] :
( ( ( set_c2 @ Xs )
= bot_bot_set_c )
= ( Xs = nil_c ) ) ).
% set_empty
thf(fact_821_le__alg__mult__imp__pdivides,axiom,
! [X: a,P: list_a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P @ 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 ) @ P ) ) ) ) ).
% le_alg_mult_imp_pdivides
thf(fact_822_alg__multE_I2_J,axiom,
! [X: a,P: list_a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P @ X ) ) ) ) ) ) ).
% alg_multE(2)
thf(fact_823_monom__in__carrier,axiom,
! [A: a,N: nat] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( monom_a_b @ r @ A @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% monom_in_carrier
thf(fact_824_ring_Oalg__mult_Ocong,axiom,
polyno4422430861927485590lt_a_b = polyno4422430861927485590lt_a_b ).
% ring.alg_mult.cong
thf(fact_825_subfieldE_I4_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit,K1: list_a,K22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_a @ K1 @ K )
=> ( ( member_list_a @ K22 @ K )
=> ( ( mult_l7073676228092353617t_unit @ R @ K1 @ K22 )
= ( mult_l7073676228092353617t_unit @ R @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_826_subfieldE_I4_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K @ R )
=> ( ( member_a @ K1 @ K )
=> ( ( member_a @ K22 @ K )
=> ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
= ( mult_a_ring_ext_a_b @ R @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_827_subfieldE_I1_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( subrin6918843898125473962t_unit @ K @ R ) ) ).
% subfieldE(1)
thf(fact_828_subfieldE_I1_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( subring_a_b @ K @ R ) ) ).
% subfieldE(1)
thf(fact_829_finite__list,axiom,
! [A2: set_c] :
( ( finite_finite_c @ A2 )
=> ? [Xs2: list_c] :
( ( set_c2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_830_finite__list,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ? [Xs2: list_a] :
( ( set_a2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_831_ex__map__conv,axiom,
! [Ys: list_list_a,F: a > list_a] :
( ( ? [Xs4: list_a] :
( Ys
= ( map_a_list_a @ F @ Xs4 ) ) )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Ys ) )
=> ? [Y2: a] :
( X2
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_832_ex__map__conv,axiom,
! [Ys: list_a,F: a > a] :
( ( ? [Xs4: list_a] :
( Ys
= ( map_a_a @ F @ Xs4 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ? [Y2: a] :
( X2
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_833_map__cong,axiom,
! [Xs: list_a,Ys: list_a,F: a > list_a,G2: a > list_a] :
( ( Xs = Ys )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Ys ) )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( map_a_list_a @ F @ Xs )
= ( map_a_list_a @ G2 @ Ys ) ) ) ) ).
% map_cong
thf(fact_834_map__cong,axiom,
! [Xs: list_a,Ys: list_a,F: a > a,G2: a > a] :
( ( Xs = Ys )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Ys ) )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( map_a_a @ F @ Xs )
= ( map_a_a @ G2 @ Ys ) ) ) ) ).
% map_cong
thf(fact_835_map__idI,axiom,
! [Xs: list_list_a,F: list_a > list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_list_a_list_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_836_map__idI,axiom,
! [Xs: list_c,F: c > c] :
( ! [X3: c] :
( ( member_c @ X3 @ ( set_c2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_c_c @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_837_map__idI,axiom,
! [Xs: list_list_list_a,F: list_list_a > list_list_a] :
( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ ( set_list_list_a2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_li8713736314956022724list_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_838_map__idI,axiom,
! [Xs: list_a,F: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_a_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_839_map__ext,axiom,
! [Xs: list_a,F: a > list_a,G2: a > list_a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( map_a_list_a @ F @ Xs )
= ( map_a_list_a @ G2 @ Xs ) ) ) ).
% map_ext
thf(fact_840_map__ext,axiom,
! [Xs: list_a,F: a > a,G2: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( map_a_a @ F @ Xs )
= ( map_a_a @ G2 @ Xs ) ) ) ).
% map_ext
thf(fact_841_list_Omap__ident__strong,axiom,
! [T: list_list_a,F: list_a > list_a] :
( ! [Z2: list_a] :
( ( member_list_a @ Z2 @ ( set_list_a2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_list_a_list_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_842_list_Omap__ident__strong,axiom,
! [T: list_c,F: c > c] :
( ! [Z2: c] :
( ( member_c @ Z2 @ ( set_c2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_c_c @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_843_list_Omap__ident__strong,axiom,
! [T: list_list_list_a,F: list_list_a > list_list_a] :
( ! [Z2: list_list_a] :
( ( member_list_list_a @ Z2 @ ( set_list_list_a2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_li8713736314956022724list_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_844_list_Omap__ident__strong,axiom,
! [T: list_a,F: a > a] :
( ! [Z2: a] :
( ( member_a @ Z2 @ ( set_a2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_a_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_845_list_Oinj__map__strong,axiom,
! [X: list_a,Xa2: list_a,F: a > list_a,Fa: a > list_a] :
( ! [Z2: a,Za: a] :
( ( member_a @ Z2 @ ( set_a2 @ X ) )
=> ( ( member_a @ Za @ ( set_a2 @ Xa2 ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_a_list_a @ F @ X )
= ( map_a_list_a @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_846_list_Oinj__map__strong,axiom,
! [X: list_a,Xa2: list_a,F: a > a,Fa: a > a] :
( ! [Z2: a,Za: a] :
( ( member_a @ Z2 @ ( set_a2 @ X ) )
=> ( ( member_a @ Za @ ( set_a2 @ Xa2 ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_a_a @ F @ X )
= ( map_a_a @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_847_list_Omap__cong0,axiom,
! [X: list_a,F: a > list_a,G2: a > list_a] :
( ! [Z2: a] :
( ( member_a @ Z2 @ ( set_a2 @ X ) )
=> ( ( F @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( map_a_list_a @ F @ X )
= ( map_a_list_a @ G2 @ X ) ) ) ).
% list.map_cong0
thf(fact_848_list_Omap__cong0,axiom,
! [X: list_a,F: a > a,G2: a > a] :
( ! [Z2: a] :
( ( member_a @ Z2 @ ( set_a2 @ X ) )
=> ( ( F @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( map_a_a @ F @ X )
= ( map_a_a @ G2 @ X ) ) ) ).
% list.map_cong0
thf(fact_849_list_Omap__cong,axiom,
! [X: list_a,Ya: list_a,F: a > list_a,G2: a > list_a] :
( ( X = Ya )
=> ( ! [Z2: a] :
( ( member_a @ Z2 @ ( set_a2 @ Ya ) )
=> ( ( F @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( map_a_list_a @ F @ X )
= ( map_a_list_a @ G2 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_850_list_Omap__cong,axiom,
! [X: list_a,Ya: list_a,F: a > a,G2: a > a] :
( ( X = Ya )
=> ( ! [Z2: a] :
( ( member_a @ Z2 @ ( set_a2 @ Ya ) )
=> ( ( F @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( map_a_a @ F @ X )
= ( map_a_a @ G2 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_851_subfieldE_I2_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( subcring_a_b @ K @ R ) ) ).
% subfieldE(2)
thf(fact_852_subfield_Oaxioms_I1_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( subdomain_a_b @ K @ R ) ) ).
% subfield.axioms(1)
thf(fact_853_subfieldE_I3_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subfieldE(3)
thf(fact_854_subfieldE_I3_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subfieldE(3)
thf(fact_855_subfieldE_I3_J,axiom,
! [K: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subfie4546268998243038636t_unit @ K @ R )
=> ( ord_le8488217952732425610list_a @ K @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% subfieldE(3)
thf(fact_856_subfieldE_I5_J,axiom,
! [K: set_set_list_a,R: partia7496981018696276118t_unit,K1: set_list_a,K22: set_list_a] :
( ( subfie4339374749748326226t_unit @ K @ R )
=> ( ( member_set_list_a @ K1 @ K )
=> ( ( member_set_list_a @ K22 @ K )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ K1 @ K22 )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( K1
= ( zero_s2910681146719230829t_unit @ R ) )
| ( K22
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_857_subfieldE_I5_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit,K1: list_a,K22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_a @ K1 @ K )
=> ( ( member_list_a @ K22 @ K )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ K1 @ K22 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( K1
= ( zero_l4142658623432671053t_unit @ R ) )
| ( K22
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_858_subfieldE_I5_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K @ R )
=> ( ( member_a @ K1 @ K )
=> ( ( member_a @ K22 @ K )
=> ( ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
= ( zero_a_b @ R ) )
=> ( ( K1
= ( zero_a_b @ R ) )
| ( K22
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_859_subfieldE_I6_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% subfieldE(6)
thf(fact_860_subfieldE_I6_J,axiom,
! [K: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subfie4339374749748326226t_unit @ K @ R )
=> ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% subfieldE(6)
thf(fact_861_subfieldE_I6_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% subfieldE(6)
thf(fact_862_field_Ocarrier__is__subfield,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( subfie4339374749748326226t_unit @ ( partia141011252114345353t_unit @ R ) @ R ) ) ).
% field.carrier_is_subfield
thf(fact_863_field_Ocarrier__is__subfield,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( subfield_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% field.carrier_is_subfield
thf(fact_864_field_Ocarrier__is__subfield,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( subfie1779122896746047282t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).
% field.carrier_is_subfield
thf(fact_865_field_Ocarrier__is__subfield,axiom,
! [R: partia2956882679547061052t_unit] :
( ( field_1861437471013600865t_unit @ R )
=> ( subfie4546268998243038636t_unit @ ( partia2464479390973590831t_unit @ R ) @ R ) ) ).
% field.carrier_is_subfield
thf(fact_866_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_867_empty__set,axiom,
( bot_bot_set_c
= ( set_c2 @ nil_c ) ) ).
% empty_set
thf(fact_868_domain_OpprimeE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
=> ( P != nil_a ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_869_domain_OpprimeE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
=> ( P != nil_list_a ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_870_domain_Opprime__iff__pirreducible,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_871_domain_Opprime__iff__pirreducible,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_872_domain_OpprimeE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).
% domain.pprimeE(2)
thf(fact_873_domain_OpprimeE_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
=> ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).
% domain.pprimeE(2)
thf(fact_874_cring_Oconst__term__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( cring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( const_term_a_b @ R @ P ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% cring.const_term_simprules(1)
thf(fact_875_cring_Oconst__term__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ R @ P ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% cring.const_term_simprules(1)
thf(fact_876_cring_Oconst__term__simprules_I1_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( const_6243872422735025855t_unit @ R @ P ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).
% cring.const_term_simprules(1)
thf(fact_877_domain_OpprimeE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a,R3: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q @ R3 ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P @ Q )
| ( polyno5814909790663948098es_a_b @ R @ P @ R3 ) ) ) ) ) ) ) ) ) ).
% domain.pprimeE(3)
thf(fact_878_domain_OpprimeE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a,R3: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q @ R3 ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P @ Q )
| ( polyno8016796738000020810t_unit @ R @ P @ R3 ) ) ) ) ) ) ) ) ) ).
% domain.pprimeE(3)
thf(fact_879_ring__hom__ring_Oeval__hom_H,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,A: a,P: list_a] :
( ( ring_h5357930050666032198t_unit @ R @ S @ H2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( eval_a_b @ R @ P @ A ) )
= ( eval_l34571156754992824t_unit @ S @ ( map_a_list_a @ H2 @ P ) @ ( H2 @ A ) ) ) ) ) ) ).
% ring_hom_ring.eval_hom'
thf(fact_880_ring__hom__ring_Oeval__hom_H,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a,A: a,P: list_a] :
( ( ring_h4024360765257340990_b_a_b @ R @ S @ H2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( eval_a_b @ R @ P @ A ) )
= ( eval_a_b @ S @ ( map_a_a @ H2 @ P ) @ ( H2 @ A ) ) ) ) ) ) ).
% ring_hom_ring.eval_hom'
thf(fact_881_ring__hom__ring_Oeval__hom_H,axiom,
! [R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H2: list_a > list_a,A: list_a,P: list_list_a] :
( ( ring_h1334922693953046990t_unit @ R @ S @ H2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( eval_l34571156754992824t_unit @ R @ P @ A ) )
= ( eval_l34571156754992824t_unit @ S @ ( map_list_a_list_a @ H2 @ P ) @ ( H2 @ A ) ) ) ) ) ) ).
% ring_hom_ring.eval_hom'
thf(fact_882_ring__hom__ring_Oeval__hom_H,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,A: list_a,P: list_list_a] :
( ( ring_h7848885096329822662it_a_b @ R @ S @ H2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( eval_l34571156754992824t_unit @ R @ P @ A ) )
= ( eval_a_b @ S @ ( map_list_a_a @ H2 @ P ) @ ( H2 @ A ) ) ) ) ) ) ).
% ring_hom_ring.eval_hom'
thf(fact_883_ring__hom__ring_Oeval__hom_H,axiom,
! [R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,H2: list_list_a > list_a,A: list_list_a,P: list_list_list_a] :
( ( ring_h4589914651911841480t_unit @ R @ S @ H2 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( eval_l1088911609197519410t_unit @ R @ P @ A ) )
= ( eval_l34571156754992824t_unit @ S @ ( map_li1646474281249396926list_a @ H2 @ P ) @ ( H2 @ A ) ) ) ) ) ) ).
% ring_hom_ring.eval_hom'
thf(fact_884_ring__hom__ring_Oeval__hom_H,axiom,
! [R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,H2: list_list_a > a,A: list_list_a,P: list_list_list_a] :
( ( ring_h3841606220870141376it_a_b @ R @ S @ H2 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( eval_l1088911609197519410t_unit @ R @ P @ A ) )
= ( eval_a_b @ S @ ( map_list_list_a_a @ H2 @ P ) @ ( H2 @ A ) ) ) ) ) ) ).
% ring_hom_ring.eval_hom'
thf(fact_885_domain_Ole__alg__mult__imp__pdivides,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,P: list_set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno1088517687229135038t_unit @ R @ P @ X ) )
=> ( polyno9075941895896075626t_unit @ R @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X ) @ nil_set_list_a ) ) @ N ) @ P ) ) ) ) ) ).
% domain.le_alg_mult_imp_pdivides
thf(fact_886_domain_Ole__alg__mult__imp__pdivides,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,P: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ R @ P @ X ) )
=> ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ N ) @ P ) ) ) ) ) ).
% domain.le_alg_mult_imp_pdivides
thf(fact_887_domain_Ole__alg__mult__imp__pdivides,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,P: list_list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno1672195411705137432t_unit @ R @ P @ X ) )
=> ( polyno4453881341673752516t_unit @ R @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ nil_list_list_a ) ) @ N ) @ P ) ) ) ) ) ).
% domain.le_alg_mult_imp_pdivides
thf(fact_888_domain_Ole__alg__mult__imp__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,P: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ R @ P @ 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 ) @ P ) ) ) ) ) ).
% domain.le_alg_mult_imp_pdivides
thf(fact_889_domain_Oalg__multE_I2_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,P: list_set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( P != nil_set_list_a )
=> ( ( polyno9075941895896075626t_unit @ R @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X ) @ nil_set_list_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno1088517687229135038t_unit @ R @ P @ X ) ) ) ) ) ) ) ).
% domain.alg_multE(2)
thf(fact_890_domain_Oalg__multE_I2_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,P: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ R @ P @ X ) ) ) ) ) ) ) ).
% domain.alg_multE(2)
thf(fact_891_domain_Oalg__multE_I2_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,P: list_list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( P != nil_list_list_a )
=> ( ( polyno4453881341673752516t_unit @ R @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ nil_list_list_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno1672195411705137432t_unit @ R @ P @ X ) ) ) ) ) ) ) ).
% domain.alg_multE(2)
thf(fact_892_domain_Oalg__multE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,P: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( P != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ X ) @ nil_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ R @ P @ X ) ) ) ) ) ) ) ).
% domain.alg_multE(2)
thf(fact_893_domain_Opolynomial__pow__division,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,N: nat,M: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno4453881341673752516t_unit @ R @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ N ) @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ M ) ) ) ) ) ).
% domain.polynomial_pow_division
thf(fact_894_domain_Opolynomial__pow__division,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,N: nat,M: nat] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ M ) ) ) ) ) ).
% domain.polynomial_pow_division
thf(fact_895_domain_Opolynomial__pow__division,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,N: nat,M: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ N ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ M ) ) ) ) ) ).
% domain.polynomial_pow_division
thf(fact_896_domain_OpprimeI,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ! [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q2 @ R4 ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P @ Q2 )
| ( polyno5814909790663948098es_a_b @ R @ P @ R4 ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) ) ) ) ) ) ) ).
% domain.pprimeI
thf(fact_897_domain_OpprimeI,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( P != nil_list_a )
=> ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ! [Q2: list_list_a,R4: list_list_a] :
( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q2 @ R4 ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P @ Q2 )
| ( polyno8016796738000020810t_unit @ R @ P @ R4 ) ) ) ) )
=> ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) ) ) ) ) ) ) ).
% domain.pprimeI
thf(fact_898_domain_Opirreducible__pow__pdivides__iff,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a,R3: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
=> ( ~ ( polyno8016796738000020810t_unit @ R @ P @ Q )
=> ( ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ N ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q @ R3 ) )
= ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ N ) @ R3 ) ) ) ) ) ) ) ) ) ).
% domain.pirreducible_pow_pdivides_iff
thf(fact_899_domain_Opirreducible__pow__pdivides__iff,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a,R3: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
=> ( ~ ( polyno5814909790663948098es_a_b @ R @ P @ Q )
=> ( ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K ) @ P @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q @ R3 ) )
= ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K ) @ P @ N ) @ R3 ) ) ) ) ) ) ) ) ) ).
% domain.pirreducible_pow_pdivides_iff
thf(fact_900_domain_Oalg__multE_I1_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,P: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( P != nil_set_list_a )
=> ( polyno9075941895896075626t_unit @ R @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X ) @ nil_set_list_a ) ) @ ( polyno1088517687229135038t_unit @ R @ P @ X ) ) @ P ) ) ) ) ) ).
% domain.alg_multE(1)
thf(fact_901_domain_Oalg__multE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( P != nil_list_a )
=> ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ ( polyno4259638811958763678t_unit @ R @ P @ X ) ) @ P ) ) ) ) ) ).
% domain.alg_multE(1)
thf(fact_902_domain_Oalg__multE_I1_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( P != nil_list_list_a )
=> ( polyno4453881341673752516t_unit @ R @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ nil_list_list_a ) ) @ ( polyno1672195411705137432t_unit @ R @ P @ X ) ) @ P ) ) ) ) ) ).
% domain.alg_multE(1)
thf(fact_903_domain_Oalg__multE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( P != nil_a )
=> ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ X ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ R @ P @ X ) ) @ P ) ) ) ) ) ).
% domain.alg_multE(1)
thf(fact_904_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_905_exists__unique__long__division,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ? [X3: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ X3 )
& ! [Y5: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ Y5 )
=> ( Y5 = X3 ) ) ) ) ) ) ) ).
% exists_unique_long_division
thf(fact_906_exp__base__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% exp_base_closed
thf(fact_907_rupture__is__field__iff__pirreducible,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).
% rupture_is_field_iff_pirreducible
thf(fact_908_long__division__a__inv_I1_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% long_division_a_inv(1)
thf(fact_909_long__division__closed_I1_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_910_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_911_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_912_finite__ring__finite__units,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( finite_finite_a @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% finite_ring_finite_units
thf(fact_913_ring_Opdiv_Ocong,axiom,
polynomial_pdiv_a_b = polynomial_pdiv_a_b ).
% ring.pdiv.cong
thf(fact_914_ring_Oexp__base_Ocong,axiom,
polyno2922411391617481336se_a_b = polyno2922411391617481336se_a_b ).
% ring.exp_base.cong
thf(fact_915_ring_Olong__divides_Ocong,axiom,
polyno2806191415236617128es_a_b = polyno2806191415236617128es_a_b ).
% ring.long_divides.cong
thf(fact_916_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_917_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_918_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_919_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_920_principal__domain_Oprimeness__condition,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( ring_p715737262848045090t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ P )
= ( ring_r5437400583859147359t_unit @ R @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_921_principal__domain_Oprimeness__condition,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ P )
= ( ring_ring_prime_a_b @ R @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_922_principal__domain_Oprimeness__condition,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ P )
= ( ring_r6430282645014804837t_unit @ R @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_923_domain_Olong__division__closed_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( 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 @ P @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_924_domain_Olong__division__closed_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ R @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_925_domain_Olong__division__zero_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ R @ nil_list_a @ Q )
= nil_list_a ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_926_domain_Olong__division__zero_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_927_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_928_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_929_domain_Olong__division__a__inv_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( 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 ) @ P ) @ Q )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno5893782122288709345t_unit @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_930_domain_Olong__division__a__inv_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ R @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ ( polynomial_pdiv_a_b @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_931_domain_Orupture__is__field__iff__pirreducible,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( field_1540243473349940225t_unit @ ( polyno859807163042199155t_unit @ R @ K @ P ) )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) ) ) ) ) ).
% domain.rupture_is_field_iff_pirreducible
thf(fact_932_domain_Orupture__is__field__iff__pirreducible,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ R @ K @ P ) )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) ) ) ) ) ).
% domain.rupture_is_field_iff_pirreducible
thf(fact_933_domain_Oexists__unique__long__division,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( Q != nil_list_a )
=> ? [X3: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ R @ P @ Q @ X3 )
& ! [Y5: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ R @ P @ Q @ Y5 )
=> ( Y5 = X3 ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_934_domain_Oexists__unique__long__division,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( Q != nil_a )
=> ? [X3: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ R @ P @ Q @ X3 )
& ! [Y5: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ R @ P @ Q @ Y5 )
=> ( Y5 = X3 ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_935_pdiv__pmod,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% pdiv_pmod
thf(fact_936_poly__mult__var_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( var_a_b @ r ) @ P )
= ( normalize_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ).
% poly_mult_var'(1)
thf(fact_937_poly__mult__var_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ ( var_a_b @ r ) )
= ( normalize_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ).
% poly_mult_var'(2)
thf(fact_938_factors__mult,axiom,
! [Fa: list_a,A: a,Fb: list_a,B: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fa @ A )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% factors_mult
thf(fact_939_poly__mult_Osimps_I1_J,axiom,
! [P22: list_a] :
( ( poly_mult_a_b @ r @ nil_a @ P22 )
= nil_a ) ).
% poly_mult.simps(1)
thf(fact_940_poly__mult__in__carrier,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_mult_a_b @ r @ P1 @ P22 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_mult_in_carrier
thf(fact_941_poly__mult__comm,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ P22 )
= ( poly_mult_a_b @ r @ P22 @ P1 ) ) ) ) ).
% poly_mult_comm
thf(fact_942_factors__closed,axiom,
! [Fs: list_a,A: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% factors_closed
thf(fact_943_long__division__closed_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(2)
thf(fact_944_poly__mult__zero_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ nil_a @ P )
= nil_a ) ) ).
% poly_mult_zero(1)
thf(fact_945_poly__mult__zero_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ nil_a )
= nil_a ) ) ).
% poly_mult_zero(2)
thf(fact_946_poly__mult__normalize,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ P22 )
= ( poly_mult_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ P22 ) ) ) ) ).
% poly_mult_normalize
thf(fact_947_poly__mult__monom__assoc,axiom,
! [P: list_a,Q: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P ) @ Q )
= ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ ( poly_mult_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% poly_mult_monom_assoc
thf(fact_948_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_949_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_950_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_951_long__division__a__inv_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% long_division_a_inv(2)
thf(fact_952_poly__mult__semiassoc,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ P ) @ Q )
= ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ ( poly_mult_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% poly_mult_semiassoc
thf(fact_953_eval__poly__mult,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_mult_a_b @ r @ P @ Q ) @ A )
= ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_poly_mult
thf(fact_954_const__term__simprules_I2_J,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( poly_mult_a_b @ r @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ).
% const_term_simprules(2)
thf(fact_955_pmod__zero__iff__pdivides,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ P @ Q )
= nil_a )
= ( polyno5814909790663948098es_a_b @ r @ Q @ P ) ) ) ) ) ).
% pmod_zero_iff_pdivides
thf(fact_956_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_957_poly__mult__one_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_mult_one'(2)
thf(fact_958_poly__mult__one_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_mult_one'(1)
thf(fact_959_poly__mult__append__zero,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Q )
= ( normalize_a_b @ r @ ( append_a @ ( poly_mult_a_b @ r @ P @ Q ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_mult_append_zero
thf(fact_960_ring_Opoly__mult_Ocong,axiom,
poly_mult_a_b = poly_mult_a_b ).
% ring.poly_mult.cong
thf(fact_961_ring_Opmod_Ocong,axiom,
polynomial_pmod_a_b = polynomial_pmod_a_b ).
% ring.pmod.cong
thf(fact_962_univ__poly__mult,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) )
= ( poly_m7087347720095500472t_unit @ R ) ) ).
% univ_poly_mult
thf(fact_963_univ__poly__mult,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) )
= ( poly_mult_a_b @ R ) ) ).
% univ_poly_mult
thf(fact_964_domain_Opoly__mult__comm,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P22: list_a] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ P1 @ P22 )
= ( poly_mult_a_b @ R @ P22 @ P1 ) ) ) ) ) ).
% domain.poly_mult_comm
thf(fact_965_domain_Opoly__mult__comm,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P22: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ P1 @ P22 )
= ( poly_m7087347720095500472t_unit @ R @ P22 @ P1 ) ) ) ) ) ).
% domain.poly_mult_comm
thf(fact_966_domain_Opoly__mult__comm,axiom,
! [R: partia2956882679547061052t_unit,P1: list_list_list_a,P22: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P1 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P22 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_m5930143454064270386t_unit @ R @ P1 @ P22 )
= ( poly_m5930143454064270386t_unit @ R @ P22 @ P1 ) ) ) ) ) ).
% domain.poly_mult_comm
thf(fact_967_domain_Opoly__mult__monom__assoc,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,A: a,N: nat] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ ( poly_mult_a_b @ R @ ( monom_a_b @ R @ A @ N ) @ P ) @ Q )
= ( poly_mult_a_b @ R @ ( monom_a_b @ R @ A @ N ) @ ( poly_mult_a_b @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.poly_mult_monom_assoc
thf(fact_968_domain_Opoly__mult__monom__assoc,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,A: list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ ( poly_m7087347720095500472t_unit @ R @ ( monom_7446464087056152608t_unit @ R @ A @ N ) @ P ) @ Q )
= ( poly_m7087347720095500472t_unit @ R @ ( monom_7446464087056152608t_unit @ R @ A @ N ) @ ( poly_m7087347720095500472t_unit @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.poly_mult_monom_assoc
thf(fact_969_domain_Opoly__mult__monom__assoc,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a,A: list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Q ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_m5930143454064270386t_unit @ R @ ( poly_m5930143454064270386t_unit @ R @ ( monom_4043874212805408666t_unit @ R @ A @ N ) @ P ) @ Q )
= ( poly_m5930143454064270386t_unit @ R @ ( monom_4043874212805408666t_unit @ R @ A @ N ) @ ( poly_m5930143454064270386t_unit @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.poly_mult_monom_assoc
thf(fact_970_domain_Olong__division__closed_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member_list_list_a @ ( polyno1727750685288865234t_unit @ R @ P @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(2)
thf(fact_971_domain_Olong__division__closed_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ R @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(2)
thf(fact_972_domain_Opoly__mult__semiassoc,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,A: a] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ ( poly_mult_a_b @ R @ ( cons_a @ A @ nil_a ) @ P ) @ Q )
= ( poly_mult_a_b @ R @ ( cons_a @ A @ nil_a ) @ ( poly_mult_a_b @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.poly_mult_semiassoc
thf(fact_973_domain_Opoly__mult__semiassoc,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,A: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ ( poly_m7087347720095500472t_unit @ R @ ( cons_list_a @ A @ nil_list_a ) @ P ) @ Q )
= ( poly_m7087347720095500472t_unit @ R @ ( cons_list_a @ A @ nil_list_a ) @ ( poly_m7087347720095500472t_unit @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.poly_mult_semiassoc
thf(fact_974_domain_Opoly__mult__semiassoc,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a,A: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Q ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_m5930143454064270386t_unit @ R @ ( poly_m5930143454064270386t_unit @ R @ ( cons_list_list_a @ A @ nil_list_list_a ) @ P ) @ Q )
= ( poly_m5930143454064270386t_unit @ R @ ( cons_list_list_a @ A @ nil_list_list_a ) @ ( poly_m5930143454064270386t_unit @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.poly_mult_semiassoc
thf(fact_975_cring_Oeval__poly__mult,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,A: a] :
( ( cring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( poly_mult_a_b @ R @ P @ Q ) @ A )
= ( mult_a_ring_ext_a_b @ R @ ( eval_a_b @ R @ P @ A ) @ ( eval_a_b @ R @ Q @ A ) ) ) ) ) ) ) ).
% cring.eval_poly_mult
thf(fact_976_cring_Oeval__poly__mult,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,A: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( poly_m7087347720095500472t_unit @ R @ P @ Q ) @ A )
= ( mult_l7073676228092353617t_unit @ R @ ( eval_l34571156754992824t_unit @ R @ P @ A ) @ ( eval_l34571156754992824t_unit @ R @ Q @ A ) ) ) ) ) ) ) ).
% cring.eval_poly_mult
thf(fact_977_cring_Oeval__poly__mult,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a,A: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Q ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( eval_l1088911609197519410t_unit @ R @ ( poly_m5930143454064270386t_unit @ R @ P @ Q ) @ A )
= ( mult_l4853965630390486993t_unit @ R @ ( eval_l1088911609197519410t_unit @ R @ P @ A ) @ ( eval_l1088911609197519410t_unit @ R @ Q @ A ) ) ) ) ) ) ) ).
% cring.eval_poly_mult
thf(fact_978_ring__hom__ring_Opoly__mult__hom_H,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,P: list_a,Q: list_a] :
( ( ring_h5357930050666032198t_unit @ R @ S @ H2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( normal637505603836502915t_unit @ S @ ( map_a_list_a @ H2 @ ( poly_mult_a_b @ R @ P @ Q ) ) )
= ( poly_m7087347720095500472t_unit @ S @ ( map_a_list_a @ H2 @ P ) @ ( map_a_list_a @ H2 @ Q ) ) ) ) ) ) ).
% ring_hom_ring.poly_mult_hom'
thf(fact_979_ring__hom__ring_Opoly__mult__hom_H,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a,P: list_a,Q: list_a] :
( ( ring_h4024360765257340990_b_a_b @ R @ S @ H2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( normalize_a_b @ S @ ( map_a_a @ H2 @ ( poly_mult_a_b @ R @ P @ Q ) ) )
= ( poly_mult_a_b @ S @ ( map_a_a @ H2 @ P ) @ ( map_a_a @ H2 @ Q ) ) ) ) ) ) ).
% ring_hom_ring.poly_mult_hom'
thf(fact_980_ring__hom__ring_Opoly__mult__hom_H,axiom,
! [R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H2: list_a > list_a,P: list_list_a,Q: list_list_a] :
( ( ring_h1334922693953046990t_unit @ R @ S @ H2 )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( normal637505603836502915t_unit @ S @ ( map_list_a_list_a @ H2 @ ( poly_m7087347720095500472t_unit @ R @ P @ Q ) ) )
= ( poly_m7087347720095500472t_unit @ S @ ( map_list_a_list_a @ H2 @ P ) @ ( map_list_a_list_a @ H2 @ Q ) ) ) ) ) ) ).
% ring_hom_ring.poly_mult_hom'
thf(fact_981_ring__hom__ring_Opoly__mult__hom_H,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,P: list_list_a,Q: list_list_a] :
( ( ring_h7848885096329822662it_a_b @ R @ S @ H2 )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( normalize_a_b @ S @ ( map_list_a_a @ H2 @ ( poly_m7087347720095500472t_unit @ R @ P @ Q ) ) )
= ( poly_mult_a_b @ S @ ( map_list_a_a @ H2 @ P ) @ ( map_list_a_a @ H2 @ Q ) ) ) ) ) ) ).
% ring_hom_ring.poly_mult_hom'
thf(fact_982_ring__hom__ring_Opoly__mult__hom_H,axiom,
! [R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,H2: list_list_a > list_a,P: list_list_list_a,Q: list_list_list_a] :
( ( ring_h4589914651911841480t_unit @ R @ S @ H2 )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Q ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( normal637505603836502915t_unit @ S @ ( map_li1646474281249396926list_a @ H2 @ ( poly_m5930143454064270386t_unit @ R @ P @ Q ) ) )
= ( poly_m7087347720095500472t_unit @ S @ ( map_li1646474281249396926list_a @ H2 @ P ) @ ( map_li1646474281249396926list_a @ H2 @ Q ) ) ) ) ) ) ).
% ring_hom_ring.poly_mult_hom'
thf(fact_983_ring__hom__ring_Opoly__mult__hom_H,axiom,
! [R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,H2: list_list_a > a,P: list_list_list_a,Q: list_list_list_a] :
( ( ring_h3841606220870141376it_a_b @ R @ S @ H2 )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Q ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( normalize_a_b @ S @ ( map_list_list_a_a @ H2 @ ( poly_m5930143454064270386t_unit @ R @ P @ Q ) ) )
= ( poly_mult_a_b @ S @ ( map_list_list_a_a @ H2 @ P ) @ ( map_list_list_a_a @ H2 @ Q ) ) ) ) ) ) ).
% ring_hom_ring.poly_mult_hom'
thf(fact_984_cring_Oconst__term__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( cring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( const_term_a_b @ R @ ( poly_mult_a_b @ R @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ R @ ( const_term_a_b @ R @ P ) @ ( const_term_a_b @ R @ Q ) ) ) ) ) ) ).
% cring.const_term_simprules(2)
thf(fact_985_cring_Oconst__term__simprules_I2_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( poly_m7087347720095500472t_unit @ R @ P @ Q ) )
= ( mult_l7073676228092353617t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P ) @ ( const_6738166269504826821t_unit @ R @ Q ) ) ) ) ) ) ).
% cring.const_term_simprules(2)
thf(fact_986_cring_Oconst__term__simprules_I2_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Q ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( const_6243872422735025855t_unit @ R @ ( poly_m5930143454064270386t_unit @ R @ P @ Q ) )
= ( mult_l4853965630390486993t_unit @ R @ ( const_6243872422735025855t_unit @ R @ P ) @ ( const_6243872422735025855t_unit @ R @ Q ) ) ) ) ) ) ).
% cring.const_term_simprules(2)
thf(fact_987_domain_Olong__division__zero_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ R @ nil_list_a @ Q )
= nil_list_a ) ) ) ) ).
% domain.long_division_zero(2)
thf(fact_988_domain_Olong__division__zero_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% domain.long_division_zero(2)
thf(fact_989_domain_Olong__division__add__iff,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,C: 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 @ C @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R @ A @ Q )
= ( polyno1727750685288865234t_unit @ R @ B @ Q ) )
= ( ( polyno1727750685288865234t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ A @ C ) @ Q )
= ( polyno1727750685288865234t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ) ).
% domain.long_division_add_iff
thf(fact_990_domain_Olong__division__add__iff,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,C: 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 @ 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 ) ) ) ) ) ) ) ) ) ).
% domain.long_division_add_iff
thf(fact_991_domain_Olong__division__add_I2_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 ) ) )
=> ( ( polyno1727750685288865234t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ A @ B ) @ Q )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno1727750685288865234t_unit @ R @ A @ Q ) @ ( polyno1727750685288865234t_unit @ R @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(2)
thf(fact_992_domain_Olong__division__add_I2_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_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 ) ) ) ) ) ) ) ) ).
% domain.long_division_add(2)
thf(fact_993_domain_Olong__division__a__inv_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ R @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) @ Q )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno1727750685288865234t_unit @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(2)
thf(fact_994_domain_Olong__division__a__inv_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polynomial_pmod_a_b @ R @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ ( polynomial_pmod_a_b @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(2)
thf(fact_995_domain_Opoly__mult__one_H_I1_J,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ( poly_m1537542421183396056t_unit @ R @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ nil_set_list_a ) @ P )
= ( normal4052021864830707619t_unit @ R @ P ) ) ) ) ).
% domain.poly_mult_one'(1)
thf(fact_996_domain_Opoly__mult__one_H_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) @ P )
= ( normalize_a_b @ R @ P ) ) ) ) ).
% domain.poly_mult_one'(1)
thf(fact_997_domain_Opoly__mult__one_H_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ nil_list_a ) @ P )
= ( normal637505603836502915t_unit @ R @ P ) ) ) ) ).
% domain.poly_mult_one'(1)
thf(fact_998_domain_Opoly__mult__one_H_I1_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_m5930143454064270386t_unit @ R @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ nil_list_list_a ) @ P )
= ( normal1297324897130370429t_unit @ R @ P ) ) ) ) ).
% domain.poly_mult_one'(1)
thf(fact_999_domain_Opoly__mult__one_H_I2_J,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ( poly_m1537542421183396056t_unit @ R @ P @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ nil_set_list_a ) )
= ( normal4052021864830707619t_unit @ R @ P ) ) ) ) ).
% domain.poly_mult_one'(2)
thf(fact_1000_domain_Opoly__mult__one_H_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ P @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) )
= ( normalize_a_b @ R @ P ) ) ) ) ).
% domain.poly_mult_one'(2)
thf(fact_1001_domain_Opoly__mult__one_H_I2_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ P @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ nil_list_a ) )
= ( normal637505603836502915t_unit @ R @ P ) ) ) ) ).
% domain.poly_mult_one'(2)
thf(fact_1002_domain_Opoly__mult__one_H_I2_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_m5930143454064270386t_unit @ R @ P @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ nil_list_list_a ) )
= ( normal1297324897130370429t_unit @ R @ P ) ) ) ) ).
% domain.poly_mult_one'(2)
thf(fact_1003_domain_Opmod__zero__iff__pdivides,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R @ P @ Q )
= nil_list_a )
= ( polyno8016796738000020810t_unit @ R @ Q @ P ) ) ) ) ) ) ).
% domain.pmod_zero_iff_pdivides
thf(fact_1004_domain_Opmod__zero__iff__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ R @ P @ Q )
= nil_a )
= ( polyno5814909790663948098es_a_b @ R @ Q @ P ) ) ) ) ) ) ).
% domain.pmod_zero_iff_pdivides
thf(fact_1005_domain_Osame__pmod__iff__pdivides,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 ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R @ A @ Q )
= ( polyno1727750685288865234t_unit @ R @ B @ Q ) )
= ( polyno8016796738000020810t_unit @ R @ Q @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ A @ B ) ) ) ) ) ) ) ) ).
% domain.same_pmod_iff_pdivides
thf(fact_1006_domain_Osame__pmod__iff__pdivides,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_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 ) ) ) ) ) ) ) ) ).
% domain.same_pmod_iff_pdivides
thf(fact_1007_domain_Opdiv__pmod,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( P
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q @ ( polyno5893782122288709345t_unit @ R @ P @ Q ) ) @ ( polyno1727750685288865234t_unit @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.pdiv_pmod
thf(fact_1008_domain_Opdiv__pmod,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q @ ( polynomial_pdiv_a_b @ R @ P @ Q ) ) @ ( polynomial_pmod_a_b @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.pdiv_pmod
thf(fact_1009_domain_Opoly__mult__var_H_I2_J,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ( poly_m1537542421183396056t_unit @ R @ P @ ( var_se6008125447796440765t_unit @ R ) )
= ( normal4052021864830707619t_unit @ R @ ( append_set_list_a @ P @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ nil_set_list_a ) ) ) ) ) ) ).
% domain.poly_mult_var'(2)
thf(fact_1010_domain_Opoly__mult__var_H_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ P @ ( var_a_b @ R ) )
= ( normalize_a_b @ R @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) ) ) ) ) ).
% domain.poly_mult_var'(2)
thf(fact_1011_domain_Opoly__mult__var_H_I2_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ P @ ( var_li8453953174693405341t_unit @ R ) )
= ( normal637505603836502915t_unit @ R @ ( append_list_a @ P @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) ) ) ) ) ).
% domain.poly_mult_var'(2)
thf(fact_1012_domain_Opoly__mult__var_H_I2_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_m5930143454064270386t_unit @ R @ P @ ( var_li3532061862469730199t_unit @ R ) )
= ( normal1297324897130370429t_unit @ R @ ( append_list_list_a @ P @ ( cons_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ nil_list_list_a ) ) ) ) ) ) ).
% domain.poly_mult_var'(2)
thf(fact_1013_domain_Opoly__mult__var_H_I1_J,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ( poly_m1537542421183396056t_unit @ R @ ( var_se6008125447796440765t_unit @ R ) @ P )
= ( normal4052021864830707619t_unit @ R @ ( append_set_list_a @ P @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ nil_set_list_a ) ) ) ) ) ) ).
% domain.poly_mult_var'(1)
thf(fact_1014_domain_Opoly__mult__var_H_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ ( var_a_b @ R ) @ P )
= ( normalize_a_b @ R @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) ) ) ) ) ).
% domain.poly_mult_var'(1)
thf(fact_1015_domain_Opoly__mult__var_H_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ ( var_li8453953174693405341t_unit @ R ) @ P )
= ( normal637505603836502915t_unit @ R @ ( append_list_a @ P @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) ) ) ) ) ).
% domain.poly_mult_var'(1)
thf(fact_1016_domain_Opoly__mult__var_H_I1_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_m5930143454064270386t_unit @ R @ ( var_li3532061862469730199t_unit @ R ) @ P )
= ( normal1297324897130370429t_unit @ R @ ( append_list_list_a @ P @ ( cons_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ nil_list_list_a ) ) ) ) ) ) ).
% domain.poly_mult_var'(1)
thf(fact_1017_long__divisionE,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ).
% long_divisionE
thf(fact_1018_long__divisionI,axiom,
! [K: set_a,P: list_a,Q: list_a,B: list_a,R3: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B @ R3 ) )
=> ( ( produc6837034575241423639list_a @ B @ R3 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ) ).
% long_divisionI
thf(fact_1019_poly__mult__monom_H,axiom,
! [P: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P )
= ( normalize_a_b @ r @ ( append_a @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ A ) @ P ) @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ) ) ) ).
% poly_mult_monom'
thf(fact_1020_eval__append,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P @ Q ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( pow_a_1026414303147256608_b_nat @ r @ A @ ( size_size_list_a @ Q ) ) ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_append
thf(fact_1021_poly__add_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
~ ! [P12: list_a,P23: list_a] :
( X
!= ( produc6837034575241423639list_a @ P12 @ P23 ) ) ).
% poly_add.cases
thf(fact_1022_poly__mult_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [P23: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ P23 ) )
=> ~ ! [V: a,Va: list_a,P23: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ Va ) @ P23 ) ) ) ).
% poly_mult.cases
thf(fact_1023_combine_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [K4: a,Ks: list_a,U2: a,Us2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ K4 @ Ks ) @ ( cons_a @ U2 @ Us2 ) ) )
=> ( ! [Us2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Us2 ) )
=> ~ ! [Ks: list_a] :
( X
!= ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).
% combine.cases
thf(fact_1024_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_1025_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_1026_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_1027_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_1028_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_1029_normalize__replicate__zero,axiom,
! [N: nat,P: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) )
= ( normalize_a_b @ r @ P ) ) ).
% normalize_replicate_zero
thf(fact_1030_local_Omonom__def,axiom,
! [A: a,N: nat] :
( ( monom_a_b @ r @ A @ N )
= ( cons_a @ A @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ).
% local.monom_def
thf(fact_1031_eval__monom,axiom,
! [B: a,A: a,N: nat] :
( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( monom_a_b @ r @ B @ N ) @ A )
= ( mult_a_ring_ext_a_b @ r @ B @ ( pow_a_1026414303147256608_b_nat @ r @ A @ N ) ) ) ) ) ).
% eval_monom
thf(fact_1032_exists__long__division,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ~ ! [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 @ P @ Q @ ( produc6837034575241423639list_a @ B2 @ R4 ) ) ) ) ) ) ) ) ).
% exists_long_division
thf(fact_1033_map__replicate,axiom,
! [F: a > list_a,N: nat,X: a] :
( ( map_a_list_a @ F @ ( replicate_a @ N @ X ) )
= ( replicate_list_a @ N @ ( F @ X ) ) ) ).
% map_replicate
thf(fact_1034_map__replicate,axiom,
! [F: a > a,N: nat,X: a] :
( ( map_a_a @ F @ ( replicate_a @ N @ X ) )
= ( replicate_a @ N @ ( F @ X ) ) ) ).
% map_replicate
thf(fact_1035_poly__mult__replicate__zero_I1_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= nil_a ) ) ).
% poly_mult_replicate_zero(1)
thf(fact_1036_poly__mult__replicate__zero_I2_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= nil_a ) ) ).
% poly_mult_replicate_zero(2)
thf(fact_1037_poly__mult__prepend__replicate__zero,axiom,
! [P1: list_a,P22: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ P22 )
= ( poly_mult_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P1 ) @ P22 ) ) ) ) ).
% poly_mult_prepend_replicate_zero
thf(fact_1038_eval__replicate,axiom,
! [P: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) @ A )
= ( eval_a_b @ r @ P @ A ) ) ) ) ).
% eval_replicate
thf(fact_1039_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_1040_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_1041_shuffles_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ( ! [Xs2: list_a] :
( X
!= ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ~ ! [X3: a,Xs2: list_a,Y4: a,Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1042_replicate__append__same,axiom,
! [I: nat,X: a] :
( ( append_a @ ( replicate_a @ I @ X ) @ ( cons_a @ X @ nil_a ) )
= ( cons_a @ X @ ( replicate_a @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_1043_domain_Opoly__mult__replicate__zero_I1_J,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ( poly_m1537542421183396056t_unit @ R @ ( replicate_set_list_a @ N @ ( zero_s2910681146719230829t_unit @ R ) ) @ P )
= nil_set_list_a ) ) ) ).
% domain.poly_mult_replicate_zero(1)
thf(fact_1044_domain_Opoly__mult__replicate__zero_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ ( replicate_a @ N @ ( zero_a_b @ R ) ) @ P )
= nil_a ) ) ) ).
% domain.poly_mult_replicate_zero(1)
thf(fact_1045_domain_Opoly__mult__replicate__zero_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) @ P )
= nil_list_a ) ) ) ).
% domain.poly_mult_replicate_zero(1)
thf(fact_1046_domain_Opoly__mult__replicate__zero_I1_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_m5930143454064270386t_unit @ R @ ( replic3997036819131463498list_a @ N @ ( zero_l347298301471573063t_unit @ R ) ) @ P )
= nil_list_list_a ) ) ) ).
% domain.poly_mult_replicate_zero(1)
thf(fact_1047_domain_Opoly__mult__replicate__zero_I2_J,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ( poly_m1537542421183396056t_unit @ R @ P @ ( replicate_set_list_a @ N @ ( zero_s2910681146719230829t_unit @ R ) ) )
= nil_set_list_a ) ) ) ).
% domain.poly_mult_replicate_zero(2)
thf(fact_1048_domain_Opoly__mult__replicate__zero_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ P @ ( replicate_a @ N @ ( zero_a_b @ R ) ) )
= nil_a ) ) ) ).
% domain.poly_mult_replicate_zero(2)
thf(fact_1049_domain_Opoly__mult__replicate__zero_I2_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ P @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) )
= nil_list_a ) ) ) ).
% domain.poly_mult_replicate_zero(2)
thf(fact_1050_domain_Opoly__mult__replicate__zero_I2_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_m5930143454064270386t_unit @ R @ P @ ( replic3997036819131463498list_a @ N @ ( zero_l347298301471573063t_unit @ R ) ) )
= nil_list_list_a ) ) ) ).
% domain.poly_mult_replicate_zero(2)
thf(fact_1051_domain_Oexists__long__division,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( Q != nil_list_a )
=> ~ ! [B2: list_list_a] :
( ( member_list_list_a @ B2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ! [R4: list_list_a] :
( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ~ ( polyno6947042923167803568t_unit @ R @ P @ Q @ ( produc8696003437204565271list_a @ B2 @ R4 ) ) ) ) ) ) ) ) ) ).
% domain.exists_long_division
thf(fact_1052_domain_Oexists__long__division,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( Q != nil_a )
=> ~ ! [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 @ P @ Q @ ( produc6837034575241423639list_a @ B2 @ R4 ) ) ) ) ) ) ) ) ) ).
% domain.exists_long_division
thf(fact_1053_domain_Olong__divisionI,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a,B: list_list_a,R3: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno6947042923167803568t_unit @ R @ P @ Q @ ( produc8696003437204565271list_a @ B @ R3 ) )
=> ( ( produc8696003437204565271list_a @ B @ R3 )
= ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ R @ P @ Q ) @ ( polyno1727750685288865234t_unit @ R @ P @ Q ) ) ) ) ) ) ) ) ) ).
% domain.long_divisionI
thf(fact_1054_domain_Olong__divisionI,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a,B: list_a,R3: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ R @ P @ Q @ ( produc6837034575241423639list_a @ B @ R3 ) )
=> ( ( produc6837034575241423639list_a @ B @ R3 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ R @ P @ Q ) @ ( polynomial_pmod_a_b @ R @ P @ Q ) ) ) ) ) ) ) ) ) ).
% domain.long_divisionI
thf(fact_1055_domain_Olong__divisionE,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( Q != nil_list_a )
=> ( polyno6947042923167803568t_unit @ R @ P @ Q @ ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ R @ P @ Q ) @ ( polyno1727750685288865234t_unit @ R @ P @ Q ) ) ) ) ) ) ) ) ).
% domain.long_divisionE
thf(fact_1056_domain_Olong__divisionE,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( Q != nil_a )
=> ( polyno2806191415236617128es_a_b @ R @ P @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ R @ P @ Q ) @ ( polynomial_pmod_a_b @ R @ P @ Q ) ) ) ) ) ) ) ) ).
% domain.long_divisionE
thf(fact_1057_domain_Opoly__mult__monom_H,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,A: set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( poly_m1537542421183396056t_unit @ R @ ( monom_317758005976320064t_unit @ R @ A @ N ) @ P )
= ( normal4052021864830707619t_unit @ R @ ( append_set_list_a @ ( map_se2668659675339852484list_a @ ( mult_s7802724872828879953t_unit @ R @ A ) @ P ) @ ( replicate_set_list_a @ N @ ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ) ).
% domain.poly_mult_monom'
thf(fact_1058_domain_Opoly__mult__monom_H,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,A: a,N: nat] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ ( monom_a_b @ R @ A @ N ) @ P )
= ( normalize_a_b @ R @ ( append_a @ ( map_a_a @ ( mult_a_ring_ext_a_b @ R @ A ) @ P ) @ ( replicate_a @ N @ ( zero_a_b @ R ) ) ) ) ) ) ) ) ).
% domain.poly_mult_monom'
thf(fact_1059_domain_Opoly__mult__monom_H,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,A: list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ ( monom_7446464087056152608t_unit @ R @ A @ N ) @ P )
= ( normal637505603836502915t_unit @ R @ ( append_list_a @ ( map_list_a_list_a @ ( mult_l7073676228092353617t_unit @ R @ A ) @ P ) @ ( replicate_list_a @ N @ ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ) ).
% domain.poly_mult_monom'
thf(fact_1060_domain_Opoly__mult__monom_H,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,A: list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_m5930143454064270386t_unit @ R @ ( monom_4043874212805408666t_unit @ R @ A @ N ) @ P )
= ( normal1297324897130370429t_unit @ R @ ( append_list_list_a @ ( map_li8713736314956022724list_a @ ( mult_l4853965630390486993t_unit @ R @ A ) @ P ) @ ( replic3997036819131463498list_a @ N @ ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ) ).
% domain.poly_mult_monom'
thf(fact_1061_poly__add__append__replicate,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P @ ( replicate_a @ ( size_size_list_a @ Q ) @ ( zero_a_b @ r ) ) ) @ Q )
= ( normalize_a_b @ r @ ( append_a @ P @ Q ) ) ) ) ) ).
% poly_add_append_replicate
thf(fact_1062_poly__add__append__zero,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ ( append_a @ Q @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
= ( normalize_a_b @ r @ ( append_a @ ( poly_add_a_b @ r @ P @ Q ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_add_append_zero
thf(fact_1063_normalize__trick,axiom,
! [P: list_a] :
( P
= ( append_a @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) ) @ ( zero_a_b @ r ) ) @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_trick
thf(fact_1064_ee__sym,axiom,
! [As: list_a,Bs: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( essent8953798148185448568xt_a_b @ r @ Bs @ As ) ) ) ) ).
% ee_sym
thf(fact_1065_ee__length,axiom,
! [As: list_a,Bs: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
=> ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) ) ) ).
% ee_length
thf(fact_1066_poly__add__in__carrier,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_add_a_b @ r @ P1 @ P22 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_add_in_carrier
thf(fact_1067_poly__add__comm,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P22 )
= ( poly_add_a_b @ r @ P22 @ P1 ) ) ) ) ).
% poly_add_comm
thf(fact_1068_ee__trans,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
=> ( ( essent8953798148185448568xt_a_b @ r @ Bs @ Cs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( essent8953798148185448568xt_a_b @ r @ As @ Cs ) ) ) ) ) ) ).
% ee_trans
thf(fact_1069_poly__mult__l__distr_H,axiom,
! [P1: list_a,P22: list_a,P32: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P32 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_add_a_b @ r @ P1 @ P22 ) @ P32 )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P32 ) @ ( poly_mult_a_b @ r @ P22 @ P32 ) ) ) ) ) ) ).
% poly_mult_l_distr'
thf(fact_1070_poly__mult__r__distr_H,axiom,
! [P1: list_a,P22: list_a,P32: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P32 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ ( poly_add_a_b @ r @ P22 @ P32 ) )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P22 ) @ ( poly_mult_a_b @ r @ P1 @ P32 ) ) ) ) ) ) ).
% poly_mult_r_distr'
thf(fact_1071_poly__add__normalize_I3_J,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P22 )
= ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ ( normalize_a_b @ r @ P22 ) ) ) ) ) ).
% poly_add_normalize(3)
thf(fact_1072_poly__add__normalize_I2_J,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P22 )
= ( poly_add_a_b @ r @ P1 @ ( normalize_a_b @ r @ P22 ) ) ) ) ) ).
% poly_add_normalize(2)
thf(fact_1073_poly__add__normalize__aux,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P22 )
= ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ P22 ) ) ) ) ).
% poly_add_normalize_aux
thf(fact_1074_eval__poly__add,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) @ A )
= ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_poly_add
thf(fact_1075_poly__add__zero_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ nil_a @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_zero'(2)
thf(fact_1076_poly__add__zero_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P @ nil_a )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_zero'(1)
thf(fact_1077_const__term__simprules_I3_J,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) )
= ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ).
% const_term_simprules(3)
thf(fact_1078_eval__poly__add__aux,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) @ A )
= ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ) ).
% eval_poly_add_aux
thf(fact_1079_poly__add__replicate__zero_H_I1_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_replicate_zero'(1)
thf(fact_1080_poly__add__replicate__zero_H_I2_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_replicate_zero'(2)
thf(fact_1081_ee__refl,axiom,
! [As: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( essent8953798148185448568xt_a_b @ r @ As @ As ) ) ).
% ee_refl
thf(fact_1082_ring_Opoly__add_Ocong,axiom,
poly_add_a_b = poly_add_a_b ).
% ring.poly_add.cong
thf(fact_1083_univ__poly__add,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) )
= ( poly_a7601779127272115787t_unit @ R ) ) ).
% univ_poly_add
thf(fact_1084_univ__poly__add,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) )
= ( poly_add_a_b @ R ) ) ).
% univ_poly_add
thf(fact_1085_domain_Opoly__mult__r__distr_H,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P22: list_a,P32: list_a] :
( ( domain_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P32 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( poly_mult_a_b @ R @ P1 @ ( poly_add_a_b @ R @ P22 @ P32 ) )
= ( poly_add_a_b @ R @ ( poly_mult_a_b @ R @ P1 @ P22 ) @ ( poly_mult_a_b @ R @ P1 @ P32 ) ) ) ) ) ) ) ).
% domain.poly_mult_r_distr'
thf(fact_1086_domain_Opoly__mult__r__distr_H,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P22: list_list_a,P32: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P32 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( poly_m7087347720095500472t_unit @ R @ P1 @ ( poly_a7601779127272115787t_unit @ R @ P22 @ P32 ) )
= ( poly_a7601779127272115787t_unit @ R @ ( poly_m7087347720095500472t_unit @ R @ P1 @ P22 ) @ ( poly_m7087347720095500472t_unit @ R @ P1 @ P32 ) ) ) ) ) ) ) ).
% domain.poly_mult_r_distr'
thf(fact_1087_domain_Opoly__mult__r__distr_H,axiom,
! [R: partia2956882679547061052t_unit,P1: list_list_list_a,P22: list_list_list_a,P32: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P1 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P22 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P32 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( poly_m5930143454064270386t_unit @ R @ P1 @ ( poly_a7341706734723628101t_unit @ R @ P22 @ P32 ) )
= ( poly_a7341706734723628101t_unit @ R @ ( poly_m5930143454064270386t_unit @ R @ P1 @ P22 ) @ ( poly_m5930143454064270386t_unit @ R @ P1 @ P32 ) ) ) ) ) ) ) ).
% domain.poly_mult_r_distr'
thf(fact_1088_cring_Oconst__term__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( cring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( const_term_a_b @ R @ ( poly_add_a_b @ R @ P @ Q ) )
= ( add_a_b @ R @ ( const_term_a_b @ R @ P ) @ ( const_term_a_b @ R @ Q ) ) ) ) ) ) ).
% cring.const_term_simprules(3)
thf(fact_1089_cring_Oconst__term__simprules_I3_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( poly_a7601779127272115787t_unit @ R @ P @ Q ) )
= ( add_li7652885771158616974t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P ) @ ( const_6738166269504826821t_unit @ R @ Q ) ) ) ) ) ) ).
% cring.const_term_simprules(3)
thf(fact_1090_cring_Oconst__term__simprules_I3_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Q ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( const_6243872422735025855t_unit @ R @ ( poly_a7341706734723628101t_unit @ R @ P @ Q ) )
= ( add_li174743652000525320t_unit @ R @ ( const_6243872422735025855t_unit @ R @ P ) @ ( const_6243872422735025855t_unit @ R @ Q ) ) ) ) ) ) ).
% cring.const_term_simprules(3)
thf(fact_1091_ring__hom__ring_Opoly__add__hom_H,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,P: list_a,Q: list_a] :
( ( ring_h5357930050666032198t_unit @ R @ S @ H2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( normal637505603836502915t_unit @ S @ ( map_a_list_a @ H2 @ ( poly_add_a_b @ R @ P @ Q ) ) )
= ( poly_a7601779127272115787t_unit @ S @ ( map_a_list_a @ H2 @ P ) @ ( map_a_list_a @ H2 @ Q ) ) ) ) ) ) ).
% ring_hom_ring.poly_add_hom'
thf(fact_1092_ring__hom__ring_Opoly__add__hom_H,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a,P: list_a,Q: list_a] :
( ( ring_h4024360765257340990_b_a_b @ R @ S @ H2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( normalize_a_b @ S @ ( map_a_a @ H2 @ ( poly_add_a_b @ R @ P @ Q ) ) )
= ( poly_add_a_b @ S @ ( map_a_a @ H2 @ P ) @ ( map_a_a @ H2 @ Q ) ) ) ) ) ) ).
% ring_hom_ring.poly_add_hom'
thf(fact_1093_ring__hom__ring_Opoly__add__hom_H,axiom,
! [R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H2: list_a > list_a,P: list_list_a,Q: list_list_a] :
( ( ring_h1334922693953046990t_unit @ R @ S @ H2 )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( normal637505603836502915t_unit @ S @ ( map_list_a_list_a @ H2 @ ( poly_a7601779127272115787t_unit @ R @ P @ Q ) ) )
= ( poly_a7601779127272115787t_unit @ S @ ( map_list_a_list_a @ H2 @ P ) @ ( map_list_a_list_a @ H2 @ Q ) ) ) ) ) ) ).
% ring_hom_ring.poly_add_hom'
thf(fact_1094_ring__hom__ring_Opoly__add__hom_H,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,P: list_list_a,Q: list_list_a] :
( ( ring_h7848885096329822662it_a_b @ R @ S @ H2 )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( normalize_a_b @ S @ ( map_list_a_a @ H2 @ ( poly_a7601779127272115787t_unit @ R @ P @ Q ) ) )
= ( poly_add_a_b @ S @ ( map_list_a_a @ H2 @ P ) @ ( map_list_a_a @ H2 @ Q ) ) ) ) ) ) ).
% ring_hom_ring.poly_add_hom'
thf(fact_1095_ring__hom__ring_Opoly__add__hom_H,axiom,
! [R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,H2: list_list_a > list_a,P: list_list_list_a,Q: list_list_list_a] :
( ( ring_h4589914651911841480t_unit @ R @ S @ H2 )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Q ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( normal637505603836502915t_unit @ S @ ( map_li1646474281249396926list_a @ H2 @ ( poly_a7341706734723628101t_unit @ R @ P @ Q ) ) )
= ( poly_a7601779127272115787t_unit @ S @ ( map_li1646474281249396926list_a @ H2 @ P ) @ ( map_li1646474281249396926list_a @ H2 @ Q ) ) ) ) ) ) ).
% ring_hom_ring.poly_add_hom'
thf(fact_1096_ring__hom__ring_Opoly__add__hom_H,axiom,
! [R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,H2: list_list_a > a,P: list_list_list_a,Q: list_list_list_a] :
( ( ring_h3841606220870141376it_a_b @ R @ S @ H2 )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Q ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( normalize_a_b @ S @ ( map_list_list_a_a @ H2 @ ( poly_a7341706734723628101t_unit @ R @ P @ Q ) ) )
= ( poly_add_a_b @ S @ ( map_list_list_a_a @ H2 @ P ) @ ( map_list_list_a_a @ H2 @ Q ) ) ) ) ) ) ).
% ring_hom_ring.poly_add_hom'
thf(fact_1097_pdivides__imp__degree__le,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).
% pdivides_imp_degree_le
thf(fact_1098_poly__sub__degree__le,axiom,
! [X: list_a,N: nat,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% poly_sub_degree_le
thf(fact_1099_pirreducible__degree,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% pirreducible_degree
thf(fact_1100_poly__add__degree__le,axiom,
! [X: list_a,N: nat,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% poly_add_degree_le
thf(fact_1101_degree__one__imp__pirreducible,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).
% degree_one_imp_pirreducible
thf(fact_1102_univ__poly__a__inv__degree,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) @ one_one_nat )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ).
% univ_poly_a_inv_degree
thf(fact_1103_degree__oneE,axiom,
! [P: list_a,K: set_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ K )
=> ( ( A3
!= ( zero_a_b @ r ) )
=> ! [B2: a] :
( ( member_a @ B2 @ K )
=> ( P
!= ( cons_a @ A3 @ ( cons_a @ B2 @ nil_a ) ) ) ) ) ) ) ) ).
% degree_oneE
thf(fact_1104_nat__pow__eone,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ one_one_nat )
= X ) ) ).
% nat_pow_eone
thf(fact_1105_empty_Oprems_I1_J,axiom,
! [X: c] :
( ( member_c @ X @ bot_bot_set_c )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( f @ X ) ) @ one_one_nat ) @ n ) ) ).
% empty.prems(1)
thf(fact_1106_assms_I2_J,axiom,
! [X: c] :
( ( member_c @ X @ a2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( f @ X ) ) @ one_one_nat ) @ n ) ) ).
% assms(2)
thf(fact_1107_domain_Odegree__one__imp__pirreducible,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) ) ) ) ) ).
% domain.degree_one_imp_pirreducible
thf(fact_1108_domain_Odegree__one__imp__pirreducible,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) ) ) ) ) ).
% domain.degree_one_imp_pirreducible
thf(fact_1109_domain_Ouniv__poly__a__inv__degree,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) ) @ one_one_nat )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% domain.univ_poly_a_inv_degree
thf(fact_1110_domain_Ouniv__poly__a__inv__degree,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) ) @ one_one_nat )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% domain.univ_poly_a_inv_degree
thf(fact_1111_domain_Opoly__add__degree__le,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_list_a,N: nat,Y: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ X @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ X ) @ one_one_nat ) @ N )
=> ( ( member5342144027231129785list_a @ Y @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( add_li5162926044081146114t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ X @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% domain.poly_add_degree_le
thf(fact_1112_domain_Opoly__add__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,X: list_a,N: nat,Y: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ X @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% domain.poly_add_degree_le
thf(fact_1113_domain_Opoly__add__degree__le,axiom,
! [R: partia2670972154091845814t_unit,X: list_list_a,N: nat,Y: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ X ) @ one_one_nat ) @ N )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ X @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% domain.poly_add_degree_le
thf(fact_1114_domain_Opoly__sub__degree__le,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_list_a,N: nat,Y: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ X @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ X ) @ one_one_nat ) @ N )
=> ( ( member5342144027231129785list_a @ Y @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ X @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% domain.poly_sub_degree_le
thf(fact_1115_domain_Opoly__sub__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,X: list_a,N: nat,Y: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ X @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% domain.poly_sub_degree_le
thf(fact_1116_domain_Opoly__sub__degree__le,axiom,
! [R: partia2670972154091845814t_unit,X: list_list_a,N: nat,Y: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ X ) @ one_one_nat ) @ N )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ X @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ) ).
% domain.poly_sub_degree_le
thf(fact_1117_domain_Opdivides__imp__degree__le,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ R @ P @ Q )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% domain.pdivides_imp_degree_le
thf(fact_1118_domain_Opdivides__imp__degree__le,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ R @ P @ Q )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% domain.pdivides_imp_degree_le
thf(fact_1119_long__dividesI,axiom,
! [B: list_a,R3: list_a,P: list_a,Q: list_a] :
( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ B ) @ R3 ) )
=> ( ( ( R3 = nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) )
=> ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B @ R3 ) ) ) ) ) ) ).
% long_dividesI
thf(fact_1120_univ__poly__units_H,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
= ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( P != nil_a )
& ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ).
% univ_poly_units'
thf(fact_1121_degree__zero__imp__not__is__root,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ).
% degree_zero_imp_not_is_root
thf(fact_1122_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_1123_nunit__factors,axiom,
! [A: a,As: list_a] :
( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( factor5638265376665762323xt_a_b @ r @ As @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ As ) ) ) ) ).
% nunit_factors
thf(fact_1124_alg__mult__gt__zero__iff__is__root,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P @ X ) )
= ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_1125_pmod__const_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
=> ( ( polynomial_pmod_a_b @ r @ P @ Q )
= P ) ) ) ) ) ).
% pmod_const(2)
thf(fact_1126_pmod__degree,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( ( polynomial_pmod_a_b @ r @ P @ Q )
= nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).
% pmod_degree
thf(fact_1127_rupture__one__not__zero,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) )
=> ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) )
!= ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) ) ) ) ) ) ).
% rupture_one_not_zero
thf(fact_1128_pmod__const_I1_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
=> ( ( polynomial_pdiv_a_b @ r @ P @ Q )
= nil_a ) ) ) ) ) ).
% pmod_const(1)
thf(fact_1129_subfield__long__division__theorem__shell,axiom,
! [K: set_a,P: list_a,B: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( B
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ? [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ B @ Q2 ) @ R4 ) )
& ( ( R4
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R4 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% subfield_long_division_theorem_shell
thf(fact_1130_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_1131_replicate__empty,axiom,
! [N: nat,X: a] :
( ( ( replicate_a @ N @ X )
= nil_a )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_1132_empty__replicate,axiom,
! [N: nat,X: a] :
( ( nil_a
= ( replicate_a @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_1133_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_1134_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_1135_finite__maxlen,axiom,
! [M2: set_list_a] :
( ( finite_finite_list_a @ M2 )
=> ? [N2: nat] :
! [X4: list_a] :
( ( member_list_a @ X4 @ M2 )
=> ( ord_less_nat @ ( size_size_list_a @ X4 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_1136_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1137_replicate__0,axiom,
! [X: a] :
( ( replicate_a @ zero_zero_nat @ X )
= nil_a ) ).
% replicate_0
thf(fact_1138_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,X: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno1672195411705137432t_unit @ R @ P @ X ) )
= ( polyno5142720416380192742t_unit @ R @ P @ X ) ) ) ) ).
% domain.alg_mult_gt_zero_iff_is_root
thf(fact_1139_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4259638811958763678t_unit @ R @ P @ X ) )
= ( polyno6951661231331188332t_unit @ R @ P @ X ) ) ) ) ).
% domain.alg_mult_gt_zero_iff_is_root
thf(fact_1140_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,X: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ R @ P @ X ) )
= ( polyno4133073214067823460ot_a_b @ R @ P @ X ) ) ) ) ).
% domain.alg_mult_gt_zero_iff_is_root
thf(fact_1141_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_1142_semiring_Onat__pow__zero,axiom,
! [R: partia7496981018696276118t_unit,N: nat] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( N != zero_zero_nat )
=> ( ( pow_se8252319793075206062it_nat @ R @ ( zero_s2910681146719230829t_unit @ R ) @ N )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% semiring.nat_pow_zero
thf(fact_1143_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_1144_domain_Orupture__one__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) )
=> ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ R @ K @ P ) )
!= ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ R @ K @ P ) ) ) ) ) ) ) ).
% domain.rupture_one_not_zero
thf(fact_1145_domain_Orupture__one__not__zero,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) )
=> ( ( one_se2489417650821308733t_unit @ ( polyno859807163042199155t_unit @ R @ K @ P ) )
!= ( zero_s2920163772466840039t_unit @ ( polyno859807163042199155t_unit @ R @ K @ P ) ) ) ) ) ) ) ).
% domain.rupture_one_not_zero
thf(fact_1146_domain_Opmod__const_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) )
=> ( ( polyno1727750685288865234t_unit @ R @ P @ Q )
= P ) ) ) ) ) ) ).
% domain.pmod_const(2)
thf(fact_1147_domain_Opmod__const_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
=> ( ( polynomial_pmod_a_b @ R @ P @ Q )
= P ) ) ) ) ) ) ).
% domain.pmod_const(2)
thf(fact_1148_domain_Odegree__zero__imp__not__is__root,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,X: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno5142720416380192742t_unit @ R @ P @ X ) ) ) ) ).
% domain.degree_zero_imp_not_is_root
thf(fact_1149_domain_Odegree__zero__imp__not__is__root,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno6951661231331188332t_unit @ R @ P @ X ) ) ) ) ).
% domain.degree_zero_imp_not_is_root
thf(fact_1150_domain_Odegree__zero__imp__not__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,X: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4133073214067823460ot_a_b @ R @ P @ X ) ) ) ) ).
% domain.degree_zero_imp_not_is_root
thf(fact_1151_domain_Opmod__degree,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( Q != nil_list_a )
=> ( ( ( polyno1727750685288865234t_unit @ R @ P @ Q )
= nil_list_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( polyno1727750685288865234t_unit @ R @ P @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% domain.pmod_degree
thf(fact_1152_domain_Opmod__degree,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( ( polynomial_pmod_a_b @ R @ P @ Q )
= nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( polynomial_pmod_a_b @ R @ P @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% domain.pmod_degree
thf(fact_1153_domain_Opmod__const_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) )
=> ( ( polyno5893782122288709345t_unit @ R @ P @ Q )
= nil_list_a ) ) ) ) ) ) ).
% domain.pmod_const(1)
thf(fact_1154_domain_Opmod__const_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
=> ( ( polynomial_pdiv_a_b @ R @ P @ Q )
= nil_a ) ) ) ) ) ) ).
% domain.pmod_const(1)
thf(fact_1155_domain_Ouniv__poly__units_H,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) )
= ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
& ( P != nil_a )
& ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ) ).
% domain.univ_poly_units'
thf(fact_1156_domain_Ouniv__poly__units_H,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
= ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
& ( P != nil_list_a )
& ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ) ).
% domain.univ_poly_units'
thf(fact_1157_domain_Ofield__long__division__theorem__shell,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,B: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( B
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ? [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
& ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
& ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ B @ Q2 ) @ R4 ) )
& ( ( R4
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R4 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% domain.field_long_division_theorem_shell
thf(fact_1158_domain_Ofield__long__division__theorem__shell,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,B: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( B
!= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ? [Q2: list_list_a,R4: list_list_a] :
( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
& ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
& ( P
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ B @ Q2 ) @ R4 ) )
& ( ( R4
= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ R4 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ) ).
% domain.field_long_division_theorem_shell
thf(fact_1159_degree__zero__imp__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).
% degree_zero_imp_splitted
thf(fact_1160_order__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_a_ring_ext_a_b @ r ) )
= ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% order_gt_0_iff_finite
thf(fact_1161_boundD__carrier,axiom,
! [N: nat,F: nat > a,M: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_a @ ( F @ M ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_1162_ring_Osplitted_Ocong,axiom,
polyno8329700637149614481ed_a_b = polyno8329700637149614481ed_a_b ).
% ring.splitted.cong
thf(fact_1163_abelian__monoid_OboundD__carrier,axiom,
! [G: partia7496981018696276118t_unit,N: nat,F: nat > set_list_a,M: nat] :
( ( abelia3322010900105369177t_unit @ G )
=> ( ( bound_set_list_a @ ( zero_s2910681146719230829t_unit @ G ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_set_list_a @ ( F @ M ) @ ( partia141011252114345353t_unit @ G ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_1164_abelian__monoid_OboundD__carrier,axiom,
! [G: partia2175431115845679010xt_a_b,N: nat,F: nat > a,M: nat] :
( ( abelian_monoid_a_b @ G )
=> ( ( bound_a @ ( zero_a_b @ G ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_a @ ( F @ M ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_1165_abelian__monoid_OboundD__carrier,axiom,
! [G: partia2670972154091845814t_unit,N: nat,F: nat > list_a,M: nat] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ G ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_list_a @ ( F @ M ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_1166_abelian__monoid_OboundD__carrier,axiom,
! [G: partia2956882679547061052t_unit,N: nat,F: nat > list_list_a,M: nat] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( bound_list_list_a @ ( zero_l347298301471573063t_unit @ G ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_list_list_a @ ( F @ M ) @ ( partia2464479390973590831t_unit @ G ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_1167_field_Opdivides__imp__splitted,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,Q: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( Q != nil_set_list_a )
=> ( ( polyno7858167711734664505t_unit @ R @ Q )
=> ( ( polyno9075941895896075626t_unit @ R @ P @ Q )
=> ( polyno7858167711734664505t_unit @ R @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_1168_field_Opdivides__imp__splitted,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( Q != nil_list_list_a )
=> ( ( polyno5970451904377802771t_unit @ R @ Q )
=> ( ( polyno4453881341673752516t_unit @ R @ P @ Q )
=> ( polyno5970451904377802771t_unit @ R @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_1169_field_Opdivides__imp__splitted,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno6259083269128200473t_unit @ R @ Q )
=> ( ( polyno8016796738000020810t_unit @ R @ P @ Q )
=> ( polyno6259083269128200473t_unit @ R @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_1170_field_Opdivides__imp__splitted,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ R @ Q )
=> ( ( polyno5814909790663948098es_a_b @ R @ P @ Q )
=> ( polyno8329700637149614481ed_a_b @ R @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_1171_field_Odegree__one__imp__splitted,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno8329700637149614481ed_a_b @ R @ P ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_1172_poly__mult__degree__le,axiom,
! [X: list_a,Y: list_a,N: nat,M: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ M )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ N @ M ) ) ) ) ) ) ).
% poly_mult_degree_le
thf(fact_1173_poly__mult__degree__le__1,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) ) ) ) ) ).
% poly_mult_degree_le_1
thf(fact_1174_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_1175_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_1176_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_1177_add_Oint__pow__diff,axiom,
! [X: a,N: int,M: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ ( minus_minus_int @ N @ M ) @ X )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ N @ X ) @ ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ M @ X ) ) ) ) ) ).
% add.int_pow_diff
thf(fact_1178_polynomial__pow__degree,axiom,
! [P: list_a,N: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ).
% polynomial_pow_degree
thf(fact_1179_subring__polynomial__pow__degree,axiom,
! [K: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% subring_polynomial_pow_degree
thf(fact_1180_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_1181_poly__add__monom,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( poly_add_a_b @ r @ ( monom_a_b @ r @ A @ ( size_size_list_a @ P ) ) @ P )
= ( cons_a @ A @ P ) ) ) ) ).
% poly_add_monom
thf(fact_1182_pirreducible__roots,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ( ( polynomial_roots_a_b @ r @ P )
= zero_zero_multiset_a ) ) ) ) ).
% pirreducible_roots
thf(fact_1183_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_1184_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_1185_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_1186_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_1187_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_1188_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_1189_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_1190_field__intro2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) ) )
=> ( field_a_b @ r ) ) ) ).
% field_intro2
thf(fact_1191_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_1192_degree__zero__imp__empty__roots,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polynomial_roots_a_b @ r @ P )
= zero_zero_multiset_a ) ) ) ).
% degree_zero_imp_empty_roots
thf(fact_1193_add_Oint__pow__1,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ one_one_int @ X )
= X ) ) ).
% add.int_pow_1
thf(fact_1194_euclidean__domainI,axiom,
! [Phi: a > nat] :
( ! [A3: a,B2: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ? [Q3: a,R5: a] :
( ( member_a @ Q3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ R5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( A3
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ B2 @ Q3 ) @ R5 ) )
& ( ( R5
= ( zero_a_b @ r ) )
| ( ord_less_nat @ ( Phi @ R5 ) @ ( Phi @ B2 ) ) ) ) ) )
=> ( ring_e8745995371659049232in_a_b @ r @ Phi ) ) ).
% euclidean_domainI
thf(fact_1195_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_1196_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_1197_primeideal__iff__prime,axiom,
! [P: a] :
( ( member_a @ P @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P ) @ r )
= ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% primeideal_iff_prime
thf(fact_1198_roots__inclI,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A3 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P @ A3 ) ) @ Q ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).
% roots_inclI
thf(fact_1199_maximalideal__prime,axiom,
! [I2: set_a] :
( ( maximalideal_a_b @ I2 @ r )
=> ( primeideal_a_b @ I2 @ r ) ) ).
% maximalideal_prime
thf(fact_1200_zeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroprimeideal
thf(fact_1201_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_1202_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_1203_pdivides__imp__roots__incl,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).
% pdivides_imp_roots_incl
thf(fact_1204_not__empty__rootsE,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( polynomial_roots_a_b @ r @ P )
!= zero_zero_multiset_a )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( set_mset_a @ ( polynomial_roots_a_b @ r @ P ) ) )
=> ( ( member_list_a @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A3 ) @ nil_a ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ~ ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A3 ) @ nil_a ) ) @ P ) ) ) ) ) ) ).
% not_empty_rootsE
thf(fact_1205_poly__mult__monom,axiom,
! [P: list_a,A: a,N: nat] :
( ( polynomial_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ P )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( ( P = nil_a )
=> ( ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P )
= nil_a ) )
& ( ( P != nil_a )
=> ( ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P )
= ( append_a @ ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ P ) @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ) ) ) ) ).
% poly_mult_monom
thf(fact_1206_normalize__polynomial,axiom,
! [K: set_a,P: list_a] :
( ( polynomial_a_b @ r @ K @ P )
=> ( ( normalize_a_b @ r @ P )
= P ) ) ).
% normalize_polynomial
thf(fact_1207_polynomial__incl,axiom,
! [K: set_a,P: list_a] :
( ( polynomial_a_b @ r @ K @ P )
=> ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K ) ) ).
% polynomial_incl
thf(fact_1208_poly__mult__closed,axiom,
! [K: set_a,P1: list_a,P22: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P1 )
=> ( ( polynomial_a_b @ r @ K @ P22 )
=> ( polynomial_a_b @ r @ K @ ( poly_mult_a_b @ r @ P1 @ P22 ) ) ) ) ) ).
% poly_mult_closed
thf(fact_1209_poly__add__closed,axiom,
! [K: set_a,P1: list_a,P22: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P1 )
=> ( ( polynomial_a_b @ r @ K @ P22 )
=> ( polynomial_a_b @ r @ K @ ( poly_add_a_b @ r @ P1 @ P22 ) ) ) ) ) ).
% poly_add_closed
thf(fact_1210_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_1211_normalize__gives__polynomial,axiom,
! [P: list_a,K: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K )
=> ( polynomial_a_b @ r @ K @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_gives_polynomial
thf(fact_1212_eval__poly__in__carrier,axiom,
! [K: set_a,P: list_a,X: a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% eval_poly_in_carrier
thf(fact_1213_poly__mult__integral,axiom,
! [K: set_a,P1: list_a,P22: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P1 )
=> ( ( polynomial_a_b @ r @ K @ P22 )
=> ( ( ( poly_mult_a_b @ r @ P1 @ P22 )
= nil_a )
=> ( ( P1 = nil_a )
| ( P22 = nil_a ) ) ) ) ) ) ).
% poly_mult_integral
thf(fact_1214_poly__add__zero_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_add_a_b @ r @ nil_a @ P )
= P ) ) ) ).
% poly_add_zero(2)
thf(fact_1215_poly__add__zero_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_add_a_b @ r @ P @ nil_a )
= P ) ) ) ).
% poly_add_zero(1)
thf(fact_1216_poly__mult__r__distr,axiom,
! [K: set_a,P1: list_a,P22: list_a,P32: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P1 )
=> ( ( polynomial_a_b @ r @ K @ P22 )
=> ( ( polynomial_a_b @ r @ K @ P32 )
=> ( ( poly_mult_a_b @ r @ P1 @ ( poly_add_a_b @ r @ P22 @ P32 ) )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P22 ) @ ( poly_mult_a_b @ r @ P1 @ P32 ) ) ) ) ) ) ) ).
% poly_mult_r_distr
thf(fact_1217_poly__mult__l__distr,axiom,
! [K: set_a,P1: list_a,P22: list_a,P32: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P1 )
=> ( ( polynomial_a_b @ r @ K @ P22 )
=> ( ( polynomial_a_b @ r @ K @ P32 )
=> ( ( poly_mult_a_b @ r @ ( poly_add_a_b @ r @ P1 @ P22 ) @ P32 )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P32 ) @ ( poly_mult_a_b @ r @ P22 @ P32 ) ) ) ) ) ) ) ).
% poly_mult_l_distr
thf(fact_1218_poly__mult__is__polynomial,axiom,
! [K: set_a,P1: list_a,P22: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ K )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ K )
=> ( polynomial_a_b @ r @ K @ ( poly_mult_a_b @ r @ P1 @ P22 ) ) ) ) ) ).
% poly_mult_is_polynomial
thf(fact_1219_poly__add__is__polynomial,axiom,
! [K: set_a,P1: list_a,P22: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ K )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ K )
=> ( polynomial_a_b @ r @ K @ ( poly_add_a_b @ r @ P1 @ P22 ) ) ) ) ) ).
% poly_add_is_polynomial
thf(fact_1220_poly__add__replicate__zero_I2_J,axiom,
! [K: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_add_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= P ) ) ) ).
% poly_add_replicate_zero(2)
thf(fact_1221_poly__add__replicate__zero_I1_J,axiom,
! [K: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_add_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= P ) ) ) ).
% poly_add_replicate_zero(1)
thf(fact_1222_lead__coeff__not__zero,axiom,
! [K: set_a,A: a,P: list_a] :
( ( polynomial_a_b @ r @ K @ ( cons_a @ A @ P ) )
=> ( member_a @ A @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% lead_coeff_not_zero
thf(fact_1223_poly__mult__one_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_mult_a_b @ r @ P @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) )
= P ) ) ) ).
% poly_mult_one(2)
thf(fact_1224_poly__mult__one_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_mult_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) @ P )
= P ) ) ) ).
% poly_mult_one(1)
thf(fact_1225_append__is__polynomial,axiom,
! [K: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( P != nil_a )
=> ( polynomial_a_b @ r @ K @ ( append_a @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ) ) ) ).
% append_is_polynomial
thf(fact_1226_poly__mult__append__zero__lcancel,axiom,
! [K: set_a,P: list_a,Q: list_a,R3: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( polynomial_a_b @ r @ K @ Q )
=> ( ( ( poly_mult_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Q )
= ( append_a @ R3 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
=> ( ( poly_mult_a_b @ r @ P @ Q )
= R3 ) ) ) ) ) ).
% poly_mult_append_zero_lcancel
thf(fact_1227_poly__mult__append__zero__rcancel,axiom,
! [K: set_a,P: list_a,Q: list_a,R3: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( polynomial_a_b @ r @ K @ Q )
=> ( ( ( poly_mult_a_b @ r @ P @ ( append_a @ Q @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
= ( append_a @ R3 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
=> ( ( poly_mult_a_b @ r @ P @ Q )
= R3 ) ) ) ) ) ).
% poly_mult_append_zero_rcancel
thf(fact_1228_const__term__zero,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= ( zero_a_b @ r ) )
=> ~ ! [P3: list_a] :
( ( polynomial_a_b @ r @ K @ P3 )
=> ( ( P3 != nil_a )
=> ( P
!= ( append_a @ P3 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ) ) ).
% const_term_zero
thf(fact_1229_roots__mem__iff__is__root,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a @ X @ ( set_mset_a @ ( polynomial_roots_a_b @ r @ P ) ) )
= ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ).
% roots_mem_iff_is_root
thf(fact_1230_lead__coeff__in__carrier,axiom,
! [K: set_a,A: a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ ( cons_a @ A @ P ) )
=> ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ).
% lead_coeff_in_carrier
thf(fact_1231_poly__mult__degree__eq,axiom,
! [K: set_a,P1: list_a,P22: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P1 )
=> ( ( polynomial_a_b @ r @ K @ P22 )
=> ( ( ( ( P1 = nil_a )
| ( P22 = nil_a ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( poly_mult_a_b @ r @ P1 @ P22 ) ) @ one_one_nat )
= zero_zero_nat ) )
& ( ~ ( ( P1 = nil_a )
| ( P22 = nil_a ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( poly_mult_a_b @ r @ P1 @ P22 ) ) @ one_one_nat )
= ( plus_plus_nat @ ( minus_minus_nat @ ( size_size_list_a @ P1 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ P22 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% poly_mult_degree_eq
thf(fact_1232_field__long__division__theorem,axiom,
! [K: set_a,P: list_a,B: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( polynomial_a_b @ r @ K @ B )
=> ( ( B != nil_a )
=> ? [Q2: list_a,R4: list_a] :
( ( polynomial_a_b @ r @ K @ Q2 )
& ( polynomial_a_b @ r @ K @ R4 )
& ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ B @ Q2 ) @ R4 ) )
& ( ( R4 = nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R4 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% field_long_division_theorem
thf(fact_1233_zero__is__polynomial,axiom,
! [K: set_a] : ( polynomial_a_b @ r @ K @ nil_a ) ).
% zero_is_polynomial
thf(fact_1234_carrier__polynomial,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( polynomial_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ P ) ) ) ).
% carrier_polynomial
thf(fact_1235_polynomial__in__carrier,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% polynomial_in_carrier
thf(fact_1236_one__is__polynomial,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( polynomial_a_b @ r @ K @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) ) ) ).
% one_is_polynomial
thf(fact_1237_const__is__polynomial,axiom,
! [A: a,K: set_a] :
( ( member_a @ A @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( polynomial_a_b @ r @ K @ ( cons_a @ A @ nil_a ) ) ) ).
% const_is_polynomial
thf(fact_1238_monom__is__polynomial,axiom,
! [K: set_a,A: a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( polynomial_a_b @ r @ K @ ( monom_a_b @ r @ A @ N ) ) ) ) ).
% monom_is_polynomial
thf(fact_1239_zle__add1__eq__le,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1240_zmult__zless__mono2,axiom,
! [I: int,J: int,K2: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ord_less_int @ ( times_times_int @ K2 @ I ) @ ( times_times_int @ K2 @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1241_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1242_times__int__code_I1_J,axiom,
! [K2: int] :
( ( times_times_int @ K2 @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1243_plus__int__code_I1_J,axiom,
! [K2: int] :
( ( plus_plus_int @ K2 @ zero_zero_int )
= K2 ) ).
% plus_int_code(1)
thf(fact_1244_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1245_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1246_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1247_zless__add1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z )
| ( W2 = Z ) ) ) ).
% zless_add1_eq
thf(fact_1248_int__gr__induct,axiom,
! [K2: int,I: int,P2: int > $o] :
( ( ord_less_int @ K2 @ I )
=> ( ( P2 @ ( plus_plus_int @ K2 @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K2 @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1249_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1250_int__distrib_I2_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1251_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W2 )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(1)
thf(fact_1252_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1253_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_1254_int__ge__induct,axiom,
! [K2: int,I: int,P2: int > $o] :
( ( ord_less_eq_int @ K2 @ I )
=> ( ( P2 @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1255_add1__zle__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z )
= ( ord_less_int @ W2 @ Z ) ) ).
% add1_zle_eq
thf(fact_1256_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1257_zless__imp__add1__zle,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1258_int__induct,axiom,
! [P2: int > $o,K2: int,I: int] :
( ( P2 @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K2 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_induct
thf(fact_1259_associated__polynomials__iff,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
& ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ X2 @ nil_a ) @ Q ) ) ) ) ) ) ) ) ).
% associated_polynomials_iff
thf(fact_1260_associated__polynomials__imp__same__length,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
=> ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q ) ) ) ) ) ) ).
% associated_polynomials_imp_same_length
thf(fact_1261_associated__polynomials__imp__same__roots,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q )
=> ( ( polynomial_roots_a_b @ r @ P )
= ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ).
% associated_polynomials_imp_same_roots
thf(fact_1262_associated__polynomials__imp__same__is__root,axiom,
! [P: list_a,Q: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
= ( polyno4133073214067823460ot_a_b @ r @ Q @ X ) ) ) ) ) ).
% associated_polynomials_imp_same_is_root
thf(fact_1263_cgenideal__pirreducible,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ Q )
=> ( ( member_list_a @ Q @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) ) ) ) ) ) ).
% cgenideal_pirreducible
thf(fact_1264_subring__degree__one__associatedI,axiom,
! [K: set_a,A: a,A5: a,B: a] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A @ K )
=> ( ( member_a @ A5 @ K )
=> ( ( member_a @ B @ K )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ A5 )
= ( 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 @ A5 @ B ) @ nil_a ) ) ) ) ) ) ) ) ).
% subring_degree_one_associatedI
thf(fact_1265_associated__sym,axiom,
! [A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( associ5860276527279195403xt_a_b @ r @ B @ A ) ) ).
% associated_sym
thf(fact_1266_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_1267_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_1268_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_1269_mult__cong__r,axiom,
! [B: a,B3: a,A: a] :
( ( associ5860276527279195403xt_a_b @ r @ B @ B3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B3 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_1270_mult__cong__l,axiom,
! [A: a,A5: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ A5 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A5 @ B ) ) ) ) ) ) ).
% mult_cong_l
thf(fact_1271_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_1272_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_1273_ring__associated__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
& ( A
= ( mult_a_ring_ext_a_b @ r @ X2 @ B ) ) ) ) ) ) ) ).
% ring_associated_iff
thf(fact_1274_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_1275_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_1276_associated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ A ) ) ).
% associated_refl
% Conjectures (1)
thf(conj_0,conjecture,
cring_3148771470849435808t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
%------------------------------------------------------------------------------