TPTP Problem File: SLH0090^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Finite_Fields/0006_Formal_Polynomial_Derivatives/prob_00394_013603__18280272_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1642 ( 234 unt; 364 typ; 0 def)
% Number of atoms : 3704 (1079 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 15410 ( 146 ~; 15 |; 73 &;13162 @)
% ( 0 <=>;2014 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 38 ( 37 usr)
% Number of type conns : 1358 (1358 >; 0 *; 0 +; 0 <<)
% Number of symbols : 330 ( 327 usr; 16 con; 0-4 aty)
% Number of variables : 3049 ( 280 ^;2687 !; 82 ?;3049 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:22:13.214
%------------------------------------------------------------------------------
% Could-be-implicit typings (37)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia5333488208502193986t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia6652711048488803900t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia2956882679547061052t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_It__Int__Oint_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__Int__Oint_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J_J_J,type,
partia4934656038542163276t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_Itf__a_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_J_J,type,
partia2670972154091845814t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J_J,type,
partia2175431115845679010xt_a_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_li6773872926390105121list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_li3422455791611400469list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_nat_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Int__Oint_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_set_int_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
set_list_a_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_list_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_a_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_list_a_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mtf__a_J_J,type,
set_list_list_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
set_nat_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
set_list_set_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
set_nat_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
set_a_set_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Int__Oint_J_Mtf__a_J_J,type,
set_set_int_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
list_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
set_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
set_a_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
set_list_a_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
list_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
set_nat_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (327)
thf(sy_c_AbelCoset_Oa__l__coset_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
a_l_co3970804650394549132t_unit: partia2956882679547061052t_unit > list_list_a > set_list_list_a > set_list_list_a ).
thf(sy_c_AbelCoset_Oa__l__coset_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
a_l_co7008843373686234386t_unit: partia2670972154091845814t_unit > list_a > set_list_a > set_list_a ).
thf(sy_c_AbelCoset_Oa__l__coset_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
a_l_co3504123944629134560t_unit: partia4934656038542163276t_unit > set_int > set_set_int > set_set_int ).
thf(sy_c_AbelCoset_Oa__l__coset_001tf__a_001tf__b,type,
a_l_coset_a_b: partia2175431115845679010xt_a_b > a > set_a > set_a ).
thf(sy_c_AbelCoset_Oabelian__group__hom_001tf__a_001tf__b_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
abelia5726065498384912733t_unit: partia2175431115845679010xt_a_b > partia2670972154091845814t_unit > ( a > list_a ) > $o ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J_J,type,
partia5038748322285217333t_unit: partia5333488208502193986t_unit > set_list_list_list_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J,type,
partia2464479390973590831t_unit: partia2956882679547061052t_unit > set_list_list_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J_J,type,
partia5243588073521899067t_unit: partia6652711048488803900t_unit > set_list_set_int ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_Itf__a_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_J,type,
partia5361259788508890537t_unit: partia2670972154091845814t_unit > set_list_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Set__Oset_It__Int__Oint_J_001t__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__Int__Oint_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J_J,type,
partia966996272515721803t_unit: partia4934656038542163276t_unit > set_set_int ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
partia707051561876973205xt_a_b: partia2175431115845679010xt_a_b > set_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier__update_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J,type,
partia5725451877363000582t_unit: ( set_list_list_a > set_list_list_a ) > partia2956882679547061052t_unit > partia2956882679547061052t_unit ).
thf(sy_c_Congruence_Opartial__object_Ocarrier__update_001t__List__Olist_Itf__a_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_J,type,
partia9041243232023819264t_unit: ( set_list_a > set_list_a ) > partia2670972154091845814t_unit > partia2670972154091845814t_unit ).
thf(sy_c_Congruence_Opartial__object_Ocarrier__update_001t__Set__Oset_It__Int__Oint_J_001t__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__Int__Oint_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J_J,type,
partia2914870419229397556t_unit: ( set_set_int > set_set_int ) > partia4934656038542163276t_unit > partia4934656038542163276t_unit ).
thf(sy_c_Congruence_Opartial__object_Ocarrier__update_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
partia8674076737563717228xt_a_b: ( set_a > set_a ) > partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b ).
thf(sy_c_Coset_Oorder_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
order_3240872107759947550t_unit: partia2670972154091845814t_unit > nat ).
thf(sy_c_Coset_Oorder_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
order_a_ring_ext_a_b: partia2175431115845679010xt_a_b > nat ).
thf(sy_c_Divisibility_Oassociated_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
associ5860276527279195403xt_a_b: partia2175431115845679010xt_a_b > a > a > $o ).
thf(sy_c_Divisibility_Omonoid__cancel_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
monoid4303264861975686087t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Divisibility_Oprime_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
prime_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Ofinite__dimension_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
embedd1345800358437254783t_unit: partia2670972154091845814t_unit > set_list_a > set_list_a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Ofinite__dimension_001tf__a_001tf__b,type,
embedd8708762675212832759on_a_b: partia2175431115845679010xt_a_b > set_a > set_a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
embedd5150658419831591667t_unit: partia2670972154091845814t_unit > set_list_a > list_a > set_list_a > set_list_a ).
thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001tf__a_001tf__b,type,
embedd971793762689825387on_a_b: partia2175431115845679010xt_a_b > set_a > a > set_a > set_a ).
thf(sy_c_Embedded__Algebras_Osubalgebra_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
embedd1768981623711841426t_unit: set_list_a > set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Embedded__Algebras_Osubalgebra_001tf__a_001tf__b,type,
embedd9027525575939734154ra_a_b: set_a > set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
finite1660835950917165235list_a: set_list_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
finite6197958912794628473et_int: set_set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
inj_on7005093563047325836list_a: ( int > nat > list_a ) > set_int > $o ).
thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001_062_It__Nat__Onat_Mtf__a_J,type,
inj_on_int_nat_a: ( int > nat > a ) > set_int > $o ).
thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001_062_It__Set__Oset_It__Int__Oint_J_Mtf__a_J,type,
inj_on_int_set_int_a: ( int > set_int > a ) > set_int > $o ).
thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Int__Oint,type,
inj_on_int_int: ( int > int ) > set_int > $o ).
thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__List__Olist_Itf__a_J,type,
inj_on_int_list_a: ( int > list_a ) > set_int > $o ).
thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001tf__a,type,
inj_on_int_a: ( int > a ) > set_int > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J,type,
inj_on2455837796590618731list_a: ( list_list_a > list_a ) > set_list_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a,type,
inj_on_list_list_a_a: ( list_list_a > a ) > set_list_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
inj_on4494935751713688664list_a: ( list_a > nat > list_a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001_062_It__Nat__Onat_Mtf__a_J,type,
inj_on_list_a_nat_a: ( list_a > nat > a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001_062_It__Set__Oset_It__Int__Oint_J_Mtf__a_J,type,
inj_on4987226016813664192_int_a: ( list_a > set_int > a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
inj_on7824993887354538302list_a: ( list_a > a > list_a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__Int__Oint,type,
inj_on_list_a_int: ( list_a > int ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
inj_on6538719745830000247list_a: ( list_a > list_list_a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
inj_on_list_a_list_a: ( list_a > list_a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001tf__a,type,
inj_on_list_a_a: ( list_a > a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Int__Oint,type,
inj_on_nat_int: ( nat > int ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
inj_on_nat_list_a: ( nat > list_a ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
inj_on_nat_set_int: ( nat > set_int ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001tf__a,type,
inj_on_nat_a: ( nat > a ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Int__Oint_J_001t__List__Olist_Itf__a_J,type,
inj_on8112305350124607983list_a: ( set_int > list_a ) > set_set_int > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Int__Oint_J_001tf__a,type,
inj_on_set_int_a: ( set_int > a ) > set_set_int > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
inj_on_a_nat_a: ( a > nat > a ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Int__Oint,type,
inj_on_a_int: ( a > int ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__List__Olist_Itf__a_J,type,
inj_on_a_list_a: ( a > list_a ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Group_Omonoid_Omult_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
mult_l7073676228092353617t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
mult_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Group_Omonoid_Oone_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
one_li8234411390022467901t_unit: partia2956882679547061052t_unit > list_list_a ).
thf(sy_c_Group_Omonoid_Oone_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
one_li8328186300101108157t_unit: partia2670972154091845814t_unit > list_a ).
thf(sy_c_Group_Omonoid_Oone_001t__Set__Oset_It__Int__Oint_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J,type,
one_se8065767436706823081t_unit: partia4934656038542163276t_unit > set_int ).
thf(sy_c_Group_Omonoid_Oone_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
one_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Group_Omonoid_Oone__update_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
one_up8845488168893825172t_unit: ( list_a > list_a ) > partia2670972154091845814t_unit > partia2670972154091845814t_unit ).
thf(sy_c_Group_Omonoid_Oone__update_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
one_up5219539153869309446xt_a_b: ( a > a ) > partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b ).
thf(sy_c_Group_Opow_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_001t__Nat__Onat,type,
pow_li488931774710091566it_nat: partia2956882679547061052t_unit > list_list_a > nat > list_list_a ).
thf(sy_c_Group_Opow_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_001t__Nat__Onat,type,
pow_li1142815632869257134it_nat: partia2670972154091845814t_unit > list_a > nat > list_a ).
thf(sy_c_Group_Opow_001t__Set__Oset_It__Int__Oint_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J_001t__Nat__Onat,type,
pow_se2518650051167492506it_nat: partia4934656038542163276t_unit > set_int > nat > set_int ).
thf(sy_c_Group_Opow_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
pow_a_1026414303147256608_b_nat: partia2175431115845679010xt_a_b > a > nat > a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
minus_646659088055828811list_a: set_list_a > set_list_a > set_list_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
hilber6777891363991549349at_a_a: set_nat_a > ( ( nat > a ) > a ) > a > nat > a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Int__Oint_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
hilber6310135974011704177list_a: set_int > ( int > nat > list_a ) > ( nat > list_a ) > int ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Int__Oint_001_062_It__Nat__Onat_Mtf__a_J,type,
hilber1943333713353564907_nat_a: set_int > ( int > nat > a ) > ( nat > a ) > int ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Int__Oint_001_062_It__Set__Oset_It__Int__Oint_J_Mtf__a_J,type,
hilber3251774746670828377_int_a: set_int > ( int > set_int > a ) > ( set_int > a ) > int ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Int__Oint_001t__Int__Oint,type,
hilber4630740438190819238nt_int: set_int > ( int > int ) > int > int ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Int__Oint_001t__List__Olist_Itf__a_J,type,
hilber7218023918483505546list_a: set_int > ( int > list_a ) > list_a > int ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Int__Oint_001tf__a,type,
hilber550346152434376708_int_a: set_int > ( int > a ) > a > int ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J,type,
hilber2546232707058280006list_a: set_list_list_a > ( list_list_a > list_a ) > list_a > list_list_a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__List__Olist_Itf__a_J_001t__Int__Oint,type,
hilber854936315069205348_a_int: set_list_a > ( list_a > int ) > int > list_a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
hilber6629114656297661522list_a: set_list_a > ( list_a > list_list_a ) > list_list_a > list_a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
hilber4441511847348742220list_a: set_list_a > ( list_a > list_a ) > list_a > list_a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__List__Olist_Itf__a_J_001t__Set__Oset_It__Int__Oint_J,type,
hilber3467374374478178586et_int: set_list_a > ( list_a > set_int ) > set_int > list_a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__List__Olist_Itf__a_J_001tf__a,type,
hilber8556971080906182598st_a_a: set_list_a > ( list_a > a ) > a > list_a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Set__Oset_It__Int__Oint_J_001t__List__Olist_Itf__a_J,type,
hilber3792369929691227220list_a: set_set_int > ( set_int > list_a ) > list_a > set_int ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Set__Oset_It__Int__Oint_J_001tf__a,type,
hilber773427641517404110_int_a: set_set_int > ( set_int > a ) > a > set_int ).
thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
hilber8476257288600951993list_a: set_a > ( a > nat > list_a ) > ( nat > list_a ) > a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
hilber1523447219144142899_nat_a: set_a > ( a > nat > a ) > ( nat > a ) > a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001_062_It__Set__Oset_It__Int__Oint_J_Mtf__a_J,type,
hilber9111489492337244321_int_a: set_a > ( a > set_int > a ) > ( set_int > a ) > a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
hilber4343350721326501661list_a: set_a > ( a > a > list_a ) > ( a > list_a ) > a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001t__Int__Oint,type,
hilber7984441185272261726_a_int: set_a > ( a > int ) > int > a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001t__List__Olist_Itf__a_J,type,
hilber1222834799880672978list_a: set_a > ( a > list_a ) > list_a > a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001t__Set__Oset_It__Int__Oint_J,type,
hilber386172014519280148et_int: set_a > ( a > set_int ) > set_int > a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001tf__a,type,
hilbert_inv_into_a_a: set_a > ( a > a ) > a > a ).
thf(sy_c_Ideal_Ogenideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
genide3243992037924705879t_unit: partia2670972154091845814t_unit > set_list_a > set_list_a ).
thf(sy_c_Ideal_Omaximalideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
maxima6585700282301356660t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Ideal_Omaximalideal_001tf__a_001tf__b,type,
maximalideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ideal_Oprimeideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
primei6309817859076077608t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Ideal_Oprimeideal_001tf__a_001tf__b,type,
primeideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ideal_Oprincipalideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
princi8786919440553033881t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Ideal_Oprincipalideal_001tf__a_001tf__b,type,
principalideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_If_001t__List__Olist_Itf__a_J,type,
if_list_a: $o > list_a > list_a > list_a ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_IntRing_OZFact,type,
zFact: int > partia4934656038542163276t_unit ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bot_bot_set_list_a: set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
ord_le5338140678153942166list_a: set_list_list_a > set_list_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_less_set_list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
ord_le8488217952732425610list_a: set_list_list_a > set_list_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
top_top_set_int: set_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
top_to6581514389725680442list_a: set_list_list_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
top_top_set_list_a: set_list_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Polynomials_Oring_Opoly__of__const_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
poly_o1617770581224298896t_unit: partia2956882679547061052t_unit > list_list_a > list_list_list_a ).
thf(sy_c_Polynomials_Oring_Opoly__of__const_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
poly_o8716471131768098070t_unit: partia2670972154091845814t_unit > list_a > list_list_a ).
thf(sy_c_Polynomials_Oring_Opoly__of__const_001tf__a_001tf__b,type,
poly_of_const_a_b: partia2175431115845679010xt_a_b > a > list_a ).
thf(sy_c_Polynomials_Ouniv__poly_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
univ_p2250591967980070728t_unit: partia2956882679547061052t_unit > set_list_list_a > partia5333488208502193986t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
univ_p7953238456130426574t_unit: partia2670972154091845814t_unit > set_list_a > partia2956882679547061052t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
univ_p8160232691235018276t_unit: partia4934656038542163276t_unit > set_set_int > partia6652711048488803900t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001tf__a_001tf__b,type,
univ_poly_a_b: partia2175431115845679010xt_a_b > set_a > partia2670972154091845814t_unit ).
thf(sy_c_QuotRing_Oring__iso_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_i4611353245267337884t_unit: partia2956882679547061052t_unit > partia2670972154091845814t_unit > set_li3422455791611400469list_a ).
thf(sy_c_QuotRing_Oring__iso_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_i5684343068699926420it_a_b: partia2956882679547061052t_unit > partia2175431115845679010xt_a_b > set_list_list_a_a ).
thf(sy_c_QuotRing_Oring__iso_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_i7414513579304222626t_unit: partia2670972154091845814t_unit > partia2670972154091845814t_unit > set_list_a_list_a ).
thf(sy_c_QuotRing_Oring__iso_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_i5545097000213460560t_unit: partia2670972154091845814t_unit > partia4934656038542163276t_unit > set_list_a_set_int ).
thf(sy_c_QuotRing_Oring__iso_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_i7048835797181109658it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > set_list_a_a ).
thf(sy_c_QuotRing_Oring__iso_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_i8159631783994945698t_unit: partia4934656038542163276t_unit > partia2670972154091845814t_unit > set_set_int_list_a ).
thf(sy_c_QuotRing_Oring__iso_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_i2213127016525301402it_a_b: partia4934656038542163276t_unit > partia2175431115845679010xt_a_b > set_set_int_a ).
thf(sy_c_QuotRing_Oring__iso_001tf__a_001tf__b_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_i4464730343205239444t_unit: partia2175431115845679010xt_a_b > partia2956882679547061052t_unit > set_a_list_list_a ).
thf(sy_c_QuotRing_Oring__iso_001tf__a_001tf__b_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_i4557880751517319194t_unit: partia2175431115845679010xt_a_b > partia2670972154091845814t_unit > set_a_list_a ).
thf(sy_c_QuotRing_Oring__iso_001tf__a_001tf__b_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_i4118987378922720216t_unit: partia2175431115845679010xt_a_b > partia4934656038542163276t_unit > set_a_set_int ).
thf(sy_c_QuotRing_Oring__iso_001tf__a_001tf__b_001tf__a_001tf__b,type,
ring_iso_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > set_a_a ).
thf(sy_c_RingHom_Oring__hom__ring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_h3119652981342401090t_unit: partia2956882679547061052t_unit > partia2956882679547061052t_unit > ( list_list_a > list_list_a ) > $o ).
thf(sy_c_RingHom_Oring__hom__ring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h4589914651911841480t_unit: partia2956882679547061052t_unit > partia2670972154091845814t_unit > ( list_list_a > list_a ) > $o ).
thf(sy_c_RingHom_Oring__hom__ring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h3841606220870141376it_a_b: partia2956882679547061052t_unit > partia2175431115845679010xt_a_b > ( list_list_a > a ) > $o ).
thf(sy_c_RingHom_Oring__hom__ring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_h7560679385066609224t_unit: partia2670972154091845814t_unit > partia2956882679547061052t_unit > ( list_a > list_list_a ) > $o ).
thf(sy_c_RingHom_Oring__hom__ring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h1334922693953046990t_unit: partia2670972154091845814t_unit > partia2670972154091845814t_unit > ( list_a > list_a ) > $o ).
thf(sy_c_RingHom_Oring__hom__ring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h7848885096329822662it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > ( list_a > a ) > $o ).
thf(sy_c_RingHom_Oring__hom__ring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h3064273570435554678t_unit: partia4934656038542163276t_unit > partia2670972154091845814t_unit > ( set_int > list_a ) > $o ).
thf(sy_c_RingHom_Oring__hom__ring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h5360235766848426350it_a_b: partia4934656038542163276t_unit > partia2175431115845679010xt_a_b > ( set_int > a ) > $o ).
thf(sy_c_RingHom_Oring__hom__ring_001tf__a_001tf__b_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_h2621993495375454400t_unit: partia2175431115845679010xt_a_b > partia2956882679547061052t_unit > ( a > list_list_a ) > $o ).
thf(sy_c_RingHom_Oring__hom__ring_001tf__a_001tf__b_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h5357930050666032198t_unit: partia2175431115845679010xt_a_b > partia2670972154091845814t_unit > ( a > list_a ) > $o ).
thf(sy_c_RingHom_Oring__hom__ring_001tf__a_001tf__b_001tf__a_001tf__b,type,
ring_h4024360765257340990_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > ( a > a ) > $o ).
thf(sy_c_Ring_Oa__inv_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
a_inv_5142495083975434441t_unit: partia5333488208502193986t_unit > list_list_list_a > list_list_list_a ).
thf(sy_c_Ring_Oa__inv_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
a_inv_7033018035630854991t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a ).
thf(sy_c_Ring_Oa__inv_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
a_inv_7608866662416757709t_unit: partia6652711048488803900t_unit > list_set_int > list_set_int ).
thf(sy_c_Ring_Oa__inv_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
a_inv_8944721093294617173t_unit: partia2670972154091845814t_unit > list_a > list_a ).
thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
a_inv_a_b: partia2175431115845679010xt_a_b > a > a ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
a_minu4820293213911669576t_unit: partia5333488208502193986t_unit > list_list_list_a > list_list_list_a > list_list_list_a ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
a_minu2241224857956505934t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a > list_list_a ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
a_minu3289003089969826958t_unit: partia6652711048488803900t_unit > list_set_int > list_set_int > list_set_int ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
a_minu3984020753470702548t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Ring_Oa__minus_001tf__a_001tf__b,type,
a_minus_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oabelian__group_001tf__a_001tf__b,type,
abelian_group_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oabelian__monoid_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
abelia3641329199688042803t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring_Oabelian__monoid_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
abelia226231641709521465t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Oabelian__monoid_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
abelia3815030880812984441t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Oabelian__monoid_001tf__a_001tf__b,type,
abelian_monoid_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oadd__pow_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__Int__Oint,type,
add_po2638046716968164713it_int: partia2670972154091845814t_unit > int > list_a > list_a ).
thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Int__Oint,type,
add_pow_a_b_int: partia2175431115845679010xt_a_b > int > a > a ).
thf(sy_c_Ring_Ocring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
cring_5991999922451032090t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring_Ocring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
cring_3148771470849435808t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Ocring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
cring_3079150759069666002t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Ocring_001tf__a_001tf__b,type,
cring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Odomain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
domain7810152921033798211t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
domain6553523120543210313t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
domain6183376680155302761t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Odomain_001tf__a_001tf__b,type,
domain_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Ofield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
field_6388047844668329575t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Ofield_001tf__a_001tf__b,type,
field_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_l1939023646219158831t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring_Oring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_l6212528067271185461t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Oring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_s5316885176909347197t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Oring_001tf__a_001tf__b,type,
ring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
add_li174743652000525320t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a > list_list_a ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
add_li7652885771158616974t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Ring_Oring_Oadd_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
add_se5859248395121729892t_unit: partia4934656038542163276t_unit > set_int > set_int > set_int ).
thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
add_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
zero_l347298301471573063t_unit: partia2956882679547061052t_unit > list_list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
zero_l4142658623432671053t_unit: partia2670972154091845814t_unit > list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
zero_s6269048424454532197t_unit: partia4934656038542163276t_unit > set_int ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring_Oring_Ozero__update_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
zero_u1196785550890449590t_unit: ( list_a > list_a ) > partia2670972154091845814t_unit > partia2670972154091845814t_unit ).
thf(sy_c_Ring_Oring_Ozero__update_001tf__a_001tf__b,type,
zero_update_a_b: ( a > a ) > partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h5031276006722532742t_unit: partia2956882679547061052t_unit > partia2670972154091845814t_unit > set_li3422455791611400469list_a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h8078271382950527358it_a_b: partia2956882679547061052t_unit > partia2175431115845679010xt_a_b > set_list_list_a_a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_h8002040739877300486t_unit: partia2670972154091845814t_unit > partia2956882679547061052t_unit > set_li6773872926390105121list_a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h7399960747407462284t_unit: partia2670972154091845814t_unit > partia2670972154091845814t_unit > set_list_a_list_a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h2895973938487309444it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > set_list_a_a ).
thf(sy_c_Ring_Oring__hom_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h8006428831458247992t_unit: partia4934656038542163276t_unit > partia2670972154091845814t_unit > set_set_int_list_a ).
thf(sy_c_Ring_Oring__hom_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h6431528728137583664it_a_b: partia4934656038542163276t_unit > partia2175431115845679010xt_a_b > set_set_int_a ).
thf(sy_c_Ring_Oring__hom_001tf__a_001tf__b_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h405018892823518980t_unit: partia2175431115845679010xt_a_b > partia2670972154091845814t_unit > set_a_list_a ).
thf(sy_c_Ring_Oring__hom_001tf__a_001tf__b_001tf__a_001tf__b,type,
ring_hom_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > set_a_a ).
thf(sy_c_Ring_Osemiring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
semiri2871908745932252451t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
semiring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Characteristic_Ochar_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_c667363890163664752t_unit: partia2956882679547061052t_unit > nat ).
thf(sy_c_Ring__Characteristic_Ochar_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_c500279861223467766t_unit: partia2670972154091845814t_unit > nat ).
thf(sy_c_Ring__Characteristic_Ochar_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_c6147214092195050492t_unit: partia4934656038542163276t_unit > nat ).
thf(sy_c_Ring__Characteristic_Ochar_001tf__a_001tf__b,type,
ring_char_a_b: partia2175431115845679010xt_a_b > nat ).
thf(sy_c_Ring__Characteristic_Ochar__iso_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_c7096748559348378049t_unit: partia2670972154091845814t_unit > set_int > list_a ).
thf(sy_c_Ring__Characteristic_Ochar__iso_001tf__a_001tf__b,type,
ring_char_iso_a_b: partia2175431115845679010xt_a_b > set_int > a ).
thf(sy_c_Ring__Characteristic_Oint__embed_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_i3137826187831172292t_unit: partia2956882679547061052t_unit > int > list_list_a ).
thf(sy_c_Ring__Characteristic_Oint__embed_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_i7845631192508594762t_unit: partia2670972154091845814t_unit > int > list_a ).
thf(sy_c_Ring__Characteristic_Oint__embed_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_i2743490682209504680t_unit: partia4934656038542163276t_unit > int > set_int ).
thf(sy_c_Ring__Characteristic_Oint__embed_001tf__a_001tf__b,type,
ring_int_embed_a_b: partia2175431115845679010xt_a_b > int > a ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
ring_p8404492108403472412t_unit: partia5333488208502193986t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_p715737262848045090t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_p8098905331641078952t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001tf__a_001tf__b,type,
ring_r999134135267193926le_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001tf__a_001tf__b,type,
ring_ring_prime_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
collect_nat_list_a: ( ( nat > list_a ) > $o ) > set_nat_list_a ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001_062_It__Set__Oset_It__Int__Oint_J_Mtf__a_J,type,
collect_set_int_a: ( ( set_int > a ) > $o ) > set_set_int_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
collect_a_list_a: ( ( a > list_a ) > $o ) > set_a_list_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_001tf__a,type,
image_nat_list_a_a: ( ( nat > list_a ) > a ) > set_nat_list_a > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
image_nat_a_a: ( ( nat > a ) > a ) > set_nat_a > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Set__Oset_It__Int__Oint_J_Mtf__a_J_001tf__a,type,
image_set_int_a_a: ( ( set_int > a ) > a ) > set_set_int_a > set_a ).
thf(sy_c_Set_Oimage_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J_001tf__a,type,
image_a_list_a_a: ( ( a > list_a ) > a ) > set_a_list_a > set_a ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
image_int_nat_list_a: ( int > nat > list_a ) > set_int > set_nat_list_a ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001_062_It__Nat__Onat_Mtf__a_J,type,
image_int_nat_a: ( int > nat > a ) > set_int > set_nat_a ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001_062_It__Set__Oset_It__Int__Oint_J_Mtf__a_J,type,
image_int_set_int_a: ( int > set_int > a ) > set_int > set_set_int_a ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
image_8546930381880718615list_a: ( int > list_list_a ) > set_int > set_list_list_a ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__List__Olist_Itf__a_J,type,
image_int_list_a: ( int > list_a ) > set_int > set_list_a ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001tf__a,type,
image_int_a: ( int > a ) > set_int > set_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
image_1156962946714028939list_a: ( list_list_a > list_list_list_a ) > set_list_list_a > set_list_list_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
image_5446147394702115205list_a: ( list_list_a > list_list_a ) > set_list_list_a > set_list_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J,type,
image_4177985004758493951list_a: ( list_list_a > list_a ) > set_list_list_a > set_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a,type,
image_list_list_a_a: ( list_list_a > a ) > set_list_list_a > set_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
image_5238573146875157228list_a: ( list_a > nat > list_a ) > set_list_a > set_nat_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001_062_It__Nat__Onat_Mtf__a_J,type,
image_list_a_nat_a: ( list_a > nat > a ) > set_list_a > set_nat_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001_062_It__Set__Oset_It__Int__Oint_J_Mtf__a_J,type,
image_751163810753716308_int_a: ( list_a > set_int > a ) > set_list_a > set_set_int_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
image_6252139687081310122list_a: ( list_a > a > list_a ) > set_list_a > set_a_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Int__Oint,type,
image_list_a_int: ( list_a > int ) > set_list_a > set_int ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
image_8260866953997875467list_a: ( list_a > list_list_a ) > set_list_a > set_list_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
image_list_a_list_a: ( list_a > list_a ) > set_list_a > set_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001tf__a,type,
image_list_a_a: ( list_a > a ) > set_list_a > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
image_nat_list_a: ( nat > list_a ) > set_nat > set_list_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
image_nat_set_int: ( nat > set_int ) > set_nat > set_set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__List__Olist_Itf__a_J,type,
image_set_int_list_a: ( set_int > list_a ) > set_set_int > set_list_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001tf__a,type,
image_set_int_a: ( set_int > a ) > set_set_int > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
image_a_nat_list_a: ( a > nat > list_a ) > set_a > set_nat_list_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
image_a_nat_a: ( a > nat > a ) > set_a > set_nat_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_It__Set__Oset_It__Int__Oint_J_Mtf__a_J,type,
image_a_set_int_a: ( a > set_int > a ) > set_a > set_set_int_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
image_a_a_list_a: ( a > a > list_a ) > set_a > set_a_list_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Int__Oint,type,
image_a_int: ( a > int ) > set_a > set_int ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
image_a_list_list_a: ( a > list_list_a ) > set_a > set_list_list_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_Itf__a_J,type,
image_a_list_a: ( a > list_a ) > set_a > set_list_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Subrings_Osubcring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subcri8676831449680469861t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subcri7763218559781929323t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
subcri1024317279029940167t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001tf__a_001tf__b,type,
subcring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubdomain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subdom561091866123308472t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subdom7821232466298058046t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
subdom1520866149873910708t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001tf__a_001tf__b,type,
subdomain_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
subfie2726367901829656742t_unit: set_list_list_list_a > partia5333488208502193986t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subfie4546268998243038636t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subfie1779122896746047282t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
subfie3888952257595785920t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subrin3541368690557094692t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subrin6918843898125473962t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
subrin7689096310803670856t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Subrings_Osubring_001tf__a_001tf__b,type,
subring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_UnivPoly_Obound_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
bound_list_list_a: list_list_a > nat > ( nat > list_list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001t__List__Olist_Itf__a_J,type,
bound_list_a: list_a > nat > ( nat > list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001t__Set__Oset_It__Int__Oint_J,type,
bound_set_int: set_int > nat > ( nat > set_int ) > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
up_lis8963924889346801084t_unit: partia2956882679547061052t_unit > set_nat_list_list_a ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
up_lis8464167429055313730t_unit: partia2670972154091845814t_unit > set_nat_list_a ).
thf(sy_c_UnivPoly_Oup_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
up_set1168727741560211120t_unit: partia4934656038542163276t_unit > set_nat_set_int ).
thf(sy_c_UnivPoly_Oup_001tf__a_001tf__b,type,
up_a_b: partia2175431115845679010xt_a_b > set_nat_a ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member7168557129179038582list_a: ( list_list_a > list_a ) > set_li3422455791611400469list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
member_list_list_a_a: ( list_list_a > a ) > set_list_list_a_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member6714375691612171394list_a: ( list_a > list_list_a ) > set_li6773872926390105121list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member_list_a_list_a: ( list_a > list_a ) > set_list_a_list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
member5309957867533224448et_int: ( list_a > set_int ) > set_list_a_set_int > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
member_list_a_a: ( list_a > a ) > set_list_a_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member8650753269014980122list_a: ( nat > list_list_a ) > set_nat_list_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
member_nat_list_a: ( nat > list_a ) > set_nat_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
member_nat_set_int: ( nat > set_int ) > set_nat_set_int > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Int__Oint_J_Mt__List__Olist_Itf__a_J_J,type,
member6804298160596253954list_a: ( set_int > list_a ) > set_set_int_list_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Int__Oint_J_Mtf__a_J,type,
member_set_int_a: ( set_int > a ) > set_set_int_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__Int__Oint_J_J,type,
member_a_set_int: ( a > set_int ) > set_a_set_int > $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__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member5342144027231129785list_a: list_list_list_a > set_list_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
member_list_set_int: list_set_int > set_list_set_int > $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__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_K,type,
k: set_a ).
thf(sy_v_K_H____,type,
k2: partia2175431115845679010xt_a_b ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_n____,type,
n: nat ).
% Relevant facts (1272)
thf(fact_0_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_1_assms,axiom,
subring_a_b @ k @ r ).
% assms
thf(fact_2_int__embed__consistent,axiom,
! [K: set_a,I: int] :
( ( subring_a_b @ K @ r )
=> ( ( ring_int_embed_a_b @ r @ I )
= ( ring_int_embed_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r )
@ I ) ) ) ).
% int_embed_consistent
thf(fact_3_poly__of__const__consistent,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ( poly_of_const_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) )
= ( poly_of_const_a_b @ r ) ) ) ).
% poly_of_const_consistent
thf(fact_4_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_5_int__embed__closed,axiom,
! [K2: int] : ( member_a @ ( ring_int_embed_a_b @ r @ K2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ).
% int_embed_closed
thf(fact_6_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_7_nat__pow__consistent,axiom,
! [X: a,N: nat,H: set_a] :
( ( pow_a_1026414303147256608_b_nat @ r @ X @ N )
= ( pow_a_1026414303147256608_b_nat
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r )
@ X
@ N ) ) ).
% nat_pow_consistent
thf(fact_8_int__embed__range,axiom,
! [K: set_a,K2: int] :
( ( subring_a_b @ K @ r )
=> ( member_a @ ( ring_int_embed_a_b @ r @ K2 ) @ K ) ) ).
% int_embed_range
thf(fact_9_K_H__def,axiom,
( k2
= ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r ) ) ).
% K'_def
thf(fact_10_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_11_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_12_is__abelian__group,axiom,
abelian_group_a_b @ r ).
% is_abelian_group
thf(fact_13_associated__sym,axiom,
! [A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( associ5860276527279195403xt_a_b @ r @ B @ A ) ) ).
% associated_sym
thf(fact_14_d_Oassociated__sym,axiom,
! [A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ k2 @ A @ B )
=> ( associ5860276527279195403xt_a_b @ k2 @ B @ A ) ) ).
% d.associated_sym
thf(fact_15_d_Odomain__axioms,axiom,
domain_a_b @ k2 ).
% d.domain_axioms
thf(fact_16_d_Oring__axioms,axiom,
ring_a_b @ k2 ).
% d.ring_axioms
thf(fact_17_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_18_d_Ocarrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ k2 ) @ k2 ).
% d.carrier_is_subring
thf(fact_19_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_20_assoc__subst,axiom,
! [A: a,B: a,F: a > a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ! [A2: a,B2: a] :
( ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ A2 @ B2 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ ( F @ B2 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% assoc_subst
thf(fact_21_d_Oassoc__subst,axiom,
! [A: a,B: a,F: a > a] :
( ( associ5860276527279195403xt_a_b @ k2 @ A @ B )
=> ( ! [A2: a,B2: a] :
( ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ k2 ) )
& ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ k2 ) )
& ( associ5860276527279195403xt_a_b @ k2 @ A2 @ B2 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ k2 ) )
& ( member_a @ ( F @ B2 ) @ ( partia707051561876973205xt_a_b @ k2 ) )
& ( associ5860276527279195403xt_a_b @ k2 @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( associ5860276527279195403xt_a_b @ k2 @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% d.assoc_subst
thf(fact_22_d_Oassociated__trans,axiom,
! [A: a,B: a,C: a] :
( ( associ5860276527279195403xt_a_b @ k2 @ A @ B )
=> ( ( associ5860276527279195403xt_a_b @ k2 @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( associ5860276527279195403xt_a_b @ k2 @ A @ C ) ) ) ) ) ).
% d.associated_trans
thf(fact_23_d_Oint__embed__closed,axiom,
! [K2: int] : ( member_a @ ( ring_int_embed_a_b @ k2 @ K2 ) @ ( partia707051561876973205xt_a_b @ k2 ) ) ).
% d.int_embed_closed
thf(fact_24_d_Oabelian__monoid__axioms,axiom,
abelian_monoid_a_b @ k2 ).
% d.abelian_monoid_axioms
thf(fact_25_d_Ois__abelian__group,axiom,
abelian_group_a_b @ k2 ).
% d.is_abelian_group
thf(fact_26_d_Osemiring__axioms,axiom,
semiring_a_b @ k2 ).
% d.semiring_axioms
thf(fact_27_d_Oint__embed__range,axiom,
! [K: set_a,K2: int] :
( ( subring_a_b @ K @ k2 )
=> ( member_a @ ( ring_int_embed_a_b @ k2 @ K2 ) @ K ) ) ).
% d.int_embed_range
thf(fact_28_d_Onat__pow__consistent,axiom,
! [X: a,N: nat,H: set_a] :
( ( pow_a_1026414303147256608_b_nat @ k2 @ X @ N )
= ( pow_a_1026414303147256608_b_nat
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ k2 )
@ X
@ N ) ) ).
% d.nat_pow_consistent
thf(fact_29_d_Opoly__of__const__consistent,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ k2 )
=> ( ( poly_of_const_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ k2 ) )
= ( poly_of_const_a_b @ k2 ) ) ) ).
% d.poly_of_const_consistent
thf(fact_30_d_Oint__embed__consistent,axiom,
! [K: set_a,I: int] :
( ( subring_a_b @ K @ k2 )
=> ( ( ring_int_embed_a_b @ k2 @ I )
= ( ring_int_embed_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ k2 )
@ I ) ) ) ).
% d.int_embed_consistent
thf(fact_31_subring__is__domain,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r ) ) ) ).
% subring_is_domain
thf(fact_32_d_Osubring__is__domain,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ k2 )
=> ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ k2 ) ) ) ).
% d.subring_is_domain
thf(fact_33_subring__is__ring,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( ring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r ) ) ) ).
% subring_is_ring
thf(fact_34_d_Osubring__is__ring,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ k2 )
=> ( ring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ k2 ) ) ) ).
% d.subring_is_ring
thf(fact_35_associated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ A ) ) ).
% associated_refl
thf(fact_36_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_37_d_Oassociated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( associ5860276527279195403xt_a_b @ k2 @ A @ A ) ) ).
% d.associated_refl
thf(fact_38_d_Onat__pow__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ k2 @ X @ N ) @ ( partia707051561876973205xt_a_b @ k2 ) ) ) ).
% d.nat_pow_closed
thf(fact_39_d_Oonepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ k2 ) @ k2 ).
% d.onepideal
thf(fact_40_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_41_d_Osubdomain__is__domain,axiom,
! [H: set_a] :
( ( subdomain_a_b @ H @ k2 )
=> ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ k2 ) ) ) ).
% d.subdomain_is_domain
thf(fact_42_subdomain__is__domain,axiom,
! [H: set_a] :
( ( subdomain_a_b @ H @ r )
=> ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r ) ) ) ).
% subdomain_is_domain
thf(fact_43_d_Oring__incl__imp__subring,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ k2 ) )
=> ( subring_a_b @ H @ k2 ) ) ) ).
% d.ring_incl_imp_subring
thf(fact_44_d_Osubring__iff,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( subring_a_b @ H @ k2 )
= ( ring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ k2 ) ) ) ) ).
% d.subring_iff
thf(fact_45_mem__Collect__eq,axiom,
! [A: set_int > a,P: ( set_int > a ) > $o] :
( ( member_set_int_a @ A @ ( collect_set_int_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A: nat > list_a,P: ( nat > list_a ) > $o] :
( ( member_nat_list_a @ A @ ( collect_nat_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: nat > a,P: ( nat > a ) > $o] :
( ( member_nat_a @ A @ ( collect_nat_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
! [A: a > list_a,P: ( a > list_a ) > $o] :
( ( member_a_list_a @ A @ ( collect_a_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_50_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A3: set_set_int_a] :
( ( collect_set_int_a
@ ^ [X2: set_int > a] : ( member_set_int_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A3: set_nat_list_a] :
( ( collect_nat_list_a
@ ^ [X2: nat > list_a] : ( member_nat_list_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
! [A3: set_nat_a] :
( ( collect_nat_a
@ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A3: set_a_list_a] :
( ( collect_a_list_a
@ ^ [X2: a > list_a] : ( member_a_list_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_57_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_58_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_59_subring__iff,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( subring_a_b @ H @ r )
= ( ring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r ) ) ) ) ).
% subring_iff
thf(fact_60_ring__incl__imp__subring,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r ) )
=> ( subring_a_b @ H @ r ) ) ) ).
% ring_incl_imp_subring
thf(fact_61_ring_Opoly__of__const__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( poly_of_const_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ R ) )
= ( poly_of_const_a_b @ R ) ) ) ) ).
% ring.poly_of_const_consistent
thf(fact_62_ring_Opoly__of__const__consistent,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( poly_o8716471131768098070t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ R ) )
= ( poly_o8716471131768098070t_unit @ R ) ) ) ) ).
% ring.poly_of_const_consistent
thf(fact_63_ring_Oint__embed__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,I: int] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( ring_int_embed_a_b @ R @ I )
= ( ring_int_embed_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ R )
@ I ) ) ) ) ).
% ring.int_embed_consistent
thf(fact_64_ring_Oint__embed__consistent,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,I: int] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( ring_i7845631192508594762t_unit @ R @ I )
= ( ring_i7845631192508594762t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ R )
@ I ) ) ) ) ).
% ring.int_embed_consistent
thf(fact_65_d_Ouniv__poly__consistent,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ k2 )
=> ( ( univ_poly_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ k2 ) )
= ( univ_poly_a_b @ k2 ) ) ) ).
% d.univ_poly_consistent
thf(fact_66_univ__poly__consistent,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ( univ_poly_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) )
= ( univ_poly_a_b @ r ) ) ) ).
% univ_poly_consistent
thf(fact_67_univ__poly__is__ring,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_ring
thf(fact_68_d_Ouniv__poly__is__ring,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ k2 )
=> ( ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ k2 @ K ) ) ) ).
% d.univ_poly_is_ring
thf(fact_69_p_Oint__embed__range,axiom,
! [K: set_list_a,K2: int] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( member_list_a @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ k ) @ K2 ) @ K ) ) ).
% p.int_embed_range
thf(fact_70_p_Oring__axioms,axiom,
ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ r @ k ) ).
% p.ring_axioms
thf(fact_71_subdomainI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subdomain_a_b @ H @ r ) ) ).
% subdomainI'
thf(fact_72_d_OsubdomainI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ k2 )
=> ( subdomain_a_b @ H @ k2 ) ) ).
% d.subdomainI'
thf(fact_73_p_Oint__embed__consistent,axiom,
! [K: set_list_a,I: int] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ k ) @ I )
= ( ring_i7845631192508594762t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ ( univ_poly_a_b @ r @ k ) )
@ I ) ) ) ).
% p.int_embed_consistent
thf(fact_74_p_Osubring__is__ring,axiom,
! [H: set_list_a] :
( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ k ) )
=> ( ring_l6212528067271185461t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.subring_is_ring
thf(fact_75_p_Opoly__of__const__consistent,axiom,
! [K: set_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( poly_o8716471131768098070t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ ( univ_poly_a_b @ r @ k ) ) )
= ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.poly_of_const_consistent
thf(fact_76_subdomain__iff,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( subdomain_a_b @ H @ r )
= ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r ) ) ) ) ).
% subdomain_iff
thf(fact_77_d_Osubdomain__iff,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( subdomain_a_b @ H @ k2 )
= ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ k2 ) ) ) ) ).
% d.subdomain_iff
thf(fact_78_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_79_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_80_calculation,axiom,
( ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ k ) @ ( semiri1314217659103216013at_int @ n ) )
= ( poly_of_const_a_b @ r @ ( ring_int_embed_a_b @ k2 @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% calculation
thf(fact_81_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_82_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_83_domain_Ouniv__poly__is__ring,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_ring
thf(fact_84_domain_Ouniv__poly__is__ring,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ring_l1939023646219158831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_ring
thf(fact_85_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_86_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_87_ring_Ouniv__poly__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( univ_poly_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ R ) )
= ( univ_poly_a_b @ R ) ) ) ) ).
% ring.univ_poly_consistent
thf(fact_88_ring_Ouniv__poly__consistent,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( univ_p7953238456130426574t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ R ) )
= ( univ_p7953238456130426574t_unit @ R ) ) ) ) ).
% ring.univ_poly_consistent
thf(fact_89_ring_Opoly__of__const_Ocong,axiom,
poly_of_const_a_b = poly_of_const_a_b ).
% ring.poly_of_const.cong
thf(fact_90_ring_Opoly__of__const_Ocong,axiom,
poly_o8716471131768098070t_unit = poly_o8716471131768098070t_unit ).
% ring.poly_of_const.cong
thf(fact_91_ring_Oint__embed__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K2: int] :
( ( ring_a_b @ R )
=> ( member_a @ ( ring_int_embed_a_b @ R @ K2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.int_embed_closed
thf(fact_92_ring_Oint__embed__closed,axiom,
! [R: partia2670972154091845814t_unit,K2: int] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( member_list_a @ ( ring_i7845631192508594762t_unit @ R @ K2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% ring.int_embed_closed
thf(fact_93_ring_Oint__embed__closed,axiom,
! [R: partia2956882679547061052t_unit,K2: int] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( member_list_list_a @ ( ring_i3137826187831172292t_unit @ R @ K2 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% ring.int_embed_closed
thf(fact_94_ring_Oint__embed__closed,axiom,
! [R: partia4934656038542163276t_unit,K2: int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( member_set_int @ ( ring_i2743490682209504680t_unit @ R @ K2 ) @ ( partia966996272515721803t_unit @ R ) ) ) ).
% ring.int_embed_closed
thf(fact_95_ring_Oint__embed__range,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,K2: int] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( member_a @ ( ring_int_embed_a_b @ R @ K2 ) @ K ) ) ) ).
% ring.int_embed_range
thf(fact_96_ring_Oint__embed__range,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,K2: int] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( member_list_a @ ( ring_i7845631192508594762t_unit @ R @ K2 ) @ K ) ) ) ).
% ring.int_embed_range
thf(fact_97_h_Ois__abelian__group__hom,axiom,
abelia5726065498384912733t_unit @ k2 @ ( univ_poly_a_b @ r @ k ) @ ( poly_of_const_a_b @ r ) ).
% h.is_abelian_group_hom
thf(fact_98_ring_Osubdomain__iff,axiom,
! [R: partia2956882679547061052t_unit,H: set_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( subdom561091866123308472t_unit @ H @ R )
= ( domain7810152921033798211t_unit
@ ( partia5725451877363000582t_unit
@ ^ [Uu: set_list_list_a] : H
@ R ) ) ) ) ) ).
% ring.subdomain_iff
thf(fact_99_ring_Osubdomain__iff,axiom,
! [R: partia4934656038542163276t_unit,H: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ R ) )
=> ( ( subdom1520866149873910708t_unit @ H @ R )
= ( domain6183376680155302761t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : H
@ R ) ) ) ) ) ).
% ring.subdomain_iff
thf(fact_100_ring_Osubdomain__iff,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( subdomain_a_b @ H @ R )
= ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ R ) ) ) ) ) ).
% ring.subdomain_iff
thf(fact_101_ring_Osubdomain__iff,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( subdom7821232466298058046t_unit @ H @ R )
= ( domain6553523120543210313t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ R ) ) ) ) ) ).
% ring.subdomain_iff
thf(fact_102_h_Ohom__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( member_list_a @ ( poly_of_const_a_b @ r @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% h.hom_closed
thf(fact_103_h_Ohom__nat__pow,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( poly_of_const_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ k2 @ X @ N ) )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ ( poly_of_const_a_b @ r @ X ) @ N ) ) ) ).
% h.hom_nat_pow
thf(fact_104_h_Oinduced__ring__hom,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ k2 )
=> ( ring_h5357930050666032198t_unit
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ k2 )
@ ( univ_poly_a_b @ r @ k )
@ ( poly_of_const_a_b @ r ) ) ) ).
% h.induced_ring_hom
thf(fact_105_ring_Oring__incl__imp__subring,axiom,
! [R: partia2956882679547061052t_unit,H: set_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_l1939023646219158831t_unit
@ ( partia5725451877363000582t_unit
@ ^ [Uu: set_list_list_a] : H
@ R ) )
=> ( subrin3541368690557094692t_unit @ H @ R ) ) ) ) ).
% ring.ring_incl_imp_subring
thf(fact_106_ring_Oring__incl__imp__subring,axiom,
! [R: partia4934656038542163276t_unit,H: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ R ) )
=> ( ( ring_s5316885176909347197t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : H
@ R ) )
=> ( subrin7689096310803670856t_unit @ H @ R ) ) ) ) ).
% ring.ring_incl_imp_subring
thf(fact_107_ring_Oring__incl__imp__subring,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ R ) )
=> ( subring_a_b @ H @ R ) ) ) ) ).
% ring.ring_incl_imp_subring
thf(fact_108_ring_Oring__incl__imp__subring,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_l6212528067271185461t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ R ) )
=> ( subrin6918843898125473962t_unit @ H @ R ) ) ) ) ).
% ring.ring_incl_imp_subring
thf(fact_109_ring_Osubring__iff,axiom,
! [R: partia2956882679547061052t_unit,H: set_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( subrin3541368690557094692t_unit @ H @ R )
= ( ring_l1939023646219158831t_unit
@ ( partia5725451877363000582t_unit
@ ^ [Uu: set_list_list_a] : H
@ R ) ) ) ) ) ).
% ring.subring_iff
thf(fact_110_ring_Osubring__iff,axiom,
! [R: partia4934656038542163276t_unit,H: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ R ) )
=> ( ( subrin7689096310803670856t_unit @ H @ R )
= ( ring_s5316885176909347197t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : H
@ R ) ) ) ) ) ).
% ring.subring_iff
thf(fact_111_ring_Osubring__iff,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( subring_a_b @ H @ R )
= ( ring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ R ) ) ) ) ) ).
% ring.subring_iff
thf(fact_112_ring_Osubring__iff,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( subrin6918843898125473962t_unit @ H @ R )
= ( ring_l6212528067271185461t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ R ) ) ) ) ) ).
% ring.subring_iff
thf(fact_113_h_Ohomh,axiom,
member_a_list_a @ ( poly_of_const_a_b @ r ) @ ( ring_h405018892823518980t_unit @ k2 @ ( univ_poly_a_b @ r @ k ) ) ).
% h.homh
thf(fact_114_p_Osemiring__axioms,axiom,
semiri2871908745932252451t_unit @ ( univ_poly_a_b @ r @ k ) ).
% p.semiring_axioms
thf(fact_115_h_Oimg__is__ring,axiom,
( ring_l6212528067271185461t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ ( partia707051561876973205xt_a_b @ k2 ) )
@ ( univ_poly_a_b @ r @ k ) ) ) ).
% h.img_is_ring
thf(fact_116_p_Ochar__consistent,axiom,
! [H: set_list_a] :
( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ k ) )
=> ( ( ring_c500279861223467766t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ ( univ_poly_a_b @ r @ k ) ) )
= ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.char_consistent
thf(fact_117_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_118_d_Ouniv__poly__is__domain,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ k2 )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ k2 @ K ) ) ) ).
% d.univ_poly_is_domain
thf(fact_119_ring__hom__restrict,axiom,
! [F: a > list_a,S: partia2670972154091845814t_unit,G: a > list_a] :
( ( member_a_list_a @ F @ ( ring_h405018892823518980t_unit @ r @ S ) )
=> ( ! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( F @ R2 )
= ( G @ R2 ) ) )
=> ( member_a_list_a @ G @ ( ring_h405018892823518980t_unit @ r @ S ) ) ) ) ).
% ring_hom_restrict
thf(fact_120_d_Oring__hom__restrict,axiom,
! [F: a > list_a,S: partia2670972154091845814t_unit,G: a > list_a] :
( ( member_a_list_a @ F @ ( ring_h405018892823518980t_unit @ k2 @ S ) )
=> ( ! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( F @ R2 )
= ( G @ R2 ) ) )
=> ( member_a_list_a @ G @ ( ring_h405018892823518980t_unit @ k2 @ S ) ) ) ) ).
% d.ring_hom_restrict
thf(fact_121_p_Osubdomain__is__domain,axiom,
! [H: set_list_a] :
( ( subdom7821232466298058046t_unit @ H @ ( univ_poly_a_b @ r @ k ) )
=> ( domain6553523120543210313t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.subdomain_is_domain
thf(fact_122_p_Ocarrier__is__subring,axiom,
subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( univ_poly_a_b @ r @ k ) ).
% p.carrier_is_subring
thf(fact_123_p_Oint__embed__closed,axiom,
! [K2: int] : ( member_list_a @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ k ) @ K2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.int_embed_closed
thf(fact_124_p_Osubdomain__iff,axiom,
! [H: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( subdom7821232466298058046t_unit @ H @ ( univ_poly_a_b @ r @ k ) )
= ( domain6553523120543210313t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.subdomain_iff
thf(fact_125_p_Onat__pow__consistent,axiom,
! [X: list_a,N: nat,H: set_list_a] :
( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N )
= ( pow_li1142815632869257134it_nat
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ ( univ_poly_a_b @ r @ k ) )
@ X
@ N ) ) ).
% p.nat_pow_consistent
thf(fact_126_p_Ouniv__poly__consistent,axiom,
! [K: set_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( univ_p7953238456130426574t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ ( univ_poly_a_b @ r @ k ) ) )
= ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.univ_poly_consistent
thf(fact_127_canonical__embedding__ring__hom,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ring_h5357930050666032198t_unit
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r )
@ ( univ_poly_a_b @ r @ K )
@ ( poly_of_const_a_b @ r ) ) ) ).
% canonical_embedding_ring_hom
thf(fact_128_d_Ocanonical__embedding__ring__hom,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ k2 )
=> ( ring_h5357930050666032198t_unit
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ k2 )
@ ( univ_poly_a_b @ k2 @ K )
@ ( poly_of_const_a_b @ k2 ) ) ) ).
% d.canonical_embedding_ring_hom
thf(fact_129_canonical__embedding__is__hom,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( member_a_list_a @ ( poly_of_const_a_b @ r )
@ ( ring_h405018892823518980t_unit
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r )
@ ( univ_poly_a_b @ r @ K ) ) ) ) ).
% canonical_embedding_is_hom
thf(fact_130_d_Ocanonical__embedding__is__hom,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ k2 )
=> ( member_a_list_a @ ( poly_of_const_a_b @ k2 )
@ ( ring_h405018892823518980t_unit
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ k2 )
@ ( univ_poly_a_b @ k2 @ K ) ) ) ) ).
% d.canonical_embedding_is_hom
thf(fact_131_h_Oring__hom__ring__axioms,axiom,
ring_h5357930050666032198t_unit @ k2 @ ( univ_poly_a_b @ r @ k ) @ ( poly_of_const_a_b @ r ) ).
% h.ring_hom_ring_axioms
thf(fact_132_p_Osubring__iff,axiom,
! [H: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ k ) )
= ( ring_l6212528067271185461t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.subring_iff
thf(fact_133_p_Oring__incl__imp__subring,axiom,
! [H: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ring_l6212528067271185461t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ ( univ_poly_a_b @ r @ k ) ) )
=> ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.ring_incl_imp_subring
thf(fact_134_h_Oimg__is__subring,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ k2 )
=> ( subrin6918843898125473962t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( univ_poly_a_b @ r @ k ) ) ) ).
% h.img_is_subring
thf(fact_135_h_Oimg__is__domain,axiom,
( ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ k ) )
=> ( domain6553523120543210313t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ ( partia707051561876973205xt_a_b @ k2 ) )
@ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% h.img_is_domain
thf(fact_136_carrier__polynomial__shell,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% carrier_polynomial_shell
thf(fact_137_d_Ocarrier__polynomial__shell,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ k2 )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ k2 @ K ) ) )
=> ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ k2 @ ( partia707051561876973205xt_a_b @ k2 ) ) ) ) ) ) ).
% d.carrier_polynomial_shell
thf(fact_138_p_Onat__pow__closed,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.nat_pow_closed
thf(fact_139_p_Ocarrier__polynomial__shell,axiom,
! [K: set_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
=> ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ) ).
% p.carrier_polynomial_shell
thf(fact_140_ring__hom__ring_Oimg__is__subring,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a,K: set_a] :
( ( ring_h4024360765257340990_b_a_b @ R @ S @ H2 )
=> ( ( subring_a_b @ K @ R )
=> ( subring_a_b @ ( image_a_a @ H2 @ K ) @ S ) ) ) ).
% ring_hom_ring.img_is_subring
thf(fact_141_ring__hom__ring_Oimg__is__subring,axiom,
! [R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H2: list_a > list_a,K: set_list_a] :
( ( ring_h1334922693953046990t_unit @ R @ S @ H2 )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( subrin6918843898125473962t_unit @ ( image_list_a_list_a @ H2 @ K ) @ S ) ) ) ).
% ring_hom_ring.img_is_subring
thf(fact_142_ring__hom__ring_Oimg__is__subring,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,K: set_a] :
( ( ring_h5357930050666032198t_unit @ R @ S @ H2 )
=> ( ( subring_a_b @ K @ R )
=> ( subrin6918843898125473962t_unit @ ( image_a_list_a @ H2 @ K ) @ S ) ) ) ).
% ring_hom_ring.img_is_subring
thf(fact_143_ring__hom__ring_Oimg__is__subring,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,K: set_list_a] :
( ( ring_h7848885096329822662it_a_b @ R @ S @ H2 )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( subring_a_b @ ( image_list_a_a @ H2 @ K ) @ S ) ) ) ).
% ring_hom_ring.img_is_subring
thf(fact_144_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_145_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_146_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_147_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_148_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_149_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_150_image__set__eqI,axiom,
! [A3: set_int,F: int > a,B3: set_a,G: a > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A3 )
=> ( member_a @ ( F @ X3 ) @ B3 ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B3 )
=> ( ( member_int @ ( G @ X3 ) @ A3 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ( image_int_a @ F @ A3 )
= B3 ) ) ) ).
% image_set_eqI
thf(fact_151_image__set__eqI,axiom,
! [A3: set_a,F: a > a,B3: set_a,G: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ ( F @ X3 ) @ B3 ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B3 )
=> ( ( member_a @ ( G @ X3 ) @ A3 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ( image_a_a @ F @ A3 )
= B3 ) ) ) ).
% image_set_eqI
thf(fact_152_image__set__eqI,axiom,
! [A3: set_int,F: int > list_a,B3: set_list_a,G: list_a > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A3 )
=> ( member_list_a @ ( F @ X3 ) @ B3 ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ B3 )
=> ( ( member_int @ ( G @ X3 ) @ A3 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ( image_int_list_a @ F @ A3 )
= B3 ) ) ) ).
% image_set_eqI
thf(fact_153_image__set__eqI,axiom,
! [A3: set_a,F: a > list_a,B3: set_list_a,G: list_a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_list_a @ ( F @ X3 ) @ B3 ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ B3 )
=> ( ( member_a @ ( G @ X3 ) @ A3 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ( image_a_list_a @ F @ A3 )
= B3 ) ) ) ).
% image_set_eqI
thf(fact_154_image__set__eqI,axiom,
! [A3: set_list_a,F: list_a > list_a,B3: set_list_a,G: list_a > list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A3 )
=> ( member_list_a @ ( F @ X3 ) @ B3 ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ B3 )
=> ( ( member_list_a @ ( G @ X3 ) @ A3 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ( image_list_a_list_a @ F @ A3 )
= B3 ) ) ) ).
% image_set_eqI
thf(fact_155_image__set__eqI,axiom,
! [A3: set_nat_a,F: ( nat > a ) > a,B3: set_a,G: a > nat > a] :
( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ A3 )
=> ( member_a @ ( F @ X3 ) @ B3 ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B3 )
=> ( ( member_nat_a @ ( G @ X3 ) @ A3 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ( image_nat_a_a @ F @ A3 )
= B3 ) ) ) ).
% image_set_eqI
thf(fact_156_image__set__eqI,axiom,
! [A3: set_a,F: a > nat > a,B3: set_nat_a,G: ( nat > a ) > a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_nat_a @ ( F @ X3 ) @ B3 ) )
=> ( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ B3 )
=> ( ( member_a @ ( G @ X3 ) @ A3 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ( image_a_nat_a @ F @ A3 )
= B3 ) ) ) ).
% image_set_eqI
thf(fact_157_image__set__eqI,axiom,
! [A3: set_set_int_a,F: ( set_int > a ) > a,B3: set_a,G: a > set_int > a] :
( ! [X3: set_int > a] :
( ( member_set_int_a @ X3 @ A3 )
=> ( member_a @ ( F @ X3 ) @ B3 ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B3 )
=> ( ( member_set_int_a @ ( G @ X3 ) @ A3 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ( image_set_int_a_a @ F @ A3 )
= B3 ) ) ) ).
% image_set_eqI
thf(fact_158_image__set__eqI,axiom,
! [A3: set_nat_list_a,F: ( nat > list_a ) > a,B3: set_a,G: a > nat > list_a] :
( ! [X3: nat > list_a] :
( ( member_nat_list_a @ X3 @ A3 )
=> ( member_a @ ( F @ X3 ) @ B3 ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B3 )
=> ( ( member_nat_list_a @ ( G @ X3 ) @ A3 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ( image_nat_list_a_a @ F @ A3 )
= B3 ) ) ) ).
% image_set_eqI
thf(fact_159_image__set__eqI,axiom,
! [A3: set_a_list_a,F: ( a > list_a ) > a,B3: set_a,G: a > a > list_a] :
( ! [X3: a > list_a] :
( ( member_a_list_a @ X3 @ A3 )
=> ( member_a @ ( F @ X3 ) @ B3 ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B3 )
=> ( ( member_a_list_a @ ( G @ X3 ) @ A3 )
& ( ( F @ ( G @ X3 ) )
= X3 ) ) )
=> ( ( image_a_list_a_a @ F @ A3 )
= B3 ) ) ) ).
% image_set_eqI
thf(fact_160_ring__hom__ring_Oinduced__ring__hom,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,K: set_a] :
( ( ring_h5357930050666032198t_unit @ R @ S @ H2 )
=> ( ( subring_a_b @ K @ R )
=> ( ring_h5357930050666032198t_unit
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ R )
@ S
@ H2 ) ) ) ).
% ring_hom_ring.induced_ring_hom
thf(fact_161_ring__hom__ring_Oinduced__ring__hom,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,K: set_list_a] :
( ( ring_h7848885096329822662it_a_b @ R @ S @ H2 )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ring_h7848885096329822662it_a_b
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ R )
@ S
@ H2 ) ) ) ).
% ring_hom_ring.induced_ring_hom
thf(fact_162_ring_Ochar__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ H @ R )
=> ( ( ring_char_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ R ) )
= ( ring_char_a_b @ R ) ) ) ) ).
% ring.char_consistent
thf(fact_163_ring_Ochar__consistent,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ( ring_c500279861223467766t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ R ) )
= ( ring_c500279861223467766t_unit @ R ) ) ) ) ).
% ring.char_consistent
thf(fact_164_domain_Ocanonical__embedding__ring__hom,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ring_h5357930050666032198t_unit
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ R )
@ ( univ_poly_a_b @ R @ K )
@ ( poly_of_const_a_b @ R ) ) ) ) ).
% domain.canonical_embedding_ring_hom
thf(fact_165_domain_Ocanonical__embedding__ring__hom,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ring_h7560679385066609224t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ R )
@ ( univ_p7953238456130426574t_unit @ R @ K )
@ ( poly_o8716471131768098070t_unit @ R ) ) ) ) ).
% domain.canonical_embedding_ring_hom
thf(fact_166_domain_Ocanonical__embedding__is__hom,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( member_a_list_a @ ( poly_of_const_a_b @ R )
@ ( ring_h405018892823518980t_unit
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ R )
@ ( univ_poly_a_b @ R @ K ) ) ) ) ) ).
% domain.canonical_embedding_is_hom
thf(fact_167_domain_Ocanonical__embedding__is__hom,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( member6714375691612171394list_a @ ( poly_o8716471131768098070t_unit @ R )
@ ( ring_h8002040739877300486t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ R )
@ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ).
% domain.canonical_embedding_is_hom
thf(fact_168_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_169_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_170_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_171_subringE_I1_J,axiom,
! [H: set_set_int,R: partia4934656038542163276t_unit] :
( ( subrin7689096310803670856t_unit @ H @ R )
=> ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ R ) ) ) ).
% subringE(1)
thf(fact_172_ring_Ocarrier__is__subring,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( subring_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% ring.carrier_is_subring
thf(fact_173_ring_Ocarrier__is__subring,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).
% ring.carrier_is_subring
thf(fact_174_ring_Ocarrier__is__subring,axiom,
! [R: partia2956882679547061052t_unit] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( subrin3541368690557094692t_unit @ ( partia2464479390973590831t_unit @ R ) @ R ) ) ).
% ring.carrier_is_subring
thf(fact_175_ring_Ocarrier__is__subring,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( subrin7689096310803670856t_unit @ ( partia966996272515721803t_unit @ R ) @ R ) ) ).
% ring.carrier_is_subring
thf(fact_176_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_177_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_178_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_179_subdomainE_I1_J,axiom,
! [H: set_set_int,R: partia4934656038542163276t_unit] :
( ( subdom1520866149873910708t_unit @ H @ R )
=> ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ R ) ) ) ).
% subdomainE(1)
thf(fact_180_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_181_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_182_ring_Osubring__is__ring,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ H @ R )
=> ( ring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ R ) ) ) ) ).
% ring.subring_is_ring
thf(fact_183_ring_Osubring__is__ring,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ring_l6212528067271185461t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ R ) ) ) ) ).
% ring.subring_is_ring
thf(fact_184_domain_Osubring__is__domain,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ H @ R )
=> ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ R ) ) ) ) ).
% domain.subring_is_domain
thf(fact_185_domain_Osubring__is__domain,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ H @ R )
=> ( domain6553523120543210313t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ R ) ) ) ) ).
% domain.subring_is_domain
thf(fact_186_ring_Osubdomain__is__domain,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( ring_a_b @ R )
=> ( ( subdomain_a_b @ H @ R )
=> ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ R ) ) ) ) ).
% ring.subdomain_is_domain
thf(fact_187_ring_Osubdomain__is__domain,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subdom7821232466298058046t_unit @ H @ R )
=> ( domain6553523120543210313t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ R ) ) ) ) ).
% ring.subdomain_is_domain
thf(fact_188_p_Oonepideal,axiom,
princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( univ_poly_a_b @ r @ k ) ).
% p.onepideal
thf(fact_189_p_Oline__extension__consistent,axiom,
! [K: set_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd5150658419831591667t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ ( univ_poly_a_b @ r @ k ) ) )
= ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.line_extension_consistent
thf(fact_190_d_Ouniv__poly__a__minus__consistent,axiom,
! [K: set_a,Q2: list_a,P2: list_a] :
( ( subring_a_b @ K @ k2 )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ k2 @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ k2 @ K ) @ P2 @ Q2 )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ k2 @ ( partia707051561876973205xt_a_b @ k2 ) ) @ P2 @ Q2 ) ) ) ) ).
% d.univ_poly_a_minus_consistent
thf(fact_191_univ__poly__a__minus__consistent,axiom,
! [K: set_a,Q2: list_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ Q2 )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q2 ) ) ) ) ).
% univ_poly_a_minus_consistent
thf(fact_192_h_Oimg__is__cring,axiom,
( ( cring_3148771470849435808t_unit @ ( univ_poly_a_b @ r @ k ) )
=> ( cring_3148771470849435808t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ ( partia707051561876973205xt_a_b @ k2 ) )
@ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% h.img_is_cring
thf(fact_193_h_Ozero__closed,axiom,
member_list_a @ ( poly_of_const_a_b @ r @ ( zero_a_b @ k2 ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).
% h.zero_closed
thf(fact_194_p_Oembed__char__eq__0,axiom,
( ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ k ) @ ( semiri1314217659103216013at_int @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ k ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.embed_char_eq_0
thf(fact_195_p_Osubalgebra__in__carrier,axiom,
! [K: set_list_a,V: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ k ) )
=> ( ord_le8861187494160871172list_a @ V @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.subalgebra_in_carrier
thf(fact_196_p_Ocarrier__is__subalgebra,axiom,
! [K: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( embedd1768981623711841426t_unit @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.carrier_is_subalgebra
thf(fact_197_char__consistent,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( ( ring_char_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r ) )
= ( ring_char_a_b @ r ) ) ) ).
% char_consistent
thf(fact_198_d_Ochar__consistent,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ k2 )
=> ( ( ring_char_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ k2 ) )
= ( ring_char_a_b @ k2 ) ) ) ).
% d.char_consistent
thf(fact_199_pow__non__zero,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
!= ( zero_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ N )
!= ( zero_a_b @ r ) ) ) ) ).
% pow_non_zero
thf(fact_200_d_Opow__non__zero,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( X
!= ( zero_a_b @ k2 ) )
=> ( ( pow_a_1026414303147256608_b_nat @ k2 @ X @ N )
!= ( zero_a_b @ k2 ) ) ) ) ).
% d.pow_non_zero
thf(fact_201_embed__char__eq__0,axiom,
( ( ring_int_embed_a_b @ r @ ( semiri1314217659103216013at_int @ ( ring_char_a_b @ r ) ) )
= ( zero_a_b @ r ) ) ).
% embed_char_eq_0
thf(fact_202_d_Oembed__char__eq__0,axiom,
( ( ring_int_embed_a_b @ k2 @ ( semiri1314217659103216013at_int @ ( ring_char_a_b @ k2 ) ) )
= ( zero_a_b @ k2 ) ) ).
% d.embed_char_eq_0
thf(fact_203_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_204_d_Ouniv__poly__is__cring,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ k2 )
=> ( cring_3148771470849435808t_unit @ ( univ_poly_a_b @ k2 @ K ) ) ) ).
% d.univ_poly_is_cring
thf(fact_205_p_Ois__cring,axiom,
cring_3148771470849435808t_unit @ ( univ_poly_a_b @ r @ k ) ).
% p.is_cring
thf(fact_206_p_Oline__extension__in__carrier,axiom,
! [K: set_list_a,A: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ A @ E ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).
% p.line_extension_in_carrier
thf(fact_207_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_208_d_Ozero__closed,axiom,
member_a @ ( zero_a_b @ k2 ) @ ( partia707051561876973205xt_a_b @ k2 ) ).
% d.zero_closed
thf(fact_209_p_Ozero__closed,axiom,
member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).
% p.zero_closed
thf(fact_210_p_Ominus__closed,axiom,
! [X: list_a,Y3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.minus_closed
thf(fact_211_p_Or__right__minus__eq,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= ( A = B ) ) ) ) ).
% p.r_right_minus_eq
thf(fact_212_h_Ohom__zero,axiom,
( ( poly_of_const_a_b @ r @ ( zero_a_b @ k2 ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% h.hom_zero
thf(fact_213_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_214_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_215_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_216_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_217_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_218_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_219_ring_Oembed__char__eq__0,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( ring_int_embed_a_b @ R @ ( semiri1314217659103216013at_int @ ( ring_char_a_b @ R ) ) )
= ( zero_a_b @ R ) ) ) ).
% ring.embed_char_eq_0
thf(fact_220_ring_Oembed__char__eq__0,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ring_i7845631192508594762t_unit @ R @ ( semiri1314217659103216013at_int @ ( ring_c500279861223467766t_unit @ R ) ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% ring.embed_char_eq_0
thf(fact_221_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a,Q2: list_list_list_a,P2: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( member5342144027231129785list_a @ Q2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
=> ( ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 @ Q2 )
= ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P2 @ Q2 ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_222_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,Q2: list_set_int,P2: list_set_int] :
( ( domain6183376680155302761t_unit @ R )
=> ( ( subrin7689096310803670856t_unit @ K @ R )
=> ( ( member_list_set_int @ Q2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ R @ K ) ) )
=> ( ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ R @ K ) @ P2 @ Q2 )
= ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ R @ ( partia966996272515721803t_unit @ R ) ) @ P2 @ Q2 ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_223_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Q2: list_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 @ Q2 )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 @ Q2 ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_224_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,Q2: list_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 @ Q2 )
= ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 @ Q2 ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_225_h_Oimg__is__subalgebra,axiom,
! [K: set_a,V: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V @ k2 )
=> ( embedd1768981623711841426t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ V ) @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% h.img_is_subalgebra
thf(fact_226_p_Osubcring__iff,axiom,
! [H: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ k ) )
= ( cring_3148771470849435808t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.subcring_iff
thf(fact_227_d_Oring__hom__imp__img__ring,axiom,
! [H2: a > a,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ k2 @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ k2 ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_a_a @ H2 @ ( partia707051561876973205xt_a_b @ k2 ) )
@ S ) ) ) ) ).
% d.ring_hom_imp_img_ring
thf(fact_228_d_Oring__hom__imp__img__ring,axiom,
! [H2: a > list_a,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ k2 @ S ) )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ k2 ) )
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ k2 ) )
@ S ) ) ) ) ).
% d.ring_hom_imp_img_ring
thf(fact_229_ring__hom__imp__img__ring,axiom,
! [H2: a > a,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ r @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ r ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_a_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
@ S ) ) ) ) ).
% ring_hom_imp_img_ring
thf(fact_230_ring__hom__imp__img__ring,axiom,
! [H2: a > list_a,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ r @ S ) )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ r ) )
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
@ S ) ) ) ) ).
% ring_hom_imp_img_ring
thf(fact_231_p_Oring__iso__imp__img__cring,axiom,
! [H2: list_a > list_a,S: partia2670972154091845814t_unit] :
( ( member_list_a_list_a @ H2 @ ( ring_i7414513579304222626t_unit @ ( univ_poly_a_b @ r @ k ) @ S ) )
=> ( cring_3148771470849435808t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
@ S ) ) ) ).
% p.ring_iso_imp_img_cring
thf(fact_232_p_Oring__iso__imp__img__cring,axiom,
! [H2: list_a > a,S: partia2175431115845679010xt_a_b] :
( ( member_list_a_a @ H2 @ ( ring_i7048835797181109658it_a_b @ ( univ_poly_a_b @ r @ k ) @ S ) )
=> ( cring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
@ S ) ) ) ).
% p.ring_iso_imp_img_cring
thf(fact_233_p_Oring__iso__imp__img__ring,axiom,
! [H2: list_a > a,S: partia2175431115845679010xt_a_b] :
( ( member_list_a_a @ H2 @ ( ring_i7048835797181109658it_a_b @ ( univ_poly_a_b @ r @ k ) @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
@ S ) ) ) ).
% p.ring_iso_imp_img_ring
thf(fact_234_p_Oring__iso__imp__img__ring,axiom,
! [H2: list_a > list_a,S: partia2670972154091845814t_unit] :
( ( member_list_a_list_a @ H2 @ ( ring_i7414513579304222626t_unit @ ( univ_poly_a_b @ r @ k ) @ S ) )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
@ S ) ) ) ).
% p.ring_iso_imp_img_ring
thf(fact_235_h_Oline__extension__hom,axiom,
! [K: set_a,A: a,E: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( poly_of_const_a_b @ r @ A ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ E ) )
= ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ ( embedd971793762689825387on_a_b @ k2 @ K @ A @ E ) ) ) ) ) ) ).
% h.line_extension_hom
thf(fact_236_p_Oa__lcos__mult__one,axiom,
! [M2: set_list_a] :
( ( ord_le8861187494160871172list_a @ M2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ M2 )
= M2 ) ) ).
% p.a_lcos_mult_one
thf(fact_237_d_Oring__primeE_I1_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ring_ring_prime_a_b @ k2 @ P2 )
=> ( P2
!= ( zero_a_b @ k2 ) ) ) ) ).
% d.ring_primeE(1)
thf(fact_238_is__cring,axiom,
cring_a_b @ r ).
% is_cring
thf(fact_239_d_Ois__cring,axiom,
cring_a_b @ k2 ).
% d.is_cring
thf(fact_240_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_241_d_Oline__extension__in__carrier,axiom,
! [K: set_a,A: a,E: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ k2 @ K @ A @ E ) @ ( partia707051561876973205xt_a_b @ k2 ) ) ) ) ) ).
% d.line_extension_in_carrier
thf(fact_242_subalgebra__in__carrier,axiom,
! [K: set_a,V: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_243_carrier__is__subalgebra,axiom,
! [K: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_244_d_Ocarrier__is__subalgebra,axiom,
! [K: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( embedd9027525575939734154ra_a_b @ K @ ( partia707051561876973205xt_a_b @ k2 ) @ k2 ) ) ).
% d.carrier_is_subalgebra
thf(fact_245_d_Osubalgebra__in__carrier,axiom,
! [K: set_a,V: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V @ k2 )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ k2 ) ) ) ).
% d.subalgebra_in_carrier
thf(fact_246_ring__primeE_I1_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( P2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_247_line__extension__consistent,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ( embedd971793762689825387on_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) )
= ( embedd971793762689825387on_a_b @ r ) ) ) ).
% line_extension_consistent
thf(fact_248_d_Oline__extension__consistent,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ k2 )
=> ( ( embedd971793762689825387on_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ k2 ) )
= ( embedd971793762689825387on_a_b @ k2 ) ) ) ).
% d.line_extension_consistent
thf(fact_249_p_Ocarrier__is__subcring,axiom,
subcri7763218559781929323t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( univ_poly_a_b @ r @ k ) ).
% p.carrier_is_subcring
thf(fact_250_p_OsubcringI_H,axiom,
! [H: set_list_a] :
( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ k ) )
=> ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.subcringI'
thf(fact_251_p_Oa__l__coset__subset__G,axiom,
! [H: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ H ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.a_l_coset_subset_G
thf(fact_252_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_253_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_254_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_255_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_256_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_257_subdomain_Oaxioms_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( subcri7763218559781929323t_unit @ H @ R ) ) ).
% subdomain.axioms(1)
thf(fact_258_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_259_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_260_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_261_subcringE_I1_J,axiom,
! [H: set_set_int,R: partia4934656038542163276t_unit] :
( ( subcri1024317279029940167t_unit @ H @ R )
=> ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ R ) ) ) ).
% subcringE(1)
thf(fact_262_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_263_cring_Ocarrier__is__subcring,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( subcri7763218559781929323t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).
% cring.carrier_is_subcring
thf(fact_264_cring_Ocarrier__is__subcring,axiom,
! [R: partia2956882679547061052t_unit] :
( ( cring_5991999922451032090t_unit @ R )
=> ( subcri8676831449680469861t_unit @ ( partia2464479390973590831t_unit @ R ) @ R ) ) ).
% cring.carrier_is_subcring
thf(fact_265_cring_Ocarrier__is__subcring,axiom,
! [R: partia4934656038542163276t_unit] :
( ( cring_3079150759069666002t_unit @ R )
=> ( subcri1024317279029940167t_unit @ ( partia966996272515721803t_unit @ R ) @ R ) ) ).
% cring.carrier_is_subcring
thf(fact_266_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_267_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_268_ring_Osubcring__iff,axiom,
! [R: partia2956882679547061052t_unit,H: set_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( subcri8676831449680469861t_unit @ H @ R )
= ( cring_5991999922451032090t_unit
@ ( partia5725451877363000582t_unit
@ ^ [Uu: set_list_list_a] : H
@ R ) ) ) ) ) ).
% ring.subcring_iff
thf(fact_269_ring_Osubcring__iff,axiom,
! [R: partia4934656038542163276t_unit,H: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ R ) )
=> ( ( subcri1024317279029940167t_unit @ H @ R )
= ( cring_3079150759069666002t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : H
@ R ) ) ) ) ) ).
% ring.subcring_iff
thf(fact_270_ring_Osubcring__iff,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( subcring_a_b @ H @ R )
= ( cring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ R ) ) ) ) ) ).
% ring.subcring_iff
thf(fact_271_ring_Osubcring__iff,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( subcri7763218559781929323t_unit @ H @ R )
= ( cring_3148771470849435808t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ R ) ) ) ) ) ).
% ring.subcring_iff
thf(fact_272_p_Oring__hom__imp__img__ring,axiom,
! [H2: list_a > a,S: partia2175431115845679010xt_a_b] :
( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ ( univ_poly_a_b @ r @ k ) @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
@ S ) ) ) ) ).
% p.ring_hom_imp_img_ring
thf(fact_273_p_Oring__hom__imp__img__ring,axiom,
! [H2: list_a > list_a,S: partia2670972154091845814t_unit] :
( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ ( univ_poly_a_b @ r @ k ) @ S ) )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_list_a_list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
@ S ) ) ) ) ).
% p.ring_hom_imp_img_ring
thf(fact_274_ring_Oring__hom__imp__img__ring,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > a,S: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ R ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_a_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) )
@ S ) ) ) ) ) ).
% ring.ring_hom_imp_img_ring
thf(fact_275_ring_Oring__hom__imp__img__ring,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > a,S: partia2175431115845679010xt_a_b] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_l4142658623432671053t_unit @ R ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R ) )
@ S ) ) ) ) ) ).
% ring.ring_hom_imp_img_ring
thf(fact_276_ring_Oring__hom__imp__img__ring,axiom,
! [R: partia2956882679547061052t_unit,H2: list_list_a > a,S: partia2175431115845679010xt_a_b] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a_a @ H2 @ ( ring_h8078271382950527358it_a_b @ R @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_l347298301471573063t_unit @ R ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_list_list_a_a @ H2 @ ( partia2464479390973590831t_unit @ R ) )
@ S ) ) ) ) ) ).
% ring.ring_hom_imp_img_ring
thf(fact_277_ring_Oring__hom__imp__img__ring,axiom,
! [R: partia4934656038542163276t_unit,H2: set_int > a,S: partia2175431115845679010xt_a_b] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int_a @ H2 @ ( ring_h6431528728137583664it_a_b @ R @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_s6269048424454532197t_unit @ R ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_set_int_a @ H2 @ ( partia966996272515721803t_unit @ R ) )
@ S ) ) ) ) ) ).
% ring.ring_hom_imp_img_ring
thf(fact_278_ring_Oring__hom__imp__img__ring,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > list_a,S: partia2670972154091845814t_unit] :
( ( ring_a_b @ R )
=> ( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ R ) )
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) )
@ S ) ) ) ) ) ).
% ring.ring_hom_imp_img_ring
thf(fact_279_ring_Oring__hom__imp__img__ring,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > list_a,S: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_l4142658623432671053t_unit @ R ) )
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_list_a_list_a @ H2 @ ( partia5361259788508890537t_unit @ R ) )
@ S ) ) ) ) ) ).
% ring.ring_hom_imp_img_ring
thf(fact_280_ring_Oring__hom__imp__img__ring,axiom,
! [R: partia2956882679547061052t_unit,H2: list_list_a > list_a,S: partia2670972154091845814t_unit] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member7168557129179038582list_a @ H2 @ ( ring_h5031276006722532742t_unit @ R @ S ) )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_l347298301471573063t_unit @ R ) )
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_4177985004758493951list_a @ H2 @ ( partia2464479390973590831t_unit @ R ) )
@ S ) ) ) ) ) ).
% ring.ring_hom_imp_img_ring
thf(fact_281_ring_Oring__hom__imp__img__ring,axiom,
! [R: partia4934656038542163276t_unit,H2: set_int > list_a,S: partia2670972154091845814t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member6804298160596253954list_a @ H2 @ ( ring_h8006428831458247992t_unit @ R @ S ) )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_s6269048424454532197t_unit @ R ) )
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_set_int_list_a @ H2 @ ( partia966996272515721803t_unit @ R ) )
@ S ) ) ) ) ) ).
% ring.ring_hom_imp_img_ring
thf(fact_282_p_Oembed__char__eq__0__iff,axiom,
! [N: int] :
( ( ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ k ) @ N )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ N ) ) ).
% p.embed_char_eq_0_iff
thf(fact_283_p_Osubset__Idl__subset,axiom,
! [I2: set_list_a,H: set_list_a] :
( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ord_le8861187494160871172list_a @ H @ I2 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ H ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ I2 ) ) ) ) ).
% p.subset_Idl_subset
thf(fact_284_p_Ogenideal__self,axiom,
! [S: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ord_le8861187494160871172list_a @ S @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ S ) ) ) ).
% p.genideal_self
thf(fact_285_d_Oring__primeI,axiom,
! [P2: a] :
( ( P2
!= ( zero_a_b @ k2 ) )
=> ( ( prime_a_ring_ext_a_b @ k2 @ P2 )
=> ( ring_ring_prime_a_b @ k2 @ P2 ) ) ) ).
% d.ring_primeI
thf(fact_286_ring__primeI,axiom,
! [P2: a] :
( ( P2
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P2 )
=> ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% ring_primeI
thf(fact_287_d_Oring__primeE_I3_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ring_ring_prime_a_b @ k2 @ P2 )
=> ( prime_a_ring_ext_a_b @ k2 @ P2 ) ) ) ).
% d.ring_primeE(3)
thf(fact_288_carrier__is__subcring,axiom,
subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subcring
thf(fact_289_d_Ocarrier__is__subcring,axiom,
subcring_a_b @ ( partia707051561876973205xt_a_b @ k2 ) @ k2 ).
% d.carrier_is_subcring
thf(fact_290_subcringI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subcring_a_b @ H @ r ) ) ).
% subcringI'
thf(fact_291_d_OsubcringI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ k2 )
=> ( subcring_a_b @ H @ k2 ) ) ).
% d.subcringI'
thf(fact_292_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_293_d_Ozero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ k2 @ ( zero_a_b @ k2 ) ).
% d.zero_is_prime(1)
thf(fact_294_ring__primeE_I3_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( prime_a_ring_ext_a_b @ r @ P2 ) ) ) ).
% ring_primeE(3)
thf(fact_295_subcring__iff,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( subcring_a_b @ H @ r )
= ( cring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r ) ) ) ) ).
% subcring_iff
thf(fact_296_d_Osubcring__iff,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( subcring_a_b @ H @ k2 )
= ( cring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ k2 ) ) ) ) ).
% d.subcring_iff
thf(fact_297_embed__char__eq__0__iff,axiom,
! [N: int] :
( ( ( ring_int_embed_a_b @ r @ N )
= ( zero_a_b @ r ) )
= ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ ( ring_char_a_b @ r ) ) @ N ) ) ).
% embed_char_eq_0_iff
thf(fact_298_d_Oembed__char__eq__0__iff,axiom,
! [N: int] :
( ( ( ring_int_embed_a_b @ k2 @ N )
= ( zero_a_b @ k2 ) )
= ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ ( ring_char_a_b @ k2 ) ) @ N ) ) ).
% d.embed_char_eq_0_iff
thf(fact_299_ring_Oembed__char__eq__0__iff,axiom,
! [R: partia2175431115845679010xt_a_b,N: int] :
( ( ring_a_b @ R )
=> ( ( ( ring_int_embed_a_b @ R @ N )
= ( zero_a_b @ R ) )
= ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ ( ring_char_a_b @ R ) ) @ N ) ) ) ).
% ring.embed_char_eq_0_iff
thf(fact_300_ring_Oembed__char__eq__0__iff,axiom,
! [R: partia2670972154091845814t_unit,N: int] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ( ring_i7845631192508594762t_unit @ R @ N )
= ( zero_l4142658623432671053t_unit @ R ) )
= ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ ( ring_c500279861223467766t_unit @ R ) ) @ N ) ) ) ).
% ring.embed_char_eq_0_iff
thf(fact_301_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_302_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_303_ring__iso__memE_I1_J,axiom,
! [H2: a > set_int,R: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit,X: a] :
( ( member_a_set_int @ H2 @ ( ring_i4118987378922720216t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_set_int @ ( H2 @ X ) @ ( partia966996272515721803t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_304_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_305_ring__iso__memE_I1_J,axiom,
! [H2: set_int > a,R: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,X: set_int] :
( ( member_set_int_a @ H2 @ ( ring_i2213127016525301402it_a_b @ R @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_306_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_307_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_308_ring__iso__memE_I1_J,axiom,
! [H2: list_a > set_int,R: partia2670972154091845814t_unit,S: partia4934656038542163276t_unit,X: list_a] :
( ( member5309957867533224448et_int @ H2 @ ( ring_i5545097000213460560t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_set_int @ ( H2 @ X ) @ ( partia966996272515721803t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_309_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_310_ring__iso__memE_I1_J,axiom,
! [H2: set_int > list_a,R: partia4934656038542163276t_unit,S: partia2670972154091845814t_unit,X: set_int] :
( ( member6804298160596253954list_a @ H2 @ ( ring_i8159631783994945698t_unit @ R @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
=> ( member_list_a @ ( H2 @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_311_ring_Oring__hom__restrict,axiom,
! [R: partia2175431115845679010xt_a_b,F: a > list_a,S: partia2670972154091845814t_unit,G: a > list_a] :
( ( ring_a_b @ R )
=> ( ( member_a_list_a @ F @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( F @ R2 )
= ( G @ R2 ) ) )
=> ( member_a_list_a @ G @ ( ring_h405018892823518980t_unit @ R @ S ) ) ) ) ) ).
% ring.ring_hom_restrict
thf(fact_312_ring_Oring__iso__restrict,axiom,
! [R: partia4934656038542163276t_unit,F: set_int > list_a,S: partia2670972154091845814t_unit,G: set_int > list_a] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member6804298160596253954list_a @ F @ ( ring_i8159631783994945698t_unit @ R @ S ) )
=> ( ! [R2: set_int] :
( ( member_set_int @ R2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( F @ R2 )
= ( G @ R2 ) ) )
=> ( member6804298160596253954list_a @ G @ ( ring_i8159631783994945698t_unit @ R @ S ) ) ) ) ) ).
% ring.ring_iso_restrict
thf(fact_313_ring_Oring__iso__restrict,axiom,
! [R: partia4934656038542163276t_unit,F: set_int > a,S: partia2175431115845679010xt_a_b,G: set_int > a] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int_a @ F @ ( ring_i2213127016525301402it_a_b @ R @ S ) )
=> ( ! [R2: set_int] :
( ( member_set_int @ R2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( F @ R2 )
= ( G @ R2 ) ) )
=> ( member_set_int_a @ G @ ( ring_i2213127016525301402it_a_b @ R @ S ) ) ) ) ) ).
% ring.ring_iso_restrict
thf(fact_314_ring_Oring__iso__imp__img__ring,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > a,S: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ R ) )
@ S ) ) ) ) ).
% ring.ring_iso_imp_img_ring
thf(fact_315_ring_Oring__iso__imp__img__ring,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > list_a,S: partia2670972154091845814t_unit] :
( ( ring_a_b @ R )
=> ( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ R ) )
@ S ) ) ) ) ).
% ring.ring_iso_imp_img_ring
thf(fact_316_ring_Oring__iso__imp__img__ring,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > a,S: partia2175431115845679010xt_a_b] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a_a @ H2 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_l4142658623432671053t_unit @ R ) )
@ S ) ) ) ) ).
% ring.ring_iso_imp_img_ring
thf(fact_317_ring_Oring__iso__imp__img__ring,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > list_a,S: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a_list_a @ H2 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_l4142658623432671053t_unit @ R ) )
@ S ) ) ) ) ).
% ring.ring_iso_imp_img_ring
thf(fact_318_ring_Oring__iso__imp__img__ring,axiom,
! [R: partia4934656038542163276t_unit,H2: set_int > list_a,S: partia2670972154091845814t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member6804298160596253954list_a @ H2 @ ( ring_i8159631783994945698t_unit @ R @ S ) )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_s6269048424454532197t_unit @ R ) )
@ S ) ) ) ) ).
% ring.ring_iso_imp_img_ring
thf(fact_319_ring_Oring__iso__imp__img__ring,axiom,
! [R: partia4934656038542163276t_unit,H2: set_int > a,S: partia2175431115845679010xt_a_b] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int_a @ H2 @ ( ring_i2213127016525301402it_a_b @ R @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_s6269048424454532197t_unit @ R ) )
@ S ) ) ) ) ).
% ring.ring_iso_imp_img_ring
thf(fact_320_domain_Oring__iso__imp__img__domain,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > a,S: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( domain_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ R ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_321_domain_Oring__iso__imp__img__domain,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > list_a,S: partia2670972154091845814t_unit] :
( ( domain_a_b @ R )
=> ( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( domain6553523120543210313t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ R ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_322_domain_Oring__iso__imp__img__domain,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > a,S: partia2175431115845679010xt_a_b] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a_a @ H2 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( domain_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_l4142658623432671053t_unit @ R ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_323_domain_Oring__iso__imp__img__domain,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > list_a,S: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a_list_a @ H2 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( domain6553523120543210313t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_l4142658623432671053t_unit @ R ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_324_domain_Oring__iso__imp__img__domain,axiom,
! [R: partia4934656038542163276t_unit,H2: set_int > list_a,S: partia2670972154091845814t_unit] :
( ( domain6183376680155302761t_unit @ R )
=> ( ( member6804298160596253954list_a @ H2 @ ( ring_i8159631783994945698t_unit @ R @ S ) )
=> ( domain6553523120543210313t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_s6269048424454532197t_unit @ R ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_325_domain_Oring__iso__imp__img__domain,axiom,
! [R: partia4934656038542163276t_unit,H2: set_int > a,S: partia2175431115845679010xt_a_b] :
( ( domain6183376680155302761t_unit @ R )
=> ( ( member_set_int_a @ H2 @ ( ring_i2213127016525301402it_a_b @ R @ S ) )
=> ( domain_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_s6269048424454532197t_unit @ R ) )
@ S ) ) ) ) ).
% domain.ring_iso_imp_img_domain
thf(fact_326_cring_Oring__iso__imp__img__cring,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > a,S: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( cring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ R ) )
@ S ) ) ) ) ).
% cring.ring_iso_imp_img_cring
thf(fact_327_cring_Oring__iso__imp__img__cring,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > list_a,S: partia2670972154091845814t_unit] :
( ( cring_a_b @ R )
=> ( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( cring_3148771470849435808t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ R ) )
@ S ) ) ) ) ).
% cring.ring_iso_imp_img_cring
thf(fact_328_cring_Oring__iso__imp__img__cring,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > list_a,S: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a_list_a @ H2 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( cring_3148771470849435808t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_l4142658623432671053t_unit @ R ) )
@ S ) ) ) ) ).
% cring.ring_iso_imp_img_cring
thf(fact_329_cring_Oring__iso__imp__img__cring,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > a,S: partia2175431115845679010xt_a_b] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a_a @ H2 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( cring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_l4142658623432671053t_unit @ R ) )
@ S ) ) ) ) ).
% cring.ring_iso_imp_img_cring
thf(fact_330_cring_Oring__iso__imp__img__cring,axiom,
! [R: partia4934656038542163276t_unit,H2: set_int > list_a,S: partia2670972154091845814t_unit] :
( ( cring_3079150759069666002t_unit @ R )
=> ( ( member6804298160596253954list_a @ H2 @ ( ring_i8159631783994945698t_unit @ R @ S ) )
=> ( cring_3148771470849435808t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_s6269048424454532197t_unit @ R ) )
@ S ) ) ) ) ).
% cring.ring_iso_imp_img_cring
thf(fact_331_cring_Oring__iso__imp__img__cring,axiom,
! [R: partia4934656038542163276t_unit,H2: set_int > a,S: partia2175431115845679010xt_a_b] :
( ( cring_3079150759069666002t_unit @ R )
=> ( ( member_set_int_a @ H2 @ ( ring_i2213127016525301402it_a_b @ R @ S ) )
=> ( cring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_s6269048424454532197t_unit @ R ) )
@ S ) ) ) ) ).
% cring.ring_iso_imp_img_cring
thf(fact_332_a__coset__hom_I1_J,axiom,
! [H2: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,I2: set_a,A: a] :
( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( image_a_list_a @ H2 @ ( a_l_coset_a_b @ R @ A @ I2 ) )
= ( a_l_co7008843373686234386t_unit @ S @ ( H2 @ A ) @ ( image_a_list_a @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_333_a__coset__hom_I1_J,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,I2: set_a,A: a] :
( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( image_a_a @ H2 @ ( a_l_coset_a_b @ R @ A @ I2 ) )
= ( a_l_coset_a_b @ S @ ( H2 @ A ) @ ( image_a_a @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_334_a__coset__hom_I1_J,axiom,
! [H2: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,I2: set_list_a,A: list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( image_list_a_list_a @ H2 @ ( a_l_co7008843373686234386t_unit @ R @ A @ I2 ) )
= ( a_l_co7008843373686234386t_unit @ S @ ( H2 @ A ) @ ( image_list_a_list_a @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_335_a__coset__hom_I1_J,axiom,
! [H2: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,I2: set_list_a,A: list_a] :
( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( image_list_a_a @ H2 @ ( a_l_co7008843373686234386t_unit @ R @ A @ I2 ) )
= ( a_l_coset_a_b @ S @ ( H2 @ A ) @ ( image_list_a_a @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_336_a__coset__hom_I1_J,axiom,
! [H2: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,I2: set_list_list_a,A: list_list_a] :
( ( member7168557129179038582list_a @ H2 @ ( ring_h5031276006722532742t_unit @ R @ S ) )
=> ( ( ord_le8488217952732425610list_a @ I2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( image_4177985004758493951list_a @ H2 @ ( a_l_co3970804650394549132t_unit @ R @ A @ I2 ) )
= ( a_l_co7008843373686234386t_unit @ S @ ( H2 @ A ) @ ( image_4177985004758493951list_a @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_337_a__coset__hom_I1_J,axiom,
! [H2: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,I2: set_list_list_a,A: list_list_a] :
( ( member_list_list_a_a @ H2 @ ( ring_h8078271382950527358it_a_b @ R @ S ) )
=> ( ( ord_le8488217952732425610list_a @ I2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( image_list_list_a_a @ H2 @ ( a_l_co3970804650394549132t_unit @ R @ A @ I2 ) )
= ( a_l_coset_a_b @ S @ ( H2 @ A ) @ ( image_list_list_a_a @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_338_a__coset__hom_I1_J,axiom,
! [H2: set_int > list_a,R: partia4934656038542163276t_unit,S: partia2670972154091845814t_unit,I2: set_set_int,A: set_int] :
( ( member6804298160596253954list_a @ H2 @ ( ring_h8006428831458247992t_unit @ R @ S ) )
=> ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R ) )
=> ( ( image_set_int_list_a @ H2 @ ( a_l_co3504123944629134560t_unit @ R @ A @ I2 ) )
= ( a_l_co7008843373686234386t_unit @ S @ ( H2 @ A ) @ ( image_set_int_list_a @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_339_a__coset__hom_I1_J,axiom,
! [H2: set_int > a,R: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,I2: set_set_int,A: set_int] :
( ( member_set_int_a @ H2 @ ( ring_h6431528728137583664it_a_b @ R @ S ) )
=> ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R ) )
=> ( ( image_set_int_a @ H2 @ ( a_l_co3504123944629134560t_unit @ R @ A @ I2 ) )
= ( a_l_coset_a_b @ S @ ( H2 @ A ) @ ( image_set_int_a @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_340_ring__hom__ring_Oimg__is__ring,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a] :
( ( ring_h4024360765257340990_b_a_b @ R @ S @ H2 )
=> ( ring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_a_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) )
@ S ) ) ) ).
% ring_hom_ring.img_is_ring
thf(fact_341_ring__hom__ring_Oimg__is__ring,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a] :
( ( ring_h7848885096329822662it_a_b @ R @ S @ H2 )
=> ( ring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R ) )
@ S ) ) ) ).
% ring_hom_ring.img_is_ring
thf(fact_342_ring__hom__ring_Oimg__is__ring,axiom,
! [R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,H2: list_list_a > a] :
( ( ring_h3841606220870141376it_a_b @ R @ S @ H2 )
=> ( ring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_list_list_a_a @ H2 @ ( partia2464479390973590831t_unit @ R ) )
@ S ) ) ) ).
% ring_hom_ring.img_is_ring
thf(fact_343_ring__hom__ring_Oimg__is__ring,axiom,
! [R: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,H2: set_int > a] :
( ( ring_h5360235766848426350it_a_b @ R @ S @ H2 )
=> ( ring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_set_int_a @ H2 @ ( partia966996272515721803t_unit @ R ) )
@ S ) ) ) ).
% ring_hom_ring.img_is_ring
thf(fact_344_ring__hom__ring_Oimg__is__ring,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a] :
( ( ring_h5357930050666032198t_unit @ R @ S @ H2 )
=> ( ring_l6212528067271185461t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) )
@ S ) ) ) ).
% ring_hom_ring.img_is_ring
thf(fact_345_ring__hom__ring_Oimg__is__ring,axiom,
! [R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H2: list_a > list_a] :
( ( ring_h1334922693953046990t_unit @ R @ S @ H2 )
=> ( ring_l6212528067271185461t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_list_a_list_a @ H2 @ ( partia5361259788508890537t_unit @ R ) )
@ S ) ) ) ).
% ring_hom_ring.img_is_ring
thf(fact_346_ring__hom__ring_Oimg__is__ring,axiom,
! [R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,H2: list_list_a > list_a] :
( ( ring_h4589914651911841480t_unit @ R @ S @ H2 )
=> ( ring_l6212528067271185461t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_4177985004758493951list_a @ H2 @ ( partia2464479390973590831t_unit @ R ) )
@ S ) ) ) ).
% ring_hom_ring.img_is_ring
thf(fact_347_ring__hom__ring_Oimg__is__ring,axiom,
! [R: partia4934656038542163276t_unit,S: partia2670972154091845814t_unit,H2: set_int > list_a] :
( ( ring_h3064273570435554678t_unit @ R @ S @ H2 )
=> ( ring_l6212528067271185461t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_set_int_list_a @ H2 @ ( partia966996272515721803t_unit @ R ) )
@ S ) ) ) ).
% ring_hom_ring.img_is_ring
thf(fact_348_ring__hom__ring_Oimg__is__domain,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a] :
( ( ring_h4024360765257340990_b_a_b @ R @ S @ H2 )
=> ( ( domain_a_b @ S )
=> ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_a_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) )
@ S ) ) ) ) ).
% ring_hom_ring.img_is_domain
thf(fact_349_ring__hom__ring_Oimg__is__domain,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a] :
( ( ring_h7848885096329822662it_a_b @ R @ S @ H2 )
=> ( ( domain_a_b @ S )
=> ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R ) )
@ S ) ) ) ) ).
% ring_hom_ring.img_is_domain
thf(fact_350_ring__hom__ring_Oimg__is__domain,axiom,
! [R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,H2: list_list_a > a] :
( ( ring_h3841606220870141376it_a_b @ R @ S @ H2 )
=> ( ( domain_a_b @ S )
=> ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_list_list_a_a @ H2 @ ( partia2464479390973590831t_unit @ R ) )
@ S ) ) ) ) ).
% ring_hom_ring.img_is_domain
thf(fact_351_ring__hom__ring_Oimg__is__domain,axiom,
! [R: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,H2: set_int > a] :
( ( ring_h5360235766848426350it_a_b @ R @ S @ H2 )
=> ( ( domain_a_b @ S )
=> ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_set_int_a @ H2 @ ( partia966996272515721803t_unit @ R ) )
@ S ) ) ) ) ).
% ring_hom_ring.img_is_domain
thf(fact_352_ring__hom__ring_Oimg__is__domain,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a] :
( ( ring_h5357930050666032198t_unit @ R @ S @ H2 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( domain6553523120543210313t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) )
@ S ) ) ) ) ).
% ring_hom_ring.img_is_domain
thf(fact_353_ring__hom__ring_Oimg__is__domain,axiom,
! [R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H2: list_a > list_a] :
( ( ring_h1334922693953046990t_unit @ R @ S @ H2 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( domain6553523120543210313t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_list_a_list_a @ H2 @ ( partia5361259788508890537t_unit @ R ) )
@ S ) ) ) ) ).
% ring_hom_ring.img_is_domain
thf(fact_354_ring__hom__ring_Oimg__is__domain,axiom,
! [R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,H2: list_list_a > list_a] :
( ( ring_h4589914651911841480t_unit @ R @ S @ H2 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( domain6553523120543210313t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_4177985004758493951list_a @ H2 @ ( partia2464479390973590831t_unit @ R ) )
@ S ) ) ) ) ).
% ring_hom_ring.img_is_domain
thf(fact_355_ring__hom__ring_Oimg__is__domain,axiom,
! [R: partia4934656038542163276t_unit,S: partia2670972154091845814t_unit,H2: set_int > list_a] :
( ( ring_h3064273570435554678t_unit @ R @ S @ H2 )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( domain6553523120543210313t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_set_int_list_a @ H2 @ ( partia966996272515721803t_unit @ R ) )
@ S ) ) ) ) ).
% ring_hom_ring.img_is_domain
thf(fact_356_ring__hom__ring_Oimg__is__cring,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a] :
( ( ring_h4024360765257340990_b_a_b @ R @ S @ H2 )
=> ( ( cring_a_b @ S )
=> ( cring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_a_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) )
@ S ) ) ) ) ).
% ring_hom_ring.img_is_cring
thf(fact_357_ring__hom__ring_Oimg__is__cring,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a] :
( ( ring_h7848885096329822662it_a_b @ R @ S @ H2 )
=> ( ( cring_a_b @ S )
=> ( cring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R ) )
@ S ) ) ) ) ).
% ring_hom_ring.img_is_cring
thf(fact_358_ring__hom__ring_Oimg__is__cring,axiom,
! [R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,H2: list_list_a > a] :
( ( ring_h3841606220870141376it_a_b @ R @ S @ H2 )
=> ( ( cring_a_b @ S )
=> ( cring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_list_list_a_a @ H2 @ ( partia2464479390973590831t_unit @ R ) )
@ S ) ) ) ) ).
% ring_hom_ring.img_is_cring
thf(fact_359_ring__hom__ring_Oimg__is__cring,axiom,
! [R: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,H2: set_int > a] :
( ( ring_h5360235766848426350it_a_b @ R @ S @ H2 )
=> ( ( cring_a_b @ S )
=> ( cring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_set_int_a @ H2 @ ( partia966996272515721803t_unit @ R ) )
@ S ) ) ) ) ).
% ring_hom_ring.img_is_cring
thf(fact_360_ring__hom__ring_Oimg__is__cring,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a] :
( ( ring_h5357930050666032198t_unit @ R @ S @ H2 )
=> ( ( cring_3148771470849435808t_unit @ S )
=> ( cring_3148771470849435808t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) )
@ S ) ) ) ) ).
% ring_hom_ring.img_is_cring
thf(fact_361_ring__hom__ring_Oimg__is__cring,axiom,
! [R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H2: list_a > list_a] :
( ( ring_h1334922693953046990t_unit @ R @ S @ H2 )
=> ( ( cring_3148771470849435808t_unit @ S )
=> ( cring_3148771470849435808t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_list_a_list_a @ H2 @ ( partia5361259788508890537t_unit @ R ) )
@ S ) ) ) ) ).
% ring_hom_ring.img_is_cring
thf(fact_362_ring__hom__ring_Oimg__is__cring,axiom,
! [R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,H2: list_list_a > list_a] :
( ( ring_h4589914651911841480t_unit @ R @ S @ H2 )
=> ( ( cring_3148771470849435808t_unit @ S )
=> ( cring_3148771470849435808t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_4177985004758493951list_a @ H2 @ ( partia2464479390973590831t_unit @ R ) )
@ S ) ) ) ) ).
% ring_hom_ring.img_is_cring
thf(fact_363_ring__hom__ring_Oimg__is__cring,axiom,
! [R: partia4934656038542163276t_unit,S: partia2670972154091845814t_unit,H2: set_int > list_a] :
( ( ring_h3064273570435554678t_unit @ R @ S @ H2 )
=> ( ( cring_3148771470849435808t_unit @ S )
=> ( cring_3148771470849435808t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_set_int_list_a @ H2 @ ( partia966996272515721803t_unit @ R ) )
@ S ) ) ) ) ).
% ring_hom_ring.img_is_cring
thf(fact_364_h_Oinj__on__domain,axiom,
( ( inj_on_a_list_a @ ( poly_of_const_a_b @ r ) @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ k ) )
=> ( domain_a_b @ k2 ) ) ) ).
% h.inj_on_domain
thf(fact_365_p_Ochar__ring__is__subcring,axiom,
subcri7763218559781929323t_unit @ ( image_int_list_a @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ k ) ) @ top_top_set_int ) @ ( univ_poly_a_b @ r @ k ) ).
% p.char_ring_is_subcring
thf(fact_366_h_Ochar__consistent,axiom,
( ( inj_on_a_list_a @ ( poly_of_const_a_b @ r ) @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ring_char_a_b @ k2 )
= ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% h.char_consistent
thf(fact_367_d_Oring__irreducibleE_I1_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ring_r999134135267193926le_a_b @ k2 @ R3 )
=> ( R3
!= ( zero_a_b @ k2 ) ) ) ) ).
% d.ring_irreducibleE(1)
thf(fact_368_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_369_p_Ochar__ring__is__subring,axiom,
subrin6918843898125473962t_unit @ ( image_int_list_a @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ k ) ) @ top_top_set_int ) @ ( univ_poly_a_b @ r @ k ) ).
% p.char_ring_is_subring
thf(fact_370_p_Oint__embed__zero,axiom,
( ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ k ) @ zero_zero_int )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.int_embed_zero
thf(fact_371_p_Oup__minus__closed,axiom,
! [P2: nat > list_a,Q2: nat > list_a] :
( ( member_nat_list_a @ P2 @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_nat_list_a @ Q2 @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_nat_list_a
@ ^ [I3: nat] : ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ ( P2 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.up_minus_closed
thf(fact_372_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_373_d_Oa__l__coset__subset__G,axiom,
! [H: set_a,X: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ k2 @ X @ H ) @ ( partia707051561876973205xt_a_b @ k2 ) ) ) ) ).
% d.a_l_coset_subset_G
thf(fact_374_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_375_d_Oa__lcos__mult__one,axiom,
! [M2: set_a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( a_l_coset_a_b @ k2 @ ( zero_a_b @ k2 ) @ M2 )
= M2 ) ) ).
% d.a_lcos_mult_one
thf(fact_376_int__embed__zero,axiom,
( ( ring_int_embed_a_b @ r @ zero_zero_int )
= ( zero_a_b @ r ) ) ).
% int_embed_zero
thf(fact_377_d_Oint__embed__zero,axiom,
( ( ring_int_embed_a_b @ k2 @ zero_zero_int )
= ( zero_a_b @ k2 ) ) ).
% d.int_embed_zero
thf(fact_378_char__ring__is__subring,axiom,
subring_a_b @ ( image_int_a @ ( ring_int_embed_a_b @ r ) @ top_top_set_int ) @ r ).
% char_ring_is_subring
thf(fact_379_d_Ochar__ring__is__subring,axiom,
subring_a_b @ ( image_int_a @ ( ring_int_embed_a_b @ k2 ) @ top_top_set_int ) @ k2 ).
% d.char_ring_is_subring
thf(fact_380_char__ring__is__subcring,axiom,
subcring_a_b @ ( image_int_a @ ( ring_int_embed_a_b @ r ) @ top_top_set_int ) @ r ).
% char_ring_is_subcring
thf(fact_381_d_Ochar__ring__is__subcring,axiom,
subcring_a_b @ ( image_int_a @ ( ring_int_embed_a_b @ k2 ) @ top_top_set_int ) @ k2 ).
% d.char_ring_is_subcring
thf(fact_382_char__ring__is__subdomain,axiom,
subdomain_a_b @ ( image_int_a @ ( ring_int_embed_a_b @ r ) @ top_top_set_int ) @ r ).
% char_ring_is_subdomain
thf(fact_383_d_Ochar__ring__is__subdomain,axiom,
subdomain_a_b @ ( image_int_a @ ( ring_int_embed_a_b @ k2 ) @ top_top_set_int ) @ k2 ).
% d.char_ring_is_subdomain
thf(fact_384_ring__iso__imp__img__ring,axiom,
! [H2: a > a,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ r @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_ring
thf(fact_385_ring__iso__imp__img__ring,axiom,
! [H2: a > list_a,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ r @ S ) )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_ring
thf(fact_386_d_Oring__iso__imp__img__ring,axiom,
! [H2: a > a,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ k2 @ S ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ k2 ) )
@ S ) ) ) ).
% d.ring_iso_imp_img_ring
thf(fact_387_d_Oring__iso__imp__img__ring,axiom,
! [H2: a > list_a,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ k2 @ S ) )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ k2 ) )
@ S ) ) ) ).
% d.ring_iso_imp_img_ring
thf(fact_388_ring__iso__imp__img__domain,axiom,
! [H2: a > a,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ r @ S ) )
=> ( domain_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_domain
thf(fact_389_ring__iso__imp__img__domain,axiom,
! [H2: a > list_a,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ r @ S ) )
=> ( domain6553523120543210313t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_domain
thf(fact_390_d_Oring__iso__imp__img__domain,axiom,
! [H2: a > a,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ k2 @ S ) )
=> ( domain_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ k2 ) )
@ S ) ) ) ).
% d.ring_iso_imp_img_domain
thf(fact_391_d_Oring__iso__imp__img__domain,axiom,
! [H2: a > list_a,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ k2 @ S ) )
=> ( domain6553523120543210313t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ k2 ) )
@ S ) ) ) ).
% d.ring_iso_imp_img_domain
thf(fact_392_ring__iso__imp__img__cring,axiom,
! [H2: a > a,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ r @ S ) )
=> ( cring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_cring
thf(fact_393_ring__iso__imp__img__cring,axiom,
! [H2: a > list_a,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ r @ S ) )
=> ( cring_3148771470849435808t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_cring
thf(fact_394_d_Oring__iso__imp__img__cring,axiom,
! [H2: a > a,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ k2 @ S ) )
=> ( cring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ k2 ) )
@ S ) ) ) ).
% d.ring_iso_imp_img_cring
thf(fact_395_d_Oring__iso__imp__img__cring,axiom,
! [H2: a > list_a,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ k2 @ S ) )
=> ( cring_3148771470849435808t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ k2 ) )
@ S ) ) ) ).
% d.ring_iso_imp_img_cring
thf(fact_396_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_397_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_398_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_399_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_400_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_401_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_402_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_403_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_404_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_405_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_406_ring__hom__ring_Ochar__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a] :
( ( ring_h5357930050666032198t_unit @ R @ S @ H2 )
=> ( ( inj_on_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_char_a_b @ R )
= ( ring_c500279861223467766t_unit @ S ) ) ) ) ).
% ring_hom_ring.char_consistent
thf(fact_407_ring__hom__ring_Ochar__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a] :
( ( ring_h4024360765257340990_b_a_b @ R @ S @ H2 )
=> ( ( inj_on_a_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_char_a_b @ R )
= ( ring_char_a_b @ S ) ) ) ) ).
% ring_hom_ring.char_consistent
thf(fact_408_ring__hom__ring_Ochar__consistent,axiom,
! [R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H2: list_a > list_a] :
( ( ring_h1334922693953046990t_unit @ R @ S @ H2 )
=> ( ( inj_on_list_a_list_a @ H2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_c500279861223467766t_unit @ R )
= ( ring_c500279861223467766t_unit @ S ) ) ) ) ).
% ring_hom_ring.char_consistent
thf(fact_409_ring__hom__ring_Ochar__consistent,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a] :
( ( ring_h7848885096329822662it_a_b @ R @ S @ H2 )
=> ( ( inj_on_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_c500279861223467766t_unit @ R )
= ( ring_char_a_b @ S ) ) ) ) ).
% ring_hom_ring.char_consistent
thf(fact_410_ring__hom__ring_Ochar__consistent,axiom,
! [R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,H2: list_list_a > list_a] :
( ( ring_h4589914651911841480t_unit @ R @ S @ H2 )
=> ( ( inj_on2455837796590618731list_a @ H2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_c667363890163664752t_unit @ R )
= ( ring_c500279861223467766t_unit @ S ) ) ) ) ).
% ring_hom_ring.char_consistent
thf(fact_411_ring__hom__ring_Ochar__consistent,axiom,
! [R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,H2: list_list_a > a] :
( ( ring_h3841606220870141376it_a_b @ R @ S @ H2 )
=> ( ( inj_on_list_list_a_a @ H2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_c667363890163664752t_unit @ R )
= ( ring_char_a_b @ S ) ) ) ) ).
% ring_hom_ring.char_consistent
thf(fact_412_ring__hom__ring_Ochar__consistent,axiom,
! [R: partia4934656038542163276t_unit,S: partia2670972154091845814t_unit,H2: set_int > list_a] :
( ( ring_h3064273570435554678t_unit @ R @ S @ H2 )
=> ( ( inj_on8112305350124607983list_a @ H2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( ring_c6147214092195050492t_unit @ R )
= ( ring_c500279861223467766t_unit @ S ) ) ) ) ).
% ring_hom_ring.char_consistent
thf(fact_413_ring__hom__ring_Ochar__consistent,axiom,
! [R: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,H2: set_int > a] :
( ( ring_h5360235766848426350it_a_b @ R @ S @ H2 )
=> ( ( inj_on_set_int_a @ H2 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( ring_c6147214092195050492t_unit @ R )
= ( ring_char_a_b @ S ) ) ) ) ).
% ring_hom_ring.char_consistent
thf(fact_414_ring_Oint__embed__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( ring_int_embed_a_b @ R @ zero_zero_int )
= ( zero_a_b @ R ) ) ) ).
% ring.int_embed_zero
thf(fact_415_ring_Oint__embed__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ring_i7845631192508594762t_unit @ R @ zero_zero_int )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% ring.int_embed_zero
thf(fact_416_ring_Ochar__ring__is__subring,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( subring_a_b @ ( image_int_a @ ( ring_int_embed_a_b @ R ) @ top_top_set_int ) @ R ) ) ).
% ring.char_ring_is_subring
thf(fact_417_ring_Ochar__ring__is__subring,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( subrin6918843898125473962t_unit @ ( image_int_list_a @ ( ring_i7845631192508594762t_unit @ R ) @ top_top_set_int ) @ R ) ) ).
% ring.char_ring_is_subring
thf(fact_418_cring_Ochar__ring__is__subcring,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( subcring_a_b @ ( image_int_a @ ( ring_int_embed_a_b @ R ) @ top_top_set_int ) @ R ) ) ).
% cring.char_ring_is_subcring
thf(fact_419_cring_Ochar__ring__is__subcring,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( subcri7763218559781929323t_unit @ ( image_int_list_a @ ( ring_i7845631192508594762t_unit @ R ) @ top_top_set_int ) @ R ) ) ).
% cring.char_ring_is_subcring
thf(fact_420_domain_Ochar__ring__is__subdomain,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( subdomain_a_b @ ( image_int_a @ ( ring_int_embed_a_b @ R ) @ top_top_set_int ) @ R ) ) ).
% domain.char_ring_is_subdomain
thf(fact_421_domain_Ochar__ring__is__subdomain,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( subdom7821232466298058046t_unit @ ( image_int_list_a @ ( ring_i7845631192508594762t_unit @ R ) @ top_top_set_int ) @ R ) ) ).
% domain.char_ring_is_subdomain
thf(fact_422_p_Obound__upD,axiom,
! [F: nat > list_a] :
( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ? [N2: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ N2 @ F ) ) ).
% p.bound_upD
thf(fact_423_p_Ochar__iso,axiom,
( member6804298160596253954list_a @ ( ring_c7096748559348378049t_unit @ ( univ_poly_a_b @ r @ k ) )
@ ( ring_i8159631783994945698t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ k ) ) ) )
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_int_list_a @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ k ) ) @ top_top_set_int )
@ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.char_iso
thf(fact_424_h_Oinv__ring__hom,axiom,
! [K: set_a] :
( ( inj_on_a_list_a @ ( poly_of_const_a_b @ r ) @ K )
=> ( ( subring_a_b @ K @ k2 )
=> ( ring_h7848885096329822662it_a_b
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K )
@ ( univ_poly_a_b @ r @ k ) )
@ k2
@ ( hilber1222834799880672978list_a @ K @ ( poly_of_const_a_b @ r ) ) ) ) ) ).
% h.inv_ring_hom
thf(fact_425_p_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ N )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.nat_pow_zero
thf(fact_426_p_Oa__lcos__m__assoc,axiom,
! [M2: set_list_a,G: list_a,H2: list_a] :
( ( ord_le8861187494160871172list_a @ M2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ G @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ H2 @ M2 ) )
= ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ G @ H2 ) @ M2 ) ) ) ) ) ).
% p.a_lcos_m_assoc
thf(fact_427_h_Oa__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( member_list_a @ ( poly_of_const_a_b @ r @ ( a_inv_a_b @ k2 @ X ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% h.a_inv_closed
thf(fact_428_p_Oadd__pow__consistent,axiom,
! [K: set_list_a,K2: list_a,I: int] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( member_list_a @ K2 @ K )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ K2 )
= ( add_po2638046716968164713it_int
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ ( univ_poly_a_b @ r @ k ) )
@ I
@ K2 ) ) ) ) ).
% p.add_pow_consistent
thf(fact_429_a__inv__inj,axiom,
inj_on_a_a @ ( a_inv_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% a_inv_inj
thf(fact_430_d_Oadd_Oinv__inj,axiom,
inj_on_a_a @ ( a_inv_a_b @ k2 ) @ ( partia707051561876973205xt_a_b @ k2 ) ).
% d.add.inv_inj
thf(fact_431_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_432_d_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ k2 @ ( zero_a_b @ k2 ) @ N )
= ( zero_a_b @ k2 ) ) ) ).
% d.nat_pow_zero
thf(fact_433_p_Oadd_Ol__cancel,axiom,
! [C: list_a,A: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ C @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ C @ B ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( A = B ) ) ) ) ) ).
% p.add.l_cancel
thf(fact_434_p_Oadd_Om__assoc,axiom,
! [X: list_a,Y3: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ Z ) ) ) ) ) ) ).
% p.add.m_assoc
thf(fact_435_p_Oadd_Om__comm,axiom,
! [X: list_a,Y3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ X ) ) ) ) ).
% p.add.m_comm
thf(fact_436_p_Oadd_Om__lcomm,axiom,
! [X: list_a,Y3: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ Z ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Z ) ) ) ) ) ) ).
% p.add.m_lcomm
thf(fact_437_p_Oadd_Or__cancel,axiom,
! [A: list_a,C: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ C )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ C ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( A = B ) ) ) ) ) ).
% p.add.r_cancel
thf(fact_438_p_Oadd_Oinj__on__cmult,axiom,
! [C: list_a] :
( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( inj_on_list_a_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ C ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.add.inj_on_cmult
thf(fact_439_p_Oadd_Oinj__on__multc,axiom,
! [C: list_a] :
( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( inj_on_list_a_list_a
@ ^ [X2: list_a] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ C )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.add.inj_on_multc
thf(fact_440_p_Oadd_Osurj__const__mult,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( image_list_a_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.add.surj_const_mult
thf(fact_441_p_Oup__add__closed,axiom,
! [P2: nat > list_a,Q2: nat > list_a] :
( ( member_nat_list_a @ P2 @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_nat_list_a @ Q2 @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_nat_list_a
@ ^ [I3: nat] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( P2 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.up_add_closed
thf(fact_442_p_Oadd_Oinv__comm,axiom,
! [X: list_a,Y3: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).
% p.add.inv_comm
thf(fact_443_p_Oadd_Ol__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X3 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.add.l_inv_ex
thf(fact_444_p_Oadd_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ U @ X3 )
= X3 ) )
=> ( U
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.add.one_unique
thf(fact_445_p_Oadd_Or__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.add.r_inv_ex
thf(fact_446_p_Ominus__unique,axiom,
! [Y3: list_a,X: list_a,Y4: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y4 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( Y3 = Y4 ) ) ) ) ) ) ).
% p.minus_unique
thf(fact_447_p_Oadd_Oint__pow__distrib,axiom,
! [X: list_a,Y3: list_a,I: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ X ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ Y3 ) ) ) ) ) ).
% p.add.int_pow_distrib
thf(fact_448_p_Oadd_Oint__pow__mult__distrib,axiom,
! [X: list_a,Y3: list_a,I: int] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ X ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ X ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ Y3 ) ) ) ) ) ) ).
% p.add.int_pow_mult_distrib
thf(fact_449_p_Oadd_Oinj__on__g,axiom,
! [H: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( inj_on_list_a_list_a
@ ^ [Y5: list_a] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y5 @ A )
@ H ) ) ) ).
% p.add.inj_on_g
thf(fact_450_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_451_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_452_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_453_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_454_d_Ominus__minus,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( a_inv_a_b @ k2 @ ( a_inv_a_b @ k2 @ X ) )
= X ) ) ).
% d.minus_minus
thf(fact_455_d_Oadd_Oinv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( member_a @ ( a_inv_a_b @ k2 @ X ) @ ( partia707051561876973205xt_a_b @ k2 ) ) ) ).
% d.add.inv_closed
thf(fact_456_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_457_d_Ominus__zero,axiom,
( ( a_inv_a_b @ k2 @ ( zero_a_b @ k2 ) )
= ( zero_a_b @ k2 ) ) ).
% d.minus_zero
thf(fact_458_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_459_d_Oadd_Oinv__eq__1__iff,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ( a_inv_a_b @ k2 @ X )
= ( zero_a_b @ k2 ) )
= ( X
= ( zero_a_b @ k2 ) ) ) ) ).
% d.add.inv_eq_1_iff
thf(fact_460_p_Oadd_Om__closed,axiom,
! [X: list_a,Y3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.add.m_closed
thf(fact_461_p_Oadd_Oright__cancel,axiom,
! [X: list_a,Y3: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Z @ X ) )
= ( Y3 = Z ) ) ) ) ) ).
% p.add.right_cancel
thf(fact_462_p_Oadd_Oint__pow__closed,axiom,
! [X: list_a,I: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.add.int_pow_closed
thf(fact_463_p_Oadd_Oint__pow__one,axiom,
! [Z: int] :
( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ Z @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.add.int_pow_one
thf(fact_464_p_Oadd_Ol__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ A )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).
% p.add.l_cancel_one
thf(fact_465_p_Oadd_Ol__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ A ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).
% p.add.l_cancel_one'
thf(fact_466_p_Oadd_Or__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ X )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).
% p.add.r_cancel_one
thf(fact_467_p_Oadd_Or__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ X ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).
% p.add.r_cancel_one'
thf(fact_468_p_Ol__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ X )
= X ) ) ).
% p.l_zero
thf(fact_469_p_Or__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= X ) ) ).
% p.r_zero
thf(fact_470_char__iso,axiom,
( member_set_int_a @ ( ring_char_iso_a_b @ r )
@ ( ring_i2213127016525301402it_a_b @ ( zFact @ ( semiri1314217659103216013at_int @ ( ring_char_a_b @ r ) ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_int_a @ ( ring_int_embed_a_b @ r ) @ top_top_set_int )
@ r ) ) ) ).
% char_iso
thf(fact_471_d_Ochar__iso,axiom,
( member_set_int_a @ ( ring_char_iso_a_b @ k2 )
@ ( ring_i2213127016525301402it_a_b @ ( zFact @ ( semiri1314217659103216013at_int @ ( ring_char_a_b @ k2 ) ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_int_a @ ( ring_int_embed_a_b @ k2 ) @ top_top_set_int )
@ k2 ) ) ) ).
% d.char_iso
thf(fact_472_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_473_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_474_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_475_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_476_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_477_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_478_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_479_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_480_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_481_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_482_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_483_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_484_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_485_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_486_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_487_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_488_ring__iso__memE_I3_J,axiom,
! [H2: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y3: a] :
( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X @ Y3 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y3 ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_489_ring__iso__memE_I3_J,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y3: a] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X @ Y3 ) )
= ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y3 ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_490_ring__iso__memE_I3_J,axiom,
! [H2: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y3: list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R @ X @ Y3 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y3 ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_491_ring__iso__memE_I3_J,axiom,
! [H2: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y3: list_a] :
( ( member_list_a_a @ H2 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R @ X @ Y3 ) )
= ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y3 ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_492_ring__iso__memE_I3_J,axiom,
! [H2: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X: list_list_a,Y3: list_list_a] :
( ( member7168557129179038582list_a @ H2 @ ( ring_i4611353245267337884t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( add_li174743652000525320t_unit @ R @ X @ Y3 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y3 ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_493_ring__iso__memE_I3_J,axiom,
! [H2: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X: list_list_a,Y3: 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 @ Y3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( add_li174743652000525320t_unit @ R @ X @ Y3 ) )
= ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y3 ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_494_ring__iso__memE_I3_J,axiom,
! [H2: set_int > list_a,R: partia4934656038542163276t_unit,S: partia2670972154091845814t_unit,X: set_int,Y3: set_int] :
( ( member6804298160596253954list_a @ H2 @ ( ring_i8159631783994945698t_unit @ R @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( H2 @ ( add_se5859248395121729892t_unit @ R @ X @ Y3 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y3 ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_495_ring__iso__memE_I3_J,axiom,
! [H2: set_int > a,R: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,X: set_int,Y3: set_int] :
( ( member_set_int_a @ H2 @ ( ring_i2213127016525301402it_a_b @ R @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( H2 @ ( add_se5859248395121729892t_unit @ R @ X @ Y3 ) )
= ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y3 ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_496_ring__iso__set__sym,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > set_int,S: partia4934656038542163276t_unit] :
( ( ring_a_b @ R )
=> ( ( member_a_set_int @ H2 @ ( ring_i4118987378922720216t_unit @ R @ S ) )
=> ( member_set_int_a @ ( hilber386172014519280148et_int @ ( partia707051561876973205xt_a_b @ R ) @ H2 ) @ ( ring_i2213127016525301402it_a_b @ S @ R ) ) ) ) ).
% ring_iso_set_sym
thf(fact_497_ring__iso__set__sym,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > set_int,S: partia4934656038542163276t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member5309957867533224448et_int @ H2 @ ( ring_i5545097000213460560t_unit @ R @ S ) )
=> ( member6804298160596253954list_a @ ( hilber3467374374478178586et_int @ ( partia5361259788508890537t_unit @ R ) @ H2 ) @ ( ring_i8159631783994945698t_unit @ S @ R ) ) ) ) ).
% ring_iso_set_sym
thf(fact_498_ring__iso__set__sym,axiom,
! [R: partia4934656038542163276t_unit,H2: set_int > list_a,S: partia2670972154091845814t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member6804298160596253954list_a @ H2 @ ( ring_i8159631783994945698t_unit @ R @ S ) )
=> ( member5309957867533224448et_int @ ( hilber3792369929691227220list_a @ ( partia966996272515721803t_unit @ R ) @ H2 ) @ ( ring_i5545097000213460560t_unit @ S @ R ) ) ) ) ).
% ring_iso_set_sym
thf(fact_499_ring__iso__set__sym,axiom,
! [R: partia4934656038542163276t_unit,H2: set_int > a,S: partia2175431115845679010xt_a_b] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( member_set_int_a @ H2 @ ( ring_i2213127016525301402it_a_b @ R @ S ) )
=> ( member_a_set_int @ ( hilber773427641517404110_int_a @ ( partia966996272515721803t_unit @ R ) @ H2 ) @ ( ring_i4118987378922720216t_unit @ S @ R ) ) ) ) ).
% ring_iso_set_sym
thf(fact_500_ring__hom__ring_Oinv__ring__hom,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a,K: set_a] :
( ( ring_h4024360765257340990_b_a_b @ R @ S @ H2 )
=> ( ( inj_on_a_a @ H2 @ K )
=> ( ( subring_a_b @ K @ R )
=> ( ring_h4024360765257340990_b_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_a_a @ H2 @ K )
@ S )
@ R
@ ( hilbert_inv_into_a_a @ K @ H2 ) ) ) ) ) ).
% ring_hom_ring.inv_ring_hom
thf(fact_501_ring__hom__ring_Oinv__ring__hom,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,K: set_list_a] :
( ( ring_h7848885096329822662it_a_b @ R @ S @ H2 )
=> ( ( inj_on_list_a_a @ H2 @ K )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ring_h5357930050666032198t_unit
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_list_a_a @ H2 @ K )
@ S )
@ R
@ ( hilber8556971080906182598st_a_a @ K @ H2 ) ) ) ) ) ).
% ring_hom_ring.inv_ring_hom
thf(fact_502_ring__hom__ring_Oinv__ring__hom,axiom,
! [R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H2: list_a > list_a,K: set_list_a] :
( ( ring_h1334922693953046990t_unit @ R @ S @ H2 )
=> ( ( inj_on_list_a_list_a @ H2 @ K )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ring_h1334922693953046990t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_list_a_list_a @ H2 @ K )
@ S )
@ R
@ ( hilber4441511847348742220list_a @ K @ H2 ) ) ) ) ) ).
% ring_hom_ring.inv_ring_hom
thf(fact_503_ring__hom__ring_Oinv__ring__hom,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,K: set_a] :
( ( ring_h5357930050666032198t_unit @ R @ S @ H2 )
=> ( ( inj_on_a_list_a @ H2 @ K )
=> ( ( subring_a_b @ K @ R )
=> ( ring_h7848885096329822662it_a_b
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_a_list_a @ H2 @ K )
@ S )
@ R
@ ( hilber1222834799880672978list_a @ K @ H2 ) ) ) ) ) ).
% ring_hom_ring.inv_ring_hom
thf(fact_504_ring_Oadd__pow__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,K2: a,I: int] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_a @ K2 @ K )
=> ( ( add_pow_a_b_int @ R @ I @ K2 )
= ( add_pow_a_b_int
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ R )
@ I
@ K2 ) ) ) ) ) ).
% ring.add_pow_consistent
thf(fact_505_ring_Oadd__pow__consistent,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,K2: list_a,I: int] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_a @ K2 @ K )
=> ( ( add_po2638046716968164713it_int @ R @ I @ K2 )
= ( add_po2638046716968164713it_int
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ R )
@ I
@ K2 ) ) ) ) ) ).
% ring.add_pow_consistent
thf(fact_506_ring_Ochar__iso,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_set_int_a @ ( ring_char_iso_a_b @ R )
@ ( ring_i2213127016525301402it_a_b @ ( zFact @ ( semiri1314217659103216013at_int @ ( ring_char_a_b @ R ) ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_int_a @ ( ring_int_embed_a_b @ R ) @ top_top_set_int )
@ R ) ) ) ) ).
% ring.char_iso
thf(fact_507_ring_Ochar__iso,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( member6804298160596253954list_a @ ( ring_c7096748559348378049t_unit @ R )
@ ( ring_i8159631783994945698t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ ( ring_c500279861223467766t_unit @ R ) ) )
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_int_list_a @ ( ring_i7845631192508594762t_unit @ R ) @ top_top_set_int )
@ R ) ) ) ) ).
% ring.char_iso
thf(fact_508_mem__upI,axiom,
! [F: nat > a,R: partia2175431115845679010xt_a_b] :
( ! [N2: nat] : ( member_a @ ( F @ N2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ? [N3: nat] : ( bound_a @ ( zero_a_b @ R ) @ N3 @ F )
=> ( member_nat_a @ F @ ( up_a_b @ R ) ) ) ) ).
% mem_upI
thf(fact_509_mem__upI,axiom,
! [F: nat > list_a,R: partia2670972154091845814t_unit] :
( ! [N2: nat] : ( member_list_a @ ( F @ N2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ? [N3: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ N3 @ F )
=> ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_510_mem__upI,axiom,
! [F: nat > list_list_a,R: partia2956882679547061052t_unit] :
( ! [N2: nat] : ( member_list_list_a @ ( F @ N2 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ? [N3: nat] : ( bound_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ N3 @ F )
=> ( member8650753269014980122list_a @ F @ ( up_lis8963924889346801084t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_511_mem__upI,axiom,
! [F: nat > set_int,R: partia4934656038542163276t_unit] :
( ! [N2: nat] : ( member_set_int @ ( F @ N2 ) @ ( partia966996272515721803t_unit @ R ) )
=> ( ? [N3: nat] : ( bound_set_int @ ( zero_s6269048424454532197t_unit @ R ) @ N3 @ F )
=> ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_512_inv__into__image__cancel,axiom,
! [F: int > list_a,A3: set_int,S: set_int] :
( ( inj_on_int_list_a @ F @ A3 )
=> ( ( ord_less_eq_set_int @ S @ A3 )
=> ( ( image_list_a_int @ ( hilber7218023918483505546list_a @ A3 @ F ) @ ( image_int_list_a @ F @ S ) )
= S ) ) ) ).
% inv_into_image_cancel
thf(fact_513_inv__into__image__cancel,axiom,
! [F: int > a,A3: set_int,S: set_int] :
( ( inj_on_int_a @ F @ A3 )
=> ( ( ord_less_eq_set_int @ S @ A3 )
=> ( ( image_a_int @ ( hilber550346152434376708_int_a @ A3 @ F ) @ ( image_int_a @ F @ S ) )
= S ) ) ) ).
% inv_into_image_cancel
thf(fact_514_inv__into__image__cancel,axiom,
! [F: a > int,A3: set_a,S: set_a] :
( ( inj_on_a_int @ F @ A3 )
=> ( ( ord_less_eq_set_a @ S @ A3 )
=> ( ( image_int_a @ ( hilber7984441185272261726_a_int @ A3 @ F ) @ ( image_a_int @ F @ S ) )
= S ) ) ) ).
% inv_into_image_cancel
thf(fact_515_inv__into__image__cancel,axiom,
! [F: a > a,A3: set_a,S: set_a] :
( ( inj_on_a_a @ F @ A3 )
=> ( ( ord_less_eq_set_a @ S @ A3 )
=> ( ( image_a_a @ ( hilbert_inv_into_a_a @ A3 @ F ) @ ( image_a_a @ F @ S ) )
= S ) ) ) ).
% inv_into_image_cancel
thf(fact_516_inv__into__image__cancel,axiom,
! [F: a > list_a,A3: set_a,S: set_a] :
( ( inj_on_a_list_a @ F @ A3 )
=> ( ( ord_less_eq_set_a @ S @ A3 )
=> ( ( image_list_a_a @ ( hilber1222834799880672978list_a @ A3 @ F ) @ ( image_a_list_a @ F @ S ) )
= S ) ) ) ).
% inv_into_image_cancel
thf(fact_517_inv__into__image__cancel,axiom,
! [F: list_a > int,A3: set_list_a,S: set_list_a] :
( ( inj_on_list_a_int @ F @ A3 )
=> ( ( ord_le8861187494160871172list_a @ S @ A3 )
=> ( ( image_int_list_a @ ( hilber854936315069205348_a_int @ A3 @ F ) @ ( image_list_a_int @ F @ S ) )
= S ) ) ) ).
% inv_into_image_cancel
thf(fact_518_inv__into__image__cancel,axiom,
! [F: list_a > a,A3: set_list_a,S: set_list_a] :
( ( inj_on_list_a_a @ F @ A3 )
=> ( ( ord_le8861187494160871172list_a @ S @ A3 )
=> ( ( image_a_list_a @ ( hilber8556971080906182598st_a_a @ A3 @ F ) @ ( image_list_a_a @ F @ S ) )
= S ) ) ) ).
% inv_into_image_cancel
thf(fact_519_inv__into__image__cancel,axiom,
! [F: list_a > list_a,A3: set_list_a,S: set_list_a] :
( ( inj_on_list_a_list_a @ F @ A3 )
=> ( ( ord_le8861187494160871172list_a @ S @ A3 )
=> ( ( image_list_a_list_a @ ( hilber4441511847348742220list_a @ A3 @ F ) @ ( image_list_a_list_a @ F @ S ) )
= S ) ) ) ).
% inv_into_image_cancel
thf(fact_520_inv__into__image__cancel,axiom,
! [F: list_a > list_list_a,A3: set_list_a,S: set_list_a] :
( ( inj_on6538719745830000247list_a @ F @ A3 )
=> ( ( ord_le8861187494160871172list_a @ S @ A3 )
=> ( ( image_4177985004758493951list_a @ ( hilber6629114656297661522list_a @ A3 @ F ) @ ( image_8260866953997875467list_a @ F @ S ) )
= S ) ) ) ).
% inv_into_image_cancel
thf(fact_521_p_Ochar__bound_I1_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ k ) @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ord_less_eq_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ k ) ) @ X ) ) ) ).
% p.char_bound(1)
thf(fact_522_p_Ochar__bound_I2_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ k ) @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.char_bound(2)
thf(fact_523_d_Ochar__bound_I1_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_int_embed_a_b @ k2 @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_a_b @ k2 ) )
=> ( ord_less_eq_nat @ ( ring_char_a_b @ k2 ) @ X ) ) ) ).
% d.char_bound(1)
thf(fact_524_char__bound_I1_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_int_embed_a_b @ r @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_a_b @ r ) )
=> ( ord_less_eq_nat @ ( ring_char_a_b @ r ) @ X ) ) ) ).
% char_bound(1)
thf(fact_525_inv__identity,axiom,
( ( hilber4630740438190819238nt_int @ top_top_set_int
@ ^ [A4: int] : A4 )
= ( ^ [A4: int] : A4 ) ) ).
% inv_identity
thf(fact_526_inv__identity,axiom,
( ( hilbert_inv_into_a_a @ top_top_set_a
@ ^ [A4: a] : A4 )
= ( ^ [A4: a] : A4 ) ) ).
% inv_identity
thf(fact_527_inv__identity,axiom,
( ( hilber4441511847348742220list_a @ top_top_set_list_a
@ ^ [A4: list_a] : A4 )
= ( ^ [A4: list_a] : A4 ) ) ).
% inv_identity
thf(fact_528_up__add__closed,axiom,
! [P2: nat > a,Q2: nat > a] :
( ( member_nat_a @ P2 @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( add_a_b @ r @ ( P2 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_add_closed
thf(fact_529_d_Oup__add__closed,axiom,
! [P2: nat > a,Q2: nat > a] :
( ( member_nat_a @ P2 @ ( up_a_b @ k2 ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ k2 ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( add_a_b @ k2 @ ( P2 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_a_b @ k2 ) ) ) ) ).
% d.up_add_closed
thf(fact_530_add_Oint__pow__mult__distrib,axiom,
! [X: a,Y3: a,I: int] :
( ( ( add_a_b @ r @ X @ Y3 )
= ( add_a_b @ r @ Y3 @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y3 ) )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y3 ) ) ) ) ) ) ).
% add.int_pow_mult_distrib
thf(fact_531_add_Oint__pow__distrib,axiom,
! [X: a,Y3: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y3 ) )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y3 ) ) ) ) ) ).
% add.int_pow_distrib
thf(fact_532_a__lcomm,axiom,
! [X: a,Y3: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y3 @ Z ) )
= ( add_a_b @ r @ Y3 @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_533_a__comm,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ Y3 )
= ( add_a_b @ r @ Y3 @ X ) ) ) ) ).
% a_comm
thf(fact_534_a__assoc,axiom,
! [X: a,Y3: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y3 ) @ Z )
= ( add_a_b @ r @ X @ ( add_a_b @ r @ Y3 @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_535_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_536_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_537_d_Oadd_Oint__pow__distrib,axiom,
! [X: a,Y3: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( add_pow_a_b_int @ k2 @ I @ ( add_a_b @ k2 @ X @ Y3 ) )
= ( add_a_b @ k2 @ ( add_pow_a_b_int @ k2 @ I @ X ) @ ( add_pow_a_b_int @ k2 @ I @ Y3 ) ) ) ) ) ).
% d.add.int_pow_distrib
thf(fact_538_d_Oadd_Oint__pow__mult__distrib,axiom,
! [X: a,Y3: a,I: int] :
( ( ( add_a_b @ k2 @ X @ Y3 )
= ( add_a_b @ k2 @ Y3 @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( add_pow_a_b_int @ k2 @ I @ ( add_a_b @ k2 @ X @ Y3 ) )
= ( add_a_b @ k2 @ ( add_pow_a_b_int @ k2 @ I @ X ) @ ( add_pow_a_b_int @ k2 @ I @ Y3 ) ) ) ) ) ) ).
% d.add.int_pow_mult_distrib
thf(fact_539_d_Oadd_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( add_a_b @ k2 @ C @ A )
= ( add_a_b @ k2 @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( A = B ) ) ) ) ) ).
% d.add.l_cancel
thf(fact_540_d_Oadd_Om__assoc,axiom,
! [X: a,Y3: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( add_a_b @ k2 @ ( add_a_b @ k2 @ X @ Y3 ) @ Z )
= ( add_a_b @ k2 @ X @ ( add_a_b @ k2 @ Y3 @ Z ) ) ) ) ) ) ).
% d.add.m_assoc
thf(fact_541_d_Oadd_Om__comm,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( add_a_b @ k2 @ X @ Y3 )
= ( add_a_b @ k2 @ Y3 @ X ) ) ) ) ).
% d.add.m_comm
thf(fact_542_d_Oadd_Om__lcomm,axiom,
! [X: a,Y3: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( add_a_b @ k2 @ X @ ( add_a_b @ k2 @ Y3 @ Z ) )
= ( add_a_b @ k2 @ Y3 @ ( add_a_b @ k2 @ X @ Z ) ) ) ) ) ) ).
% d.add.m_lcomm
thf(fact_543_d_Oadd_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( add_a_b @ k2 @ A @ C )
= ( add_a_b @ k2 @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( A = B ) ) ) ) ) ).
% d.add.r_cancel
thf(fact_544_bound__upD,axiom,
! [F: nat > a] :
( ( member_nat_a @ F @ ( up_a_b @ r ) )
=> ? [N2: nat] : ( bound_a @ ( zero_a_b @ r ) @ N2 @ F ) ) ).
% bound_upD
thf(fact_545_d_Obound__upD,axiom,
! [F: nat > a] :
( ( member_nat_a @ F @ ( up_a_b @ k2 ) )
=> ? [N2: nat] : ( bound_a @ ( zero_a_b @ k2 ) @ N2 @ F ) ) ).
% d.bound_upD
thf(fact_546_add_Osurj__const__mult,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( image_a_a @ ( add_a_b @ r @ A ) @ ( partia707051561876973205xt_a_b @ r ) )
= ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.surj_const_mult
thf(fact_547_d_Oadd_Osurj__const__mult,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( image_a_a @ ( add_a_b @ k2 @ A ) @ ( partia707051561876973205xt_a_b @ k2 ) )
= ( partia707051561876973205xt_a_b @ k2 ) ) ) ).
% d.add.surj_const_mult
thf(fact_548_up__a__inv__closed,axiom,
! [P2: nat > a] :
( ( member_nat_a @ P2 @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( a_inv_a_b @ r @ ( P2 @ I3 ) )
@ ( up_a_b @ r ) ) ) ).
% up_a_inv_closed
thf(fact_549_d_Oup__a__inv__closed,axiom,
! [P2: nat > a] :
( ( member_nat_a @ P2 @ ( up_a_b @ k2 ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( a_inv_a_b @ k2 @ ( P2 @ I3 ) )
@ ( up_a_b @ k2 ) ) ) ).
% d.up_a_inv_closed
thf(fact_550_local_Ominus__unique,axiom,
! [Y3: a,X: a,Y4: a] :
( ( ( add_a_b @ r @ Y3 @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y4 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y3 = Y4 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_551_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_552_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_553_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_554_add_Oinv__comm,axiom,
! [X: a,Y3: a] :
( ( ( add_a_b @ r @ X @ Y3 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y3 @ X )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_555_d_Oadd_Oinv__comm,axiom,
! [X: a,Y3: a] :
( ( ( add_a_b @ k2 @ X @ Y3 )
= ( zero_a_b @ k2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( add_a_b @ k2 @ Y3 @ X )
= ( zero_a_b @ k2 ) ) ) ) ) ).
% d.add.inv_comm
thf(fact_556_d_Oadd_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ k2 ) )
& ( ( add_a_b @ k2 @ X3 @ X )
= ( zero_a_b @ k2 ) ) ) ) ).
% d.add.l_inv_ex
thf(fact_557_d_Oadd_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( add_a_b @ k2 @ U @ X3 )
= X3 ) )
=> ( U
= ( zero_a_b @ k2 ) ) ) ) ).
% d.add.one_unique
thf(fact_558_d_Oadd_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ k2 ) )
& ( ( add_a_b @ k2 @ X @ X3 )
= ( zero_a_b @ k2 ) ) ) ) ).
% d.add.r_inv_ex
thf(fact_559_d_Ominus__unique,axiom,
! [Y3: a,X: a,Y4: a] :
( ( ( add_a_b @ k2 @ Y3 @ X )
= ( zero_a_b @ k2 ) )
=> ( ( ( add_a_b @ k2 @ X @ Y4 )
= ( zero_a_b @ k2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( Y3 = Y4 ) ) ) ) ) ) ).
% d.minus_unique
thf(fact_560_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_561_r__neg2,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y3 ) )
= Y3 ) ) ) ).
% r_neg2
thf(fact_562_r__neg1,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y3 ) )
= Y3 ) ) ) ).
% r_neg1
thf(fact_563_local_Ominus__add,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y3 ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y3 ) ) ) ) ) ).
% local.minus_add
thf(fact_564_a__transpose__inv,axiom,
! [X: a,Y3: a,Z: a] :
( ( ( add_a_b @ r @ X @ Y3 )
= Z )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z )
= Y3 ) ) ) ) ) ).
% a_transpose_inv
thf(fact_565_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_566_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_567_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_568_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_569_add_Oinv__mult__group,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y3 ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y3 ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_570_d_Oadd_Oint__pow__inv,axiom,
! [X: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( add_pow_a_b_int @ k2 @ I @ ( a_inv_a_b @ k2 @ X ) )
= ( a_inv_a_b @ k2 @ ( add_pow_a_b_int @ k2 @ I @ X ) ) ) ) ).
% d.add.int_pow_inv
thf(fact_571_d_Oa__transpose__inv,axiom,
! [X: a,Y3: a,Z: a] :
( ( ( add_a_b @ k2 @ X @ Y3 )
= Z )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( add_a_b @ k2 @ ( a_inv_a_b @ k2 @ X ) @ Z )
= Y3 ) ) ) ) ) ).
% d.a_transpose_inv
thf(fact_572_d_Oadd_Oinv__mult__group,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( a_inv_a_b @ k2 @ ( add_a_b @ k2 @ X @ Y3 ) )
= ( add_a_b @ k2 @ ( a_inv_a_b @ k2 @ Y3 ) @ ( a_inv_a_b @ k2 @ X ) ) ) ) ) ).
% d.add.inv_mult_group
thf(fact_573_d_Oadd_Oinv__solve__left,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( A
= ( add_a_b @ k2 @ ( a_inv_a_b @ k2 @ B ) @ C ) )
= ( C
= ( add_a_b @ k2 @ B @ A ) ) ) ) ) ) ).
% d.add.inv_solve_left
thf(fact_574_d_Oadd_Oinv__solve__left_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ( add_a_b @ k2 @ ( a_inv_a_b @ k2 @ B ) @ C )
= A )
= ( C
= ( add_a_b @ k2 @ B @ A ) ) ) ) ) ) ).
% d.add.inv_solve_left'
thf(fact_575_d_Oadd_Oinv__solve__right,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( A
= ( add_a_b @ k2 @ B @ ( a_inv_a_b @ k2 @ C ) ) )
= ( B
= ( add_a_b @ k2 @ A @ C ) ) ) ) ) ) ).
% d.add.inv_solve_right
thf(fact_576_d_Oadd_Oinv__solve__right_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ( add_a_b @ k2 @ B @ ( a_inv_a_b @ k2 @ C ) )
= A )
= ( B
= ( add_a_b @ k2 @ A @ C ) ) ) ) ) ) ).
% d.add.inv_solve_right'
thf(fact_577_d_Ominus__add,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( a_inv_a_b @ k2 @ ( add_a_b @ k2 @ X @ Y3 ) )
= ( add_a_b @ k2 @ ( a_inv_a_b @ k2 @ X ) @ ( a_inv_a_b @ k2 @ Y3 ) ) ) ) ) ).
% d.minus_add
thf(fact_578_d_Or__neg1,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( add_a_b @ k2 @ ( a_inv_a_b @ k2 @ X ) @ ( add_a_b @ k2 @ X @ Y3 ) )
= Y3 ) ) ) ).
% d.r_neg1
thf(fact_579_d_Or__neg2,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( add_a_b @ k2 @ X @ ( add_a_b @ k2 @ ( a_inv_a_b @ k2 @ X ) @ Y3 ) )
= Y3 ) ) ) ).
% d.r_neg2
thf(fact_580_poly__of__const__inj,axiom,
inj_on_a_list_a @ ( poly_of_const_a_b @ r ) @ top_top_set_a ).
% poly_of_const_inj
thf(fact_581_d_Opoly__of__const__inj,axiom,
inj_on_a_list_a @ ( poly_of_const_a_b @ k2 ) @ top_top_set_a ).
% d.poly_of_const_inj
thf(fact_582_add__pow__consistent,axiom,
! [K: set_a,K2: a,I: int] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ K2 @ K )
=> ( ( add_pow_a_b_int @ r @ I @ K2 )
= ( add_pow_a_b_int
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r )
@ I
@ K2 ) ) ) ) ).
% add_pow_consistent
thf(fact_583_d_Oadd__pow__consistent,axiom,
! [K: set_a,K2: a,I: int] :
( ( subring_a_b @ K @ k2 )
=> ( ( member_a @ K2 @ K )
=> ( ( add_pow_a_b_int @ k2 @ I @ K2 )
= ( add_pow_a_b_int
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ k2 )
@ I
@ K2 ) ) ) ) ).
% d.add_pow_consistent
thf(fact_584_add_Oinj__on__multc,axiom,
! [C: a] :
( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( inj_on_a_a
@ ^ [X2: a] : ( add_a_b @ r @ X2 @ C )
@ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.inj_on_multc
thf(fact_585_add_Oinj__on__cmult,axiom,
! [C: a] :
( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( inj_on_a_a @ ( add_a_b @ r @ C ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.inj_on_cmult
thf(fact_586_d_Oadd_Oinj__on__cmult,axiom,
! [C: a] :
( ( member_a @ C @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( inj_on_a_a @ ( add_a_b @ k2 @ C ) @ ( partia707051561876973205xt_a_b @ k2 ) ) ) ).
% d.add.inj_on_cmult
thf(fact_587_d_Oadd_Oinj__on__multc,axiom,
! [C: a] :
( ( member_a @ C @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( inj_on_a_a
@ ^ [X2: a] : ( add_a_b @ k2 @ X2 @ C )
@ ( partia707051561876973205xt_a_b @ k2 ) ) ) ).
% d.add.inj_on_multc
thf(fact_588_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_589_d_OboundD__carrier,axiom,
! [N: nat,F: nat > a,M: nat] :
( ( bound_a @ ( zero_a_b @ k2 ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_a @ ( F @ M ) @ ( partia707051561876973205xt_a_b @ k2 ) ) ) ) ).
% d.boundD_carrier
thf(fact_590_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_591_minus__equality,axiom,
! [Y3: a,X: a] :
( ( ( add_a_b @ r @ Y3 @ X )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X )
= Y3 ) ) ) ) ).
% minus_equality
thf(fact_592_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_593_sum__zero__eq__neg,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y3 )
= ( zero_a_b @ r ) )
=> ( X
= ( a_inv_a_b @ r @ Y3 ) ) ) ) ) ).
% sum_zero_eq_neg
thf(fact_594_d_Ol__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( add_a_b @ k2 @ ( a_inv_a_b @ k2 @ X ) @ X )
= ( zero_a_b @ k2 ) ) ) ).
% d.l_neg
thf(fact_595_d_Ominus__equality,axiom,
! [Y3: a,X: a] :
( ( ( add_a_b @ k2 @ Y3 @ X )
= ( zero_a_b @ k2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( a_inv_a_b @ k2 @ X )
= Y3 ) ) ) ) ).
% d.minus_equality
thf(fact_596_d_Or__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( add_a_b @ k2 @ X @ ( a_inv_a_b @ k2 @ X ) )
= ( zero_a_b @ k2 ) ) ) ).
% d.r_neg
thf(fact_597_d_Osum__zero__eq__neg,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ( add_a_b @ k2 @ X @ Y3 )
= ( zero_a_b @ k2 ) )
=> ( X
= ( a_inv_a_b @ k2 @ Y3 ) ) ) ) ) ).
% d.sum_zero_eq_neg
thf(fact_598_a__lcos__m__assoc,axiom,
! [M2: set_a,G: a,H2: a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H2 @ M2 ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H2 ) @ M2 ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_599_d_Oa__lcos__m__assoc,axiom,
! [M2: set_a,G: a,H2: a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ H2 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( a_l_coset_a_b @ k2 @ G @ ( a_l_coset_a_b @ k2 @ H2 @ M2 ) )
= ( a_l_coset_a_b @ k2 @ ( add_a_b @ k2 @ G @ H2 ) @ M2 ) ) ) ) ) ).
% d.a_lcos_m_assoc
thf(fact_600_add_Oinj__on__g,axiom,
! [H: set_a,A: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( inj_on_a_a
@ ^ [Y5: a] : ( add_a_b @ r @ Y5 @ A )
@ H ) ) ) ).
% add.inj_on_g
thf(fact_601_d_Oadd_Oinj__on__g,axiom,
! [H: set_a,A: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( inj_on_a_a
@ ^ [Y5: a] : ( add_a_b @ k2 @ Y5 @ A )
@ H ) ) ) ).
% d.add.inj_on_g
thf(fact_602_univ__poly__a__inv__consistent,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ) ).
% univ_poly_a_inv_consistent
thf(fact_603_p_Opoly__of__const__inj,axiom,
inj_on6538719745830000247list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) @ top_top_set_list_a ).
% p.poly_of_const_inj
thf(fact_604_d_Ouniv__poly__a__inv__consistent,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ k2 )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ k2 @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ k2 @ K ) @ P2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ k2 @ ( partia707051561876973205xt_a_b @ k2 ) ) @ P2 ) ) ) ) ).
% d.univ_poly_a_inv_consistent
thf(fact_605_p_Oup__a__inv__closed,axiom,
! [P2: nat > list_a] :
( ( member_nat_list_a @ P2 @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_nat_list_a
@ ^ [I3: nat] : ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( P2 @ I3 ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.up_a_inv_closed
thf(fact_606_p_Or__neg2,axiom,
! [X: list_a,Y3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ Y3 ) )
= Y3 ) ) ) ).
% p.r_neg2
thf(fact_607_p_Or__neg1,axiom,
! [X: list_a,Y3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 ) )
= Y3 ) ) ) ).
% p.r_neg1
thf(fact_608_p_Ominus__add,axiom,
! [X: list_a,Y3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 ) ) ) ) ) ).
% p.minus_add
thf(fact_609_p_Oadd_Oinv__solve__right_H,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ C ) )
= A )
= ( B
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ C ) ) ) ) ) ) ).
% p.add.inv_solve_right'
thf(fact_610_p_Oadd_Oinv__solve__right,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( A
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ C ) ) )
= ( B
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ C ) ) ) ) ) ) ).
% p.add.inv_solve_right
thf(fact_611_p_Oadd_Oinv__solve__left_H,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ B ) @ C )
= A )
= ( C
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ A ) ) ) ) ) ) ).
% p.add.inv_solve_left'
thf(fact_612_p_Oadd_Oinv__solve__left,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( A
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ B ) @ C ) )
= ( C
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ A ) ) ) ) ) ) ).
% p.add.inv_solve_left
thf(fact_613_p_Oadd_Oinv__mult__group,axiom,
! [X: list_a,Y3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) ) ) ) ) ).
% p.add.inv_mult_group
thf(fact_614_p_Oa__transpose__inv,axiom,
! [X: list_a,Y3: list_a,Z: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 )
= Z )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ Z )
= Y3 ) ) ) ) ) ).
% p.a_transpose_inv
thf(fact_615_p_Oadd_Oint__pow__inv,axiom,
! [X: list_a,I: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ X ) ) ) ) ).
% p.add.int_pow_inv
thf(fact_616_p_Ominus__eq,axiom,
! [X: list_a,Y3: list_a] :
( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 ) ) ) ).
% p.minus_eq
thf(fact_617_p_Oadd_Oinv__inj,axiom,
inj_on_list_a_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).
% p.add.inv_inj
thf(fact_618_freshmans__dream,axiom,
! [X: a,Y3: a] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( add_a_b @ r @ X @ Y3 ) @ ( ring_char_a_b @ r ) )
= ( add_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ ( ring_char_a_b @ r ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y3 @ ( ring_char_a_b @ r ) ) ) ) ) ) ) ).
% freshmans_dream
thf(fact_619_d_Ofreshmans__dream,axiom,
! [X: a,Y3: a] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ k2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( pow_a_1026414303147256608_b_nat @ k2 @ ( add_a_b @ k2 @ X @ Y3 ) @ ( ring_char_a_b @ k2 ) )
= ( add_a_b @ k2 @ ( pow_a_1026414303147256608_b_nat @ k2 @ X @ ( ring_char_a_b @ k2 ) ) @ ( pow_a_1026414303147256608_b_nat @ k2 @ Y3 @ ( ring_char_a_b @ k2 ) ) ) ) ) ) ) ).
% d.freshmans_dream
thf(fact_620_p_Osum__zero__eq__neg,axiom,
! [X: list_a,Y3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( X
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 ) ) ) ) ) ).
% p.sum_zero_eq_neg
thf(fact_621_p_Or__neg,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.r_neg
thf(fact_622_p_Ominus__equality,axiom,
! [Y3: list_a,X: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X )
= Y3 ) ) ) ) ).
% p.minus_equality
thf(fact_623_p_Ol__neg,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.l_neg
thf(fact_624_char__bound_I2_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_int_embed_a_b @ r @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_a_b @ r ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) ) ) ) ).
% char_bound(2)
thf(fact_625_d_Ochar__bound_I2_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_int_embed_a_b @ k2 @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_a_b @ k2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ k2 ) ) ) ) ).
% d.char_bound(2)
thf(fact_626_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_627_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_628_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_629_p_OboundD__carrier,axiom,
! [N: nat,F: nat > list_a,M: nat] :
( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_list_a @ ( F @ M ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.boundD_carrier
thf(fact_630_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_631_a__closed,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X @ Y3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_632_local_Oadd_Oright__cancel,axiom,
! [X: a,Y3: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y3 @ X )
= ( add_a_b @ r @ Z @ X ) )
= ( Y3 = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_633_d_Oadd_Oint__pow__closed,axiom,
! [X: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( member_a @ ( add_pow_a_b_int @ k2 @ I @ X ) @ ( partia707051561876973205xt_a_b @ k2 ) ) ) ).
% d.add.int_pow_closed
thf(fact_634_d_Oadd_Om__closed,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( member_a @ ( add_a_b @ k2 @ X @ Y3 ) @ ( partia707051561876973205xt_a_b @ k2 ) ) ) ) ).
% d.add.m_closed
thf(fact_635_d_Oadd_Oright__cancel,axiom,
! [X: a,Y3: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ( add_a_b @ k2 @ Y3 @ X )
= ( add_a_b @ k2 @ Z @ X ) )
= ( Y3 = Z ) ) ) ) ) ).
% d.add.right_cancel
thf(fact_636_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_637_d_Oadd_Oint__pow__one,axiom,
! [Z: int] :
( ( add_pow_a_b_int @ k2 @ Z @ ( zero_a_b @ k2 ) )
= ( zero_a_b @ k2 ) ) ).
% d.add.int_pow_one
thf(fact_638_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_639_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_640_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_641_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_642_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_643_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_644_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_645_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_646_d_Oadd_Ol__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ( add_a_b @ k2 @ X @ A )
= X )
= ( A
= ( zero_a_b @ k2 ) ) ) ) ) ).
% d.add.l_cancel_one
thf(fact_647_d_Oadd_Ol__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( X
= ( add_a_b @ k2 @ X @ A ) )
= ( A
= ( zero_a_b @ k2 ) ) ) ) ) ).
% d.add.l_cancel_one'
thf(fact_648_d_Oadd_Or__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ( add_a_b @ k2 @ A @ X )
= X )
= ( A
= ( zero_a_b @ k2 ) ) ) ) ) ).
% d.add.r_cancel_one
thf(fact_649_d_Oadd_Or__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( X
= ( add_a_b @ k2 @ A @ X ) )
= ( A
= ( zero_a_b @ k2 ) ) ) ) ) ).
% d.add.r_cancel_one'
thf(fact_650_d_Ol__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( add_a_b @ k2 @ ( zero_a_b @ k2 ) @ X )
= X ) ) ).
% d.l_zero
thf(fact_651_d_Or__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( add_a_b @ k2 @ X @ ( zero_a_b @ k2 ) )
= X ) ) ).
% d.r_zero
thf(fact_652_p_Ominus__minus,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) )
= X ) ) ).
% p.minus_minus
thf(fact_653_p_Oadd_Oinv__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.add.inv_closed
thf(fact_654_p_Ominus__zero,axiom,
( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.minus_zero
thf(fact_655_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_656_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_657_p_Oadd_Oinv__eq__1__iff,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= ( X
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.add.inv_eq_1_iff
thf(fact_658_h_Ohom__add,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( poly_of_const_a_b @ r @ ( add_a_b @ k2 @ X @ Y3 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( poly_of_const_a_b @ r @ X ) @ ( poly_of_const_a_b @ r @ Y3 ) ) ) ) ) ).
% h.hom_add
thf(fact_659_h_Ohom__a__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( poly_of_const_a_b @ r @ ( a_inv_a_b @ k2 @ X ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( poly_of_const_a_b @ r @ X ) ) ) ) ).
% h.hom_a_inv
thf(fact_660_linorder__neqE__nat,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_661_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_662_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_663_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_664_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_665_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_666_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_667_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_668_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_669_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_670_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_671_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_672_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_673_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_674_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_675_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_676_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_677_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_678_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_679_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_680_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_681_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_682_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_683_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_684_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_685_inj__of__nat,axiom,
inj_on_nat_int @ semiri1314217659103216013at_int @ top_top_set_nat ).
% inj_of_nat
thf(fact_686_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_687_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_688_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_689_abelian__monoid_OboundD__carrier,axiom,
! [G2: partia2175431115845679010xt_a_b,N: nat,F: nat > a,M: nat] :
( ( abelian_monoid_a_b @ G2 )
=> ( ( bound_a @ ( zero_a_b @ G2 ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_a @ ( F @ M ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_690_abelian__monoid_OboundD__carrier,axiom,
! [G2: partia2670972154091845814t_unit,N: nat,F: nat > list_a,M: nat] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ G2 ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_list_a @ ( F @ M ) @ ( partia5361259788508890537t_unit @ G2 ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_691_abelian__monoid_OboundD__carrier,axiom,
! [G2: partia2956882679547061052t_unit,N: nat,F: nat > list_list_a,M: nat] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( bound_list_list_a @ ( zero_l347298301471573063t_unit @ G2 ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_list_list_a @ ( F @ M ) @ ( partia2464479390973590831t_unit @ G2 ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_692_abelian__monoid_OboundD__carrier,axiom,
! [G2: partia4934656038542163276t_unit,N: nat,F: nat > set_int,M: nat] :
( ( abelia3815030880812984441t_unit @ G2 )
=> ( ( bound_set_int @ ( zero_s6269048424454532197t_unit @ G2 ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_set_int @ ( F @ M ) @ ( partia966996272515721803t_unit @ G2 ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_693_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
=> ( ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 )
= ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P2 ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_694_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia4934656038542163276t_unit,K: set_set_int,P2: list_set_int] :
( ( domain6183376680155302761t_unit @ R )
=> ( ( subrin7689096310803670856t_unit @ K @ R )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ R @ K ) ) )
=> ( ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ R @ K ) @ P2 )
= ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ R @ ( partia966996272515721803t_unit @ R ) ) @ P2 ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_695_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_696_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_697_domain_Ofreshmans__dream,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y3: a] :
( ( domain_a_b @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ R ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( pow_a_1026414303147256608_b_nat @ R @ ( add_a_b @ R @ X @ Y3 ) @ ( ring_char_a_b @ R ) )
= ( add_a_b @ R @ ( pow_a_1026414303147256608_b_nat @ R @ X @ ( ring_char_a_b @ R ) ) @ ( pow_a_1026414303147256608_b_nat @ R @ Y3 @ ( ring_char_a_b @ R ) ) ) ) ) ) ) ) ).
% domain.freshmans_dream
thf(fact_698_domain_Ofreshmans__dream,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ ( ring_c500279861223467766t_unit @ R ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( pow_li1142815632869257134it_nat @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y3 ) @ ( ring_c500279861223467766t_unit @ R ) )
= ( add_li7652885771158616974t_unit @ R @ ( pow_li1142815632869257134it_nat @ R @ X @ ( ring_c500279861223467766t_unit @ R ) ) @ ( pow_li1142815632869257134it_nat @ R @ Y3 @ ( ring_c500279861223467766t_unit @ R ) ) ) ) ) ) ) ) ).
% domain.freshmans_dream
thf(fact_699_domain_Ofreshmans__dream,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y3: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ ( ring_c667363890163664752t_unit @ R ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( pow_li488931774710091566it_nat @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y3 ) @ ( ring_c667363890163664752t_unit @ R ) )
= ( add_li174743652000525320t_unit @ R @ ( pow_li488931774710091566it_nat @ R @ X @ ( ring_c667363890163664752t_unit @ R ) ) @ ( pow_li488931774710091566it_nat @ R @ Y3 @ ( ring_c667363890163664752t_unit @ R ) ) ) ) ) ) ) ) ).
% domain.freshmans_dream
thf(fact_700_domain_Ofreshmans__dream,axiom,
! [R: partia4934656038542163276t_unit,X: set_int,Y3: set_int] :
( ( domain6183376680155302761t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ ( ring_c6147214092195050492t_unit @ R ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R ) )
=> ( ( pow_se2518650051167492506it_nat @ R @ ( add_se5859248395121729892t_unit @ R @ X @ Y3 ) @ ( ring_c6147214092195050492t_unit @ R ) )
= ( add_se5859248395121729892t_unit @ R @ ( pow_se2518650051167492506it_nat @ R @ X @ ( ring_c6147214092195050492t_unit @ R ) ) @ ( pow_se2518650051167492506it_nat @ R @ Y3 @ ( ring_c6147214092195050492t_unit @ R ) ) ) ) ) ) ) ) ).
% domain.freshmans_dream
thf(fact_701_ring_Ochar__bound_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_int_embed_a_b @ R @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_a_b @ R ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ R ) ) ) ) ) ).
% ring.char_bound(2)
thf(fact_702_ring_Ochar__bound_I2_J,axiom,
! [R: partia2670972154091845814t_unit,X: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i7845631192508594762t_unit @ R @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c500279861223467766t_unit @ R ) ) ) ) ) ).
% ring.char_bound(2)
thf(fact_703_mem__upD,axiom,
! [F: nat > a,R: partia2175431115845679010xt_a_b,N: nat] :
( ( member_nat_a @ F @ ( up_a_b @ R ) )
=> ( member_a @ ( F @ N ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% mem_upD
thf(fact_704_mem__upD,axiom,
! [F: nat > list_a,R: partia2670972154091845814t_unit,N: nat] :
( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) )
=> ( member_list_a @ ( F @ N ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% mem_upD
thf(fact_705_mem__upD,axiom,
! [F: nat > list_list_a,R: partia2956882679547061052t_unit,N: nat] :
( ( member8650753269014980122list_a @ F @ ( up_lis8963924889346801084t_unit @ R ) )
=> ( member_list_list_a @ ( F @ N ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% mem_upD
thf(fact_706_mem__upD,axiom,
! [F: nat > set_int,R: partia4934656038542163276t_unit,N: nat] :
( ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ R ) )
=> ( member_set_int @ ( F @ N ) @ ( partia966996272515721803t_unit @ R ) ) ) ).
% mem_upD
thf(fact_707_inv__into__injective,axiom,
! [A3: set_list_a,F: list_a > list_a,X: list_a,Y3: list_a] :
( ( ( hilber4441511847348742220list_a @ A3 @ F @ X )
= ( hilber4441511847348742220list_a @ A3 @ F @ Y3 ) )
=> ( ( member_list_a @ X @ ( image_list_a_list_a @ F @ A3 ) )
=> ( ( member_list_a @ Y3 @ ( image_list_a_list_a @ F @ A3 ) )
=> ( X = Y3 ) ) ) ) ).
% inv_into_injective
thf(fact_708_inv__into__injective,axiom,
! [A3: set_int,F: int > list_a,X: list_a,Y3: list_a] :
( ( ( hilber7218023918483505546list_a @ A3 @ F @ X )
= ( hilber7218023918483505546list_a @ A3 @ F @ Y3 ) )
=> ( ( member_list_a @ X @ ( image_int_list_a @ F @ A3 ) )
=> ( ( member_list_a @ Y3 @ ( image_int_list_a @ F @ A3 ) )
=> ( X = Y3 ) ) ) ) ).
% inv_into_injective
thf(fact_709_inv__into__injective,axiom,
! [A3: set_int,F: int > a,X: a,Y3: a] :
( ( ( hilber550346152434376708_int_a @ A3 @ F @ X )
= ( hilber550346152434376708_int_a @ A3 @ F @ Y3 ) )
=> ( ( member_a @ X @ ( image_int_a @ F @ A3 ) )
=> ( ( member_a @ Y3 @ ( image_int_a @ F @ A3 ) )
=> ( X = Y3 ) ) ) ) ).
% inv_into_injective
thf(fact_710_inv__into__injective,axiom,
! [A3: set_a,F: a > a,X: a,Y3: a] :
( ( ( hilbert_inv_into_a_a @ A3 @ F @ X )
= ( hilbert_inv_into_a_a @ A3 @ F @ Y3 ) )
=> ( ( member_a @ X @ ( image_a_a @ F @ A3 ) )
=> ( ( member_a @ Y3 @ ( image_a_a @ F @ A3 ) )
=> ( X = Y3 ) ) ) ) ).
% inv_into_injective
thf(fact_711_inv__into__injective,axiom,
! [A3: set_a,F: a > list_a,X: list_a,Y3: list_a] :
( ( ( hilber1222834799880672978list_a @ A3 @ F @ X )
= ( hilber1222834799880672978list_a @ A3 @ F @ Y3 ) )
=> ( ( member_list_a @ X @ ( image_a_list_a @ F @ A3 ) )
=> ( ( member_list_a @ Y3 @ ( image_a_list_a @ F @ A3 ) )
=> ( X = Y3 ) ) ) ) ).
% inv_into_injective
thf(fact_712_inv__into__into,axiom,
! [X: a,F: int > a,A3: set_int] :
( ( member_a @ X @ ( image_int_a @ F @ A3 ) )
=> ( member_int @ ( hilber550346152434376708_int_a @ A3 @ F @ X ) @ A3 ) ) ).
% inv_into_into
thf(fact_713_inv__into__into,axiom,
! [X: a,F: a > a,A3: set_a] :
( ( member_a @ X @ ( image_a_a @ F @ A3 ) )
=> ( member_a @ ( hilbert_inv_into_a_a @ A3 @ F @ X ) @ A3 ) ) ).
% inv_into_into
thf(fact_714_inv__into__into,axiom,
! [X: list_a,F: int > list_a,A3: set_int] :
( ( member_list_a @ X @ ( image_int_list_a @ F @ A3 ) )
=> ( member_int @ ( hilber7218023918483505546list_a @ A3 @ F @ X ) @ A3 ) ) ).
% inv_into_into
thf(fact_715_inv__into__into,axiom,
! [X: list_a,F: a > list_a,A3: set_a] :
( ( member_list_a @ X @ ( image_a_list_a @ F @ A3 ) )
=> ( member_a @ ( hilber1222834799880672978list_a @ A3 @ F @ X ) @ A3 ) ) ).
% inv_into_into
thf(fact_716_inv__into__into,axiom,
! [X: list_a,F: list_a > list_a,A3: set_list_a] :
( ( member_list_a @ X @ ( image_list_a_list_a @ F @ A3 ) )
=> ( member_list_a @ ( hilber4441511847348742220list_a @ A3 @ F @ X ) @ A3 ) ) ).
% inv_into_into
thf(fact_717_inv__into__into,axiom,
! [X: nat > a,F: a > nat > a,A3: set_a] :
( ( member_nat_a @ X @ ( image_a_nat_a @ F @ A3 ) )
=> ( member_a @ ( hilber1523447219144142899_nat_a @ A3 @ F @ X ) @ A3 ) ) ).
% inv_into_into
thf(fact_718_inv__into__into,axiom,
! [X: a,F: ( nat > a ) > a,A3: set_nat_a] :
( ( member_a @ X @ ( image_nat_a_a @ F @ A3 ) )
=> ( member_nat_a @ ( hilber6777891363991549349at_a_a @ A3 @ F @ X ) @ A3 ) ) ).
% inv_into_into
thf(fact_719_inv__into__into,axiom,
! [X: set_int > a,F: a > set_int > a,A3: set_a] :
( ( member_set_int_a @ X @ ( image_a_set_int_a @ F @ A3 ) )
=> ( member_a @ ( hilber9111489492337244321_int_a @ A3 @ F @ X ) @ A3 ) ) ).
% inv_into_into
thf(fact_720_inv__into__into,axiom,
! [X: nat > list_a,F: a > nat > list_a,A3: set_a] :
( ( member_nat_list_a @ X @ ( image_a_nat_list_a @ F @ A3 ) )
=> ( member_a @ ( hilber8476257288600951993list_a @ A3 @ F @ X ) @ A3 ) ) ).
% inv_into_into
thf(fact_721_inv__into__into,axiom,
! [X: a > list_a,F: a > a > list_a,A3: set_a] :
( ( member_a_list_a @ X @ ( image_a_a_list_a @ F @ A3 ) )
=> ( member_a @ ( hilber4343350721326501661list_a @ A3 @ F @ X ) @ A3 ) ) ).
% inv_into_into
thf(fact_722_f__inv__into__f,axiom,
! [Y3: list_a,F: list_a > list_a,A3: set_list_a] :
( ( member_list_a @ Y3 @ ( image_list_a_list_a @ F @ A3 ) )
=> ( ( F @ ( hilber4441511847348742220list_a @ A3 @ F @ Y3 ) )
= Y3 ) ) ).
% f_inv_into_f
thf(fact_723_f__inv__into__f,axiom,
! [Y3: list_a,F: int > list_a,A3: set_int] :
( ( member_list_a @ Y3 @ ( image_int_list_a @ F @ A3 ) )
=> ( ( F @ ( hilber7218023918483505546list_a @ A3 @ F @ Y3 ) )
= Y3 ) ) ).
% f_inv_into_f
thf(fact_724_f__inv__into__f,axiom,
! [Y3: a,F: int > a,A3: set_int] :
( ( member_a @ Y3 @ ( image_int_a @ F @ A3 ) )
=> ( ( F @ ( hilber550346152434376708_int_a @ A3 @ F @ Y3 ) )
= Y3 ) ) ).
% f_inv_into_f
thf(fact_725_f__inv__into__f,axiom,
! [Y3: a,F: a > a,A3: set_a] :
( ( member_a @ Y3 @ ( image_a_a @ F @ A3 ) )
=> ( ( F @ ( hilbert_inv_into_a_a @ A3 @ F @ Y3 ) )
= Y3 ) ) ).
% f_inv_into_f
thf(fact_726_f__inv__into__f,axiom,
! [Y3: list_a,F: a > list_a,A3: set_a] :
( ( member_list_a @ Y3 @ ( image_a_list_a @ F @ A3 ) )
=> ( ( F @ ( hilber1222834799880672978list_a @ A3 @ F @ Y3 ) )
= Y3 ) ) ).
% f_inv_into_f
thf(fact_727_ring_Ochar__bound_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_int_embed_a_b @ R @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_a_b @ R ) )
=> ( ord_less_eq_nat @ ( ring_char_a_b @ R ) @ X ) ) ) ) ).
% ring.char_bound(1)
thf(fact_728_ring_Ochar__bound_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i7845631192508594762t_unit @ R @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ord_less_eq_nat @ ( ring_c500279861223467766t_unit @ R ) @ X ) ) ) ) ).
% ring.char_bound(1)
thf(fact_729_bound__below,axiom,
! [Z: list_a,M: nat,F: nat > list_a,N: nat] :
( ( bound_list_a @ Z @ M @ F )
=> ( ( ( F @ N )
!= Z )
=> ( ord_less_eq_nat @ N @ M ) ) ) ).
% bound_below
thf(fact_730_bound__below,axiom,
! [Z: a,M: nat,F: nat > a,N: nat] :
( ( bound_a @ Z @ M @ F )
=> ( ( ( F @ N )
!= Z )
=> ( ord_less_eq_nat @ N @ M ) ) ) ).
% bound_below
thf(fact_731_subset__image__inj,axiom,
! [S: set_list_list_a,F: list_a > list_list_a,T2: set_list_a] :
( ( ord_le8488217952732425610list_a @ S @ ( image_8260866953997875467list_a @ F @ T2 ) )
= ( ? [U2: set_list_a] :
( ( ord_le8861187494160871172list_a @ U2 @ T2 )
& ( inj_on6538719745830000247list_a @ F @ U2 )
& ( S
= ( image_8260866953997875467list_a @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_732_subset__image__inj,axiom,
! [S: set_a,F: int > a,T2: set_int] :
( ( ord_less_eq_set_a @ S @ ( image_int_a @ F @ T2 ) )
= ( ? [U2: set_int] :
( ( ord_less_eq_set_int @ U2 @ T2 )
& ( inj_on_int_a @ F @ U2 )
& ( S
= ( image_int_a @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_733_subset__image__inj,axiom,
! [S: set_a,F: a > a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ ( image_a_a @ F @ T2 ) )
= ( ? [U2: set_a] :
( ( ord_less_eq_set_a @ U2 @ T2 )
& ( inj_on_a_a @ F @ U2 )
& ( S
= ( image_a_a @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_734_subset__image__inj,axiom,
! [S: set_a,F: list_a > a,T2: set_list_a] :
( ( ord_less_eq_set_a @ S @ ( image_list_a_a @ F @ T2 ) )
= ( ? [U2: set_list_a] :
( ( ord_le8861187494160871172list_a @ U2 @ T2 )
& ( inj_on_list_a_a @ F @ U2 )
& ( S
= ( image_list_a_a @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_735_subset__image__inj,axiom,
! [S: set_list_a,F: int > list_a,T2: set_int] :
( ( ord_le8861187494160871172list_a @ S @ ( image_int_list_a @ F @ T2 ) )
= ( ? [U2: set_int] :
( ( ord_less_eq_set_int @ U2 @ T2 )
& ( inj_on_int_list_a @ F @ U2 )
& ( S
= ( image_int_list_a @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_736_subset__image__inj,axiom,
! [S: set_list_a,F: a > list_a,T2: set_a] :
( ( ord_le8861187494160871172list_a @ S @ ( image_a_list_a @ F @ T2 ) )
= ( ? [U2: set_a] :
( ( ord_less_eq_set_a @ U2 @ T2 )
& ( inj_on_a_list_a @ F @ U2 )
& ( S
= ( image_a_list_a @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_737_subset__image__inj,axiom,
! [S: set_list_a,F: list_a > list_a,T2: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ ( image_list_a_list_a @ F @ T2 ) )
= ( ? [U2: set_list_a] :
( ( ord_le8861187494160871172list_a @ U2 @ T2 )
& ( inj_on_list_a_list_a @ F @ U2 )
& ( S
= ( image_list_a_list_a @ F @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_738_image__inv__into__cancel,axiom,
! [F: list_a > int,A3: set_list_a,A5: set_int,B4: set_int] :
( ( ( image_list_a_int @ F @ A3 )
= A5 )
=> ( ( ord_less_eq_set_int @ B4 @ A5 )
=> ( ( image_list_a_int @ F @ ( image_int_list_a @ ( hilber854936315069205348_a_int @ A3 @ F ) @ B4 ) )
= B4 ) ) ) ).
% image_inv_into_cancel
thf(fact_739_image__inv__into__cancel,axiom,
! [F: a > int,A3: set_a,A5: set_int,B4: set_int] :
( ( ( image_a_int @ F @ A3 )
= A5 )
=> ( ( ord_less_eq_set_int @ B4 @ A5 )
=> ( ( image_a_int @ F @ ( image_int_a @ ( hilber7984441185272261726_a_int @ A3 @ F ) @ B4 ) )
= B4 ) ) ) ).
% image_inv_into_cancel
thf(fact_740_image__inv__into__cancel,axiom,
! [F: list_a > a,A3: set_list_a,A5: set_a,B4: set_a] :
( ( ( image_list_a_a @ F @ A3 )
= A5 )
=> ( ( ord_less_eq_set_a @ B4 @ A5 )
=> ( ( image_list_a_a @ F @ ( image_a_list_a @ ( hilber8556971080906182598st_a_a @ A3 @ F ) @ B4 ) )
= B4 ) ) ) ).
% image_inv_into_cancel
thf(fact_741_image__inv__into__cancel,axiom,
! [F: int > a,A3: set_int,A5: set_a,B4: set_a] :
( ( ( image_int_a @ F @ A3 )
= A5 )
=> ( ( ord_less_eq_set_a @ B4 @ A5 )
=> ( ( image_int_a @ F @ ( image_a_int @ ( hilber550346152434376708_int_a @ A3 @ F ) @ B4 ) )
= B4 ) ) ) ).
% image_inv_into_cancel
thf(fact_742_image__inv__into__cancel,axiom,
! [F: a > a,A3: set_a,A5: set_a,B4: set_a] :
( ( ( image_a_a @ F @ A3 )
= A5 )
=> ( ( ord_less_eq_set_a @ B4 @ A5 )
=> ( ( image_a_a @ F @ ( image_a_a @ ( hilbert_inv_into_a_a @ A3 @ F ) @ B4 ) )
= B4 ) ) ) ).
% image_inv_into_cancel
thf(fact_743_image__inv__into__cancel,axiom,
! [F: list_a > list_a,A3: set_list_a,A5: set_list_a,B4: set_list_a] :
( ( ( image_list_a_list_a @ F @ A3 )
= A5 )
=> ( ( ord_le8861187494160871172list_a @ B4 @ A5 )
=> ( ( image_list_a_list_a @ F @ ( image_list_a_list_a @ ( hilber4441511847348742220list_a @ A3 @ F ) @ B4 ) )
= B4 ) ) ) ).
% image_inv_into_cancel
thf(fact_744_image__inv__into__cancel,axiom,
! [F: int > list_a,A3: set_int,A5: set_list_a,B4: set_list_a] :
( ( ( image_int_list_a @ F @ A3 )
= A5 )
=> ( ( ord_le8861187494160871172list_a @ B4 @ A5 )
=> ( ( image_int_list_a @ F @ ( image_list_a_int @ ( hilber7218023918483505546list_a @ A3 @ F ) @ B4 ) )
= B4 ) ) ) ).
% image_inv_into_cancel
thf(fact_745_image__inv__into__cancel,axiom,
! [F: a > list_a,A3: set_a,A5: set_list_a,B4: set_list_a] :
( ( ( image_a_list_a @ F @ A3 )
= A5 )
=> ( ( ord_le8861187494160871172list_a @ B4 @ A5 )
=> ( ( image_a_list_a @ F @ ( image_list_a_a @ ( hilber1222834799880672978list_a @ A3 @ F ) @ B4 ) )
= B4 ) ) ) ).
% image_inv_into_cancel
thf(fact_746_surj__f__inv__f,axiom,
! [F: int > int,Y3: int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ( ( F @ ( hilber4630740438190819238nt_int @ top_top_set_int @ F @ Y3 ) )
= Y3 ) ) ).
% surj_f_inv_f
thf(fact_747_surj__f__inv__f,axiom,
! [F: int > a,Y3: a] :
( ( ( image_int_a @ F @ top_top_set_int )
= top_top_set_a )
=> ( ( F @ ( hilber550346152434376708_int_a @ top_top_set_int @ F @ Y3 ) )
= Y3 ) ) ).
% surj_f_inv_f
thf(fact_748_surj__f__inv__f,axiom,
! [F: int > list_a,Y3: list_a] :
( ( ( image_int_list_a @ F @ top_top_set_int )
= top_top_set_list_a )
=> ( ( F @ ( hilber7218023918483505546list_a @ top_top_set_int @ F @ Y3 ) )
= Y3 ) ) ).
% surj_f_inv_f
thf(fact_749_surj__f__inv__f,axiom,
! [F: a > int,Y3: int] :
( ( ( image_a_int @ F @ top_top_set_a )
= top_top_set_int )
=> ( ( F @ ( hilber7984441185272261726_a_int @ top_top_set_a @ F @ Y3 ) )
= Y3 ) ) ).
% surj_f_inv_f
thf(fact_750_surj__f__inv__f,axiom,
! [F: a > a,Y3: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( F @ ( hilbert_inv_into_a_a @ top_top_set_a @ F @ Y3 ) )
= Y3 ) ) ).
% surj_f_inv_f
thf(fact_751_surj__f__inv__f,axiom,
! [F: a > list_a,Y3: list_a] :
( ( ( image_a_list_a @ F @ top_top_set_a )
= top_top_set_list_a )
=> ( ( F @ ( hilber1222834799880672978list_a @ top_top_set_a @ F @ Y3 ) )
= Y3 ) ) ).
% surj_f_inv_f
thf(fact_752_surj__f__inv__f,axiom,
! [F: list_a > int,Y3: int] :
( ( ( image_list_a_int @ F @ top_top_set_list_a )
= top_top_set_int )
=> ( ( F @ ( hilber854936315069205348_a_int @ top_top_set_list_a @ F @ Y3 ) )
= Y3 ) ) ).
% surj_f_inv_f
thf(fact_753_surj__f__inv__f,axiom,
! [F: list_a > a,Y3: a] :
( ( ( image_list_a_a @ F @ top_top_set_list_a )
= top_top_set_a )
=> ( ( F @ ( hilber8556971080906182598st_a_a @ top_top_set_list_a @ F @ Y3 ) )
= Y3 ) ) ).
% surj_f_inv_f
thf(fact_754_surj__f__inv__f,axiom,
! [F: list_a > list_a,Y3: list_a] :
( ( ( image_list_a_list_a @ F @ top_top_set_list_a )
= top_top_set_list_a )
=> ( ( F @ ( hilber4441511847348742220list_a @ top_top_set_list_a @ F @ Y3 ) )
= Y3 ) ) ).
% surj_f_inv_f
thf(fact_755_surj__iff__all,axiom,
! [F: int > int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
= ( ! [X2: int] :
( ( F @ ( hilber4630740438190819238nt_int @ top_top_set_int @ F @ X2 ) )
= X2 ) ) ) ).
% surj_iff_all
thf(fact_756_surj__iff__all,axiom,
! [F: int > a] :
( ( ( image_int_a @ F @ top_top_set_int )
= top_top_set_a )
= ( ! [X2: a] :
( ( F @ ( hilber550346152434376708_int_a @ top_top_set_int @ F @ X2 ) )
= X2 ) ) ) ).
% surj_iff_all
thf(fact_757_surj__iff__all,axiom,
! [F: int > list_a] :
( ( ( image_int_list_a @ F @ top_top_set_int )
= top_top_set_list_a )
= ( ! [X2: list_a] :
( ( F @ ( hilber7218023918483505546list_a @ top_top_set_int @ F @ X2 ) )
= X2 ) ) ) ).
% surj_iff_all
thf(fact_758_surj__iff__all,axiom,
! [F: a > int] :
( ( ( image_a_int @ F @ top_top_set_a )
= top_top_set_int )
= ( ! [X2: int] :
( ( F @ ( hilber7984441185272261726_a_int @ top_top_set_a @ F @ X2 ) )
= X2 ) ) ) ).
% surj_iff_all
thf(fact_759_surj__iff__all,axiom,
! [F: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
= ( ! [X2: a] :
( ( F @ ( hilbert_inv_into_a_a @ top_top_set_a @ F @ X2 ) )
= X2 ) ) ) ).
% surj_iff_all
thf(fact_760_surj__iff__all,axiom,
! [F: a > list_a] :
( ( ( image_a_list_a @ F @ top_top_set_a )
= top_top_set_list_a )
= ( ! [X2: list_a] :
( ( F @ ( hilber1222834799880672978list_a @ top_top_set_a @ F @ X2 ) )
= X2 ) ) ) ).
% surj_iff_all
thf(fact_761_surj__iff__all,axiom,
! [F: list_a > int] :
( ( ( image_list_a_int @ F @ top_top_set_list_a )
= top_top_set_int )
= ( ! [X2: int] :
( ( F @ ( hilber854936315069205348_a_int @ top_top_set_list_a @ F @ X2 ) )
= X2 ) ) ) ).
% surj_iff_all
thf(fact_762_surj__iff__all,axiom,
! [F: list_a > a] :
( ( ( image_list_a_a @ F @ top_top_set_list_a )
= top_top_set_a )
= ( ! [X2: a] :
( ( F @ ( hilber8556971080906182598st_a_a @ top_top_set_list_a @ F @ X2 ) )
= X2 ) ) ) ).
% surj_iff_all
thf(fact_763_surj__iff__all,axiom,
! [F: list_a > list_a] :
( ( ( image_list_a_list_a @ F @ top_top_set_list_a )
= top_top_set_list_a )
= ( ! [X2: list_a] :
( ( F @ ( hilber4441511847348742220list_a @ top_top_set_list_a @ F @ X2 ) )
= X2 ) ) ) ).
% surj_iff_all
thf(fact_764_image__f__inv__f,axiom,
! [F: int > int,A3: set_int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ( ( image_int_int @ F @ ( image_int_int @ ( hilber4630740438190819238nt_int @ top_top_set_int @ F ) @ A3 ) )
= A3 ) ) ).
% image_f_inv_f
thf(fact_765_image__f__inv__f,axiom,
! [F: int > a,A3: set_a] :
( ( ( image_int_a @ F @ top_top_set_int )
= top_top_set_a )
=> ( ( image_int_a @ F @ ( image_a_int @ ( hilber550346152434376708_int_a @ top_top_set_int @ F ) @ A3 ) )
= A3 ) ) ).
% image_f_inv_f
thf(fact_766_image__f__inv__f,axiom,
! [F: int > list_a,A3: set_list_a] :
( ( ( image_int_list_a @ F @ top_top_set_int )
= top_top_set_list_a )
=> ( ( image_int_list_a @ F @ ( image_list_a_int @ ( hilber7218023918483505546list_a @ top_top_set_int @ F ) @ A3 ) )
= A3 ) ) ).
% image_f_inv_f
thf(fact_767_image__f__inv__f,axiom,
! [F: a > int,A3: set_int] :
( ( ( image_a_int @ F @ top_top_set_a )
= top_top_set_int )
=> ( ( image_a_int @ F @ ( image_int_a @ ( hilber7984441185272261726_a_int @ top_top_set_a @ F ) @ A3 ) )
= A3 ) ) ).
% image_f_inv_f
thf(fact_768_image__f__inv__f,axiom,
! [F: a > a,A3: set_a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( image_a_a @ F @ ( image_a_a @ ( hilbert_inv_into_a_a @ top_top_set_a @ F ) @ A3 ) )
= A3 ) ) ).
% image_f_inv_f
thf(fact_769_image__f__inv__f,axiom,
! [F: a > list_a,A3: set_list_a] :
( ( ( image_a_list_a @ F @ top_top_set_a )
= top_top_set_list_a )
=> ( ( image_a_list_a @ F @ ( image_list_a_a @ ( hilber1222834799880672978list_a @ top_top_set_a @ F ) @ A3 ) )
= A3 ) ) ).
% image_f_inv_f
thf(fact_770_image__f__inv__f,axiom,
! [F: list_a > int,A3: set_int] :
( ( ( image_list_a_int @ F @ top_top_set_list_a )
= top_top_set_int )
=> ( ( image_list_a_int @ F @ ( image_int_list_a @ ( hilber854936315069205348_a_int @ top_top_set_list_a @ F ) @ A3 ) )
= A3 ) ) ).
% image_f_inv_f
thf(fact_771_image__f__inv__f,axiom,
! [F: list_a > a,A3: set_a] :
( ( ( image_list_a_a @ F @ top_top_set_list_a )
= top_top_set_a )
=> ( ( image_list_a_a @ F @ ( image_a_list_a @ ( hilber8556971080906182598st_a_a @ top_top_set_list_a @ F ) @ A3 ) )
= A3 ) ) ).
% image_f_inv_f
thf(fact_772_image__f__inv__f,axiom,
! [F: list_a > list_a,A3: set_list_a] :
( ( ( image_list_a_list_a @ F @ top_top_set_list_a )
= top_top_set_list_a )
=> ( ( image_list_a_list_a @ F @ ( image_list_a_list_a @ ( hilber4441511847348742220list_a @ top_top_set_list_a @ F ) @ A3 ) )
= A3 ) ) ).
% image_f_inv_f
thf(fact_773_surj__imp__inv__eq,axiom,
! [F: int > int,G: int > int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ( ! [X3: int] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( hilber4630740438190819238nt_int @ top_top_set_int @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_774_surj__imp__inv__eq,axiom,
! [F: int > a,G: a > int] :
( ( ( image_int_a @ F @ top_top_set_int )
= top_top_set_a )
=> ( ! [X3: int] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( hilber550346152434376708_int_a @ top_top_set_int @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_775_surj__imp__inv__eq,axiom,
! [F: int > list_a,G: list_a > int] :
( ( ( image_int_list_a @ F @ top_top_set_int )
= top_top_set_list_a )
=> ( ! [X3: int] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( hilber7218023918483505546list_a @ top_top_set_int @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_776_surj__imp__inv__eq,axiom,
! [F: a > int,G: int > a] :
( ( ( image_a_int @ F @ top_top_set_a )
= top_top_set_int )
=> ( ! [X3: a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( hilber7984441185272261726_a_int @ top_top_set_a @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_777_surj__imp__inv__eq,axiom,
! [F: a > a,G: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( ! [X3: a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( hilbert_inv_into_a_a @ top_top_set_a @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_778_surj__imp__inv__eq,axiom,
! [F: a > list_a,G: list_a > a] :
( ( ( image_a_list_a @ F @ top_top_set_a )
= top_top_set_list_a )
=> ( ! [X3: a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( hilber1222834799880672978list_a @ top_top_set_a @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_779_surj__imp__inv__eq,axiom,
! [F: list_a > int,G: int > list_a] :
( ( ( image_list_a_int @ F @ top_top_set_list_a )
= top_top_set_int )
=> ( ! [X3: list_a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( hilber854936315069205348_a_int @ top_top_set_list_a @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_780_surj__imp__inv__eq,axiom,
! [F: list_a > a,G: a > list_a] :
( ( ( image_list_a_a @ F @ top_top_set_list_a )
= top_top_set_a )
=> ( ! [X3: list_a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( hilber8556971080906182598st_a_a @ top_top_set_list_a @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_781_surj__imp__inv__eq,axiom,
! [F: list_a > list_a,G: list_a > list_a] :
( ( ( image_list_a_list_a @ F @ top_top_set_list_a )
= top_top_set_list_a )
=> ( ! [X3: list_a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( hilber4441511847348742220list_a @ top_top_set_list_a @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_782_ring_Oup__add__closed,axiom,
! [R: partia2670972154091845814t_unit,P2: nat > list_a,Q2: nat > list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_nat_list_a @ P2 @ ( up_lis8464167429055313730t_unit @ R ) )
=> ( ( member_nat_list_a @ Q2 @ ( up_lis8464167429055313730t_unit @ R ) )
=> ( member_nat_list_a
@ ^ [I3: nat] : ( add_li7652885771158616974t_unit @ R @ ( P2 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_lis8464167429055313730t_unit @ R ) ) ) ) ) ).
% ring.up_add_closed
thf(fact_783_ring_Oup__add__closed,axiom,
! [R: partia2175431115845679010xt_a_b,P2: nat > a,Q2: nat > a] :
( ( ring_a_b @ R )
=> ( ( member_nat_a @ P2 @ ( up_a_b @ R ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ R ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( add_a_b @ R @ ( P2 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_a_b @ R ) ) ) ) ) ).
% ring.up_add_closed
thf(fact_784_ring_Oup__a__inv__closed,axiom,
! [R: partia2670972154091845814t_unit,P2: nat > list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_nat_list_a @ P2 @ ( up_lis8464167429055313730t_unit @ R ) )
=> ( member_nat_list_a
@ ^ [I3: nat] : ( a_inv_8944721093294617173t_unit @ R @ ( P2 @ I3 ) )
@ ( up_lis8464167429055313730t_unit @ R ) ) ) ) ).
% ring.up_a_inv_closed
thf(fact_785_ring_Oup__a__inv__closed,axiom,
! [R: partia2175431115845679010xt_a_b,P2: nat > a] :
( ( ring_a_b @ R )
=> ( ( member_nat_a @ P2 @ ( up_a_b @ R ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( a_inv_a_b @ R @ ( P2 @ I3 ) )
@ ( up_a_b @ R ) ) ) ) ).
% ring.up_a_inv_closed
thf(fact_786_ring_Oup__minus__closed,axiom,
! [R: partia2670972154091845814t_unit,P2: nat > list_a,Q2: nat > list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_nat_list_a @ P2 @ ( up_lis8464167429055313730t_unit @ R ) )
=> ( ( member_nat_list_a @ Q2 @ ( up_lis8464167429055313730t_unit @ R ) )
=> ( member_nat_list_a
@ ^ [I3: nat] : ( a_minu3984020753470702548t_unit @ R @ ( P2 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_lis8464167429055313730t_unit @ R ) ) ) ) ) ).
% ring.up_minus_closed
thf(fact_787_ring_Oup__minus__closed,axiom,
! [R: partia2175431115845679010xt_a_b,P2: nat > a,Q2: nat > a] :
( ( ring_a_b @ R )
=> ( ( member_nat_a @ P2 @ ( up_a_b @ R ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ R ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( a_minus_a_b @ R @ ( P2 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_a_b @ R ) ) ) ) ) ).
% ring.up_minus_closed
thf(fact_788_inj__on__inv__into,axiom,
! [B3: set_a,F: int > a,A3: set_int] :
( ( ord_less_eq_set_a @ B3 @ ( image_int_a @ F @ A3 ) )
=> ( inj_on_a_int @ ( hilber550346152434376708_int_a @ A3 @ F ) @ B3 ) ) ).
% inj_on_inv_into
thf(fact_789_inj__on__inv__into,axiom,
! [B3: set_a,F: list_a > a,A3: set_list_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_list_a_a @ F @ A3 ) )
=> ( inj_on_a_list_a @ ( hilber8556971080906182598st_a_a @ A3 @ F ) @ B3 ) ) ).
% inj_on_inv_into
thf(fact_790_inj__on__inv__into,axiom,
! [B3: set_a,F: a > a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A3 ) )
=> ( inj_on_a_a @ ( hilbert_inv_into_a_a @ A3 @ F ) @ B3 ) ) ).
% inj_on_inv_into
thf(fact_791_inj__on__inv__into,axiom,
! [B3: set_list_a,F: int > list_a,A3: set_int] :
( ( ord_le8861187494160871172list_a @ B3 @ ( image_int_list_a @ F @ A3 ) )
=> ( inj_on_list_a_int @ ( hilber7218023918483505546list_a @ A3 @ F ) @ B3 ) ) ).
% inj_on_inv_into
thf(fact_792_inj__on__inv__into,axiom,
! [B3: set_list_a,F: list_a > list_a,A3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ ( image_list_a_list_a @ F @ A3 ) )
=> ( inj_on_list_a_list_a @ ( hilber4441511847348742220list_a @ A3 @ F ) @ B3 ) ) ).
% inj_on_inv_into
thf(fact_793_inj__on__inv__into,axiom,
! [B3: set_list_a,F: list_list_a > list_a,A3: set_list_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ ( image_4177985004758493951list_a @ F @ A3 ) )
=> ( inj_on6538719745830000247list_a @ ( hilber2546232707058280006list_a @ A3 @ F ) @ B3 ) ) ).
% inj_on_inv_into
thf(fact_794_inj__on__inv__into,axiom,
! [B3: set_list_a,F: a > list_a,A3: set_a] :
( ( ord_le8861187494160871172list_a @ B3 @ ( image_a_list_a @ F @ A3 ) )
=> ( inj_on_list_a_a @ ( hilber1222834799880672978list_a @ A3 @ F ) @ B3 ) ) ).
% inj_on_inv_into
thf(fact_795_inj__transfer,axiom,
! [F: int > a,P: int > $o,X: int] :
( ( inj_on_int_a @ F @ top_top_set_int )
=> ( ! [Y: a] :
( ( member_a @ Y @ ( image_int_a @ F @ top_top_set_int ) )
=> ( P @ ( hilber550346152434376708_int_a @ top_top_set_int @ F @ Y ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_796_inj__transfer,axiom,
! [F: a > a,P: a > $o,X: a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ! [Y: a] :
( ( member_a @ Y @ ( image_a_a @ F @ top_top_set_a ) )
=> ( P @ ( hilbert_inv_into_a_a @ top_top_set_a @ F @ Y ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_797_inj__transfer,axiom,
! [F: int > list_a,P: int > $o,X: int] :
( ( inj_on_int_list_a @ F @ top_top_set_int )
=> ( ! [Y: list_a] :
( ( member_list_a @ Y @ ( image_int_list_a @ F @ top_top_set_int ) )
=> ( P @ ( hilber7218023918483505546list_a @ top_top_set_int @ F @ Y ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_798_inj__transfer,axiom,
! [F: a > list_a,P: a > $o,X: a] :
( ( inj_on_a_list_a @ F @ top_top_set_a )
=> ( ! [Y: list_a] :
( ( member_list_a @ Y @ ( image_a_list_a @ F @ top_top_set_a ) )
=> ( P @ ( hilber1222834799880672978list_a @ top_top_set_a @ F @ Y ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_799_inj__transfer,axiom,
! [F: list_a > a,P: list_a > $o,X: list_a] :
( ( inj_on_list_a_a @ F @ top_top_set_list_a )
=> ( ! [Y: a] :
( ( member_a @ Y @ ( image_list_a_a @ F @ top_top_set_list_a ) )
=> ( P @ ( hilber8556971080906182598st_a_a @ top_top_set_list_a @ F @ Y ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_800_inj__transfer,axiom,
! [F: int > nat > a,P: int > $o,X: int] :
( ( inj_on_int_nat_a @ F @ top_top_set_int )
=> ( ! [Y: nat > a] :
( ( member_nat_a @ Y @ ( image_int_nat_a @ F @ top_top_set_int ) )
=> ( P @ ( hilber1943333713353564907_nat_a @ top_top_set_int @ F @ Y ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_801_inj__transfer,axiom,
! [F: a > nat > a,P: a > $o,X: a] :
( ( inj_on_a_nat_a @ F @ top_top_set_a )
=> ( ! [Y: nat > a] :
( ( member_nat_a @ Y @ ( image_a_nat_a @ F @ top_top_set_a ) )
=> ( P @ ( hilber1523447219144142899_nat_a @ top_top_set_a @ F @ Y ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_802_inj__transfer,axiom,
! [F: list_a > list_a,P: list_a > $o,X: list_a] :
( ( inj_on_list_a_list_a @ F @ top_top_set_list_a )
=> ( ! [Y: list_a] :
( ( member_list_a @ Y @ ( image_list_a_list_a @ F @ top_top_set_list_a ) )
=> ( P @ ( hilber4441511847348742220list_a @ top_top_set_list_a @ F @ Y ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_803_inj__transfer,axiom,
! [F: int > set_int > a,P: int > $o,X: int] :
( ( inj_on_int_set_int_a @ F @ top_top_set_int )
=> ( ! [Y: set_int > a] :
( ( member_set_int_a @ Y @ ( image_int_set_int_a @ F @ top_top_set_int ) )
=> ( P @ ( hilber3251774746670828377_int_a @ top_top_set_int @ F @ Y ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_804_inj__transfer,axiom,
! [F: int > nat > list_a,P: int > $o,X: int] :
( ( inj_on7005093563047325836list_a @ F @ top_top_set_int )
=> ( ! [Y: nat > list_a] :
( ( member_nat_list_a @ Y @ ( image_int_nat_list_a @ F @ top_top_set_int ) )
=> ( P @ ( hilber6310135974011704177list_a @ top_top_set_int @ F @ Y ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_805_image__inv__f__f,axiom,
! [F: int > list_a,A3: set_int] :
( ( inj_on_int_list_a @ F @ top_top_set_int )
=> ( ( image_list_a_int @ ( hilber7218023918483505546list_a @ top_top_set_int @ F ) @ ( image_int_list_a @ F @ A3 ) )
= A3 ) ) ).
% image_inv_f_f
thf(fact_806_image__inv__f__f,axiom,
! [F: int > a,A3: set_int] :
( ( inj_on_int_a @ F @ top_top_set_int )
=> ( ( image_a_int @ ( hilber550346152434376708_int_a @ top_top_set_int @ F ) @ ( image_int_a @ F @ A3 ) )
= A3 ) ) ).
% image_inv_f_f
thf(fact_807_image__inv__f__f,axiom,
! [F: a > int,A3: set_a] :
( ( inj_on_a_int @ F @ top_top_set_a )
=> ( ( image_int_a @ ( hilber7984441185272261726_a_int @ top_top_set_a @ F ) @ ( image_a_int @ F @ A3 ) )
= A3 ) ) ).
% image_inv_f_f
thf(fact_808_image__inv__f__f,axiom,
! [F: a > a,A3: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( image_a_a @ ( hilbert_inv_into_a_a @ top_top_set_a @ F ) @ ( image_a_a @ F @ A3 ) )
= A3 ) ) ).
% image_inv_f_f
thf(fact_809_image__inv__f__f,axiom,
! [F: a > list_a,A3: set_a] :
( ( inj_on_a_list_a @ F @ top_top_set_a )
=> ( ( image_list_a_a @ ( hilber1222834799880672978list_a @ top_top_set_a @ F ) @ ( image_a_list_a @ F @ A3 ) )
= A3 ) ) ).
% image_inv_f_f
thf(fact_810_image__inv__f__f,axiom,
! [F: list_a > int,A3: set_list_a] :
( ( inj_on_list_a_int @ F @ top_top_set_list_a )
=> ( ( image_int_list_a @ ( hilber854936315069205348_a_int @ top_top_set_list_a @ F ) @ ( image_list_a_int @ F @ A3 ) )
= A3 ) ) ).
% image_inv_f_f
thf(fact_811_image__inv__f__f,axiom,
! [F: list_a > a,A3: set_list_a] :
( ( inj_on_list_a_a @ F @ top_top_set_list_a )
=> ( ( image_a_list_a @ ( hilber8556971080906182598st_a_a @ top_top_set_list_a @ F ) @ ( image_list_a_a @ F @ A3 ) )
= A3 ) ) ).
% image_inv_f_f
thf(fact_812_image__inv__f__f,axiom,
! [F: list_a > list_a,A3: set_list_a] :
( ( inj_on_list_a_list_a @ F @ top_top_set_list_a )
=> ( ( image_list_a_list_a @ ( hilber4441511847348742220list_a @ top_top_set_list_a @ F ) @ ( image_list_a_list_a @ F @ A3 ) )
= A3 ) ) ).
% image_inv_f_f
thf(fact_813_image__inv__f__f,axiom,
! [F: list_a > list_list_a,A3: set_list_a] :
( ( inj_on6538719745830000247list_a @ F @ top_top_set_list_a )
=> ( ( image_4177985004758493951list_a @ ( hilber6629114656297661522list_a @ top_top_set_list_a @ F ) @ ( image_8260866953997875467list_a @ F @ A3 ) )
= A3 ) ) ).
% image_inv_f_f
thf(fact_814_inj__imp__surj__inv,axiom,
! [F: int > int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( image_int_int @ ( hilber4630740438190819238nt_int @ top_top_set_int @ F ) @ top_top_set_int )
= top_top_set_int ) ) ).
% inj_imp_surj_inv
thf(fact_815_inj__imp__surj__inv,axiom,
! [F: int > a] :
( ( inj_on_int_a @ F @ top_top_set_int )
=> ( ( image_a_int @ ( hilber550346152434376708_int_a @ top_top_set_int @ F ) @ top_top_set_a )
= top_top_set_int ) ) ).
% inj_imp_surj_inv
thf(fact_816_inj__imp__surj__inv,axiom,
! [F: int > list_a] :
( ( inj_on_int_list_a @ F @ top_top_set_int )
=> ( ( image_list_a_int @ ( hilber7218023918483505546list_a @ top_top_set_int @ F ) @ top_top_set_list_a )
= top_top_set_int ) ) ).
% inj_imp_surj_inv
thf(fact_817_inj__imp__surj__inv,axiom,
! [F: a > int] :
( ( inj_on_a_int @ F @ top_top_set_a )
=> ( ( image_int_a @ ( hilber7984441185272261726_a_int @ top_top_set_a @ F ) @ top_top_set_int )
= top_top_set_a ) ) ).
% inj_imp_surj_inv
thf(fact_818_inj__imp__surj__inv,axiom,
! [F: a > a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( image_a_a @ ( hilbert_inv_into_a_a @ top_top_set_a @ F ) @ top_top_set_a )
= top_top_set_a ) ) ).
% inj_imp_surj_inv
thf(fact_819_inj__imp__surj__inv,axiom,
! [F: a > list_a] :
( ( inj_on_a_list_a @ F @ top_top_set_a )
=> ( ( image_list_a_a @ ( hilber1222834799880672978list_a @ top_top_set_a @ F ) @ top_top_set_list_a )
= top_top_set_a ) ) ).
% inj_imp_surj_inv
thf(fact_820_inj__imp__surj__inv,axiom,
! [F: list_a > list_list_a] :
( ( inj_on6538719745830000247list_a @ F @ top_top_set_list_a )
=> ( ( image_4177985004758493951list_a @ ( hilber6629114656297661522list_a @ top_top_set_list_a @ F ) @ top_to6581514389725680442list_a )
= top_top_set_list_a ) ) ).
% inj_imp_surj_inv
thf(fact_821_inj__imp__surj__inv,axiom,
! [F: list_a > int] :
( ( inj_on_list_a_int @ F @ top_top_set_list_a )
=> ( ( image_int_list_a @ ( hilber854936315069205348_a_int @ top_top_set_list_a @ F ) @ top_top_set_int )
= top_top_set_list_a ) ) ).
% inj_imp_surj_inv
thf(fact_822_inj__imp__surj__inv,axiom,
! [F: list_a > a] :
( ( inj_on_list_a_a @ F @ top_top_set_list_a )
=> ( ( image_a_list_a @ ( hilber8556971080906182598st_a_a @ top_top_set_list_a @ F ) @ top_top_set_a )
= top_top_set_list_a ) ) ).
% inj_imp_surj_inv
thf(fact_823_inj__imp__surj__inv,axiom,
! [F: list_a > list_a] :
( ( inj_on_list_a_list_a @ F @ top_top_set_list_a )
=> ( ( image_list_a_list_a @ ( hilber4441511847348742220list_a @ top_top_set_list_a @ F ) @ top_top_set_list_a )
= top_top_set_list_a ) ) ).
% inj_imp_surj_inv
thf(fact_824_surj__imp__inj__inv,axiom,
! [F: list_list_a > list_a] :
( ( ( image_4177985004758493951list_a @ F @ top_to6581514389725680442list_a )
= top_top_set_list_a )
=> ( inj_on6538719745830000247list_a @ ( hilber2546232707058280006list_a @ top_to6581514389725680442list_a @ F ) @ top_top_set_list_a ) ) ).
% surj_imp_inj_inv
thf(fact_825_surj__imp__inj__inv,axiom,
! [F: int > int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ( inj_on_int_int @ ( hilber4630740438190819238nt_int @ top_top_set_int @ F ) @ top_top_set_int ) ) ).
% surj_imp_inj_inv
thf(fact_826_surj__imp__inj__inv,axiom,
! [F: int > a] :
( ( ( image_int_a @ F @ top_top_set_int )
= top_top_set_a )
=> ( inj_on_a_int @ ( hilber550346152434376708_int_a @ top_top_set_int @ F ) @ top_top_set_a ) ) ).
% surj_imp_inj_inv
thf(fact_827_surj__imp__inj__inv,axiom,
! [F: int > list_a] :
( ( ( image_int_list_a @ F @ top_top_set_int )
= top_top_set_list_a )
=> ( inj_on_list_a_int @ ( hilber7218023918483505546list_a @ top_top_set_int @ F ) @ top_top_set_list_a ) ) ).
% surj_imp_inj_inv
thf(fact_828_surj__imp__inj__inv,axiom,
! [F: a > int] :
( ( ( image_a_int @ F @ top_top_set_a )
= top_top_set_int )
=> ( inj_on_int_a @ ( hilber7984441185272261726_a_int @ top_top_set_a @ F ) @ top_top_set_int ) ) ).
% surj_imp_inj_inv
thf(fact_829_surj__imp__inj__inv,axiom,
! [F: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ( inj_on_a_a @ ( hilbert_inv_into_a_a @ top_top_set_a @ F ) @ top_top_set_a ) ) ).
% surj_imp_inj_inv
thf(fact_830_surj__imp__inj__inv,axiom,
! [F: a > list_a] :
( ( ( image_a_list_a @ F @ top_top_set_a )
= top_top_set_list_a )
=> ( inj_on_list_a_a @ ( hilber1222834799880672978list_a @ top_top_set_a @ F ) @ top_top_set_list_a ) ) ).
% surj_imp_inj_inv
thf(fact_831_surj__imp__inj__inv,axiom,
! [F: list_a > int] :
( ( ( image_list_a_int @ F @ top_top_set_list_a )
= top_top_set_int )
=> ( inj_on_int_list_a @ ( hilber854936315069205348_a_int @ top_top_set_list_a @ F ) @ top_top_set_int ) ) ).
% surj_imp_inj_inv
thf(fact_832_surj__imp__inj__inv,axiom,
! [F: list_a > a] :
( ( ( image_list_a_a @ F @ top_top_set_list_a )
= top_top_set_a )
=> ( inj_on_a_list_a @ ( hilber8556971080906182598st_a_a @ top_top_set_list_a @ F ) @ top_top_set_a ) ) ).
% surj_imp_inj_inv
thf(fact_833_surj__imp__inj__inv,axiom,
! [F: list_a > list_a] :
( ( ( image_list_a_list_a @ F @ top_top_set_list_a )
= top_top_set_list_a )
=> ( inj_on_list_a_list_a @ ( hilber4441511847348742220list_a @ top_top_set_list_a @ F ) @ top_top_set_list_a ) ) ).
% surj_imp_inj_inv
thf(fact_834_ring_Obound__upD,axiom,
! [R: partia2670972154091845814t_unit,F: nat > list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) )
=> ? [N2: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ N2 @ F ) ) ) ).
% ring.bound_upD
thf(fact_835_ring_Obound__upD,axiom,
! [R: partia2175431115845679010xt_a_b,F: nat > a] :
( ( ring_a_b @ R )
=> ( ( member_nat_a @ F @ ( up_a_b @ R ) )
=> ? [N2: nat] : ( bound_a @ ( zero_a_b @ R ) @ N2 @ F ) ) ) ).
% ring.bound_upD
thf(fact_836_p_Oadd_Oone__in__subset,axiom,
! [H: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( H != bot_bot_set_list_a )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X3 ) @ H ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ H )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X3 @ Xa ) @ H ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ H ) ) ) ) ) ).
% p.add.one_in_subset
thf(fact_837_p_Ofinite__carr__imp__char__ge__0,axiom,
( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.finite_carr_imp_char_ge_0
thf(fact_838_d_Ochar__ring__is__subfield,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ k2 ) )
=> ( subfield_a_b @ ( image_int_a @ ( ring_int_embed_a_b @ k2 ) @ top_top_set_int ) @ k2 ) ) ).
% d.char_ring_is_subfield
thf(fact_839_char__ring__is__subfield,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) )
=> ( subfield_a_b @ ( image_int_a @ ( ring_int_embed_a_b @ r ) @ top_top_set_int ) @ r ) ) ).
% char_ring_is_subfield
thf(fact_840_p_Oadd_Oint__pow__mult,axiom,
! [X: list_a,I: int,J: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ ( plus_plus_int @ I @ J ) @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ I @ X ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ J @ X ) ) ) ) ).
% p.add.int_pow_mult
thf(fact_841_p_Oup__one__closed,axiom,
( member_nat_list_a
@ ^ [N4: nat] : ( if_list_a @ ( N4 = zero_zero_nat ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.up_one_closed
thf(fact_842_up__minus__closed,axiom,
! [P2: nat > a,Q2: nat > a] :
( ( member_nat_a @ P2 @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( a_minus_a_b @ r @ ( P2 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_minus_closed
thf(fact_843_d_Oup__minus__closed,axiom,
! [P2: nat > a,Q2: nat > a] :
( ( member_nat_a @ P2 @ ( up_a_b @ k2 ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_b @ k2 ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( a_minus_a_b @ k2 @ ( P2 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_a_b @ k2 ) ) ) ) ).
% d.up_minus_closed
thf(fact_844_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_845_d_Ouniv__poly__is__abelian__monoid,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ k2 )
=> ( abelia226231641709521465t_unit @ ( univ_poly_a_b @ k2 @ K ) ) ) ).
% d.univ_poly_is_abelian_monoid
thf(fact_846_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_847_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_848_d_Osubring__props_I2_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ k2 )
=> ( member_a @ ( zero_a_b @ k2 ) @ K ) ) ).
% d.subring_props(2)
thf(fact_849_d_Osubring__props_I7_J,axiom,
! [K: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K @ k2 )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H22 @ K )
=> ( member_a @ ( add_a_b @ k2 @ H1 @ H22 ) @ K ) ) ) ) ).
% d.subring_props(7)
thf(fact_850_minus__eq,axiom,
! [X: a,Y3: a] :
( ( a_minus_a_b @ r @ X @ Y3 )
= ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y3 ) ) ) ).
% minus_eq
thf(fact_851_d_Ominus__eq,axiom,
! [X: a,Y3: a] :
( ( a_minus_a_b @ k2 @ X @ Y3 )
= ( add_a_b @ k2 @ X @ ( a_inv_a_b @ k2 @ Y3 ) ) ) ).
% d.minus_eq
thf(fact_852_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_853_d_Osubring__props_I5_J,axiom,
! [K: set_a,H2: a] :
( ( subfield_a_b @ K @ k2 )
=> ( ( member_a @ H2 @ K )
=> ( member_a @ ( a_inv_a_b @ k2 @ H2 ) @ K ) ) ) ).
% d.subring_props(5)
thf(fact_854_p_Oabelian__monoid__axioms,axiom,
abelia226231641709521465t_unit @ ( univ_poly_a_b @ r @ k ) ).
% p.abelian_monoid_axioms
thf(fact_855_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_856_d_Osubring__props_I1_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ k2 )
=> ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ k2 ) ) ) ).
% d.subring_props(1)
thf(fact_857_int__embed__add,axiom,
! [X: int,Y3: int] :
( ( ring_int_embed_a_b @ r @ ( plus_plus_int @ X @ Y3 ) )
= ( add_a_b @ r @ ( ring_int_embed_a_b @ r @ X ) @ ( ring_int_embed_a_b @ r @ Y3 ) ) ) ).
% int_embed_add
thf(fact_858_d_Oint__embed__add,axiom,
! [X: int,Y3: int] :
( ( ring_int_embed_a_b @ k2 @ ( plus_plus_int @ X @ Y3 ) )
= ( add_a_b @ k2 @ ( ring_int_embed_a_b @ k2 @ X ) @ ( ring_int_embed_a_b @ k2 @ Y3 ) ) ) ).
% d.int_embed_add
thf(fact_859_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_860_d_Oadd_Oint__pow__mult,axiom,
! [X: a,I: int,J: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( add_pow_a_b_int @ k2 @ ( plus_plus_int @ I @ J ) @ X )
= ( add_a_b @ k2 @ ( add_pow_a_b_int @ k2 @ I @ X ) @ ( add_pow_a_b_int @ k2 @ J @ X ) ) ) ) ).
% d.add.int_pow_mult
thf(fact_861_p_Ocarrier__not__empty,axiom,
( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) )
!= bot_bot_set_list_a ) ).
% p.carrier_not_empty
thf(fact_862_p_Oint__embed__add,axiom,
! [X: int,Y3: int] :
( ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ k ) @ ( plus_plus_int @ X @ Y3 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ ( ring_i7845631192508594762t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 ) ) ) ).
% p.int_embed_add
thf(fact_863_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_864_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_865_minus__closed,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X @ Y3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_866_d_Ominus__closed,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( member_a @ ( a_minus_a_b @ k2 @ X @ Y3 ) @ ( partia707051561876973205xt_a_b @ k2 ) ) ) ) ).
% d.minus_closed
thf(fact_867_h_Oimg__is__subfield_I1_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ k2 )
=> ( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( inj_on_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) ) ) ).
% h.img_is_subfield(1)
thf(fact_868_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_869_d_Or__right__minus__eq,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ( a_minus_a_b @ k2 @ A @ B )
= ( zero_a_b @ k2 ) )
= ( A = B ) ) ) ) ).
% d.r_right_minus_eq
thf(fact_870_p_Oone__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).
% p.one_closed
thf(fact_871_p_Onat__pow__one,axiom,
! [N: nat] :
( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ N )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.nat_pow_one
thf(fact_872_p_Onat__pow__0,axiom,
! [X: list_a] :
( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ zero_zero_nat )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.nat_pow_0
thf(fact_873_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_874_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_875_subfieldE_I6_J,axiom,
! [K: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subfie4546268998243038636t_unit @ K @ R )
=> ( ( one_li8234411390022467901t_unit @ R )
!= ( zero_l347298301471573063t_unit @ R ) ) ) ).
% subfieldE(6)
thf(fact_876_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_877_subfieldE_I1_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( subrin6918843898125473962t_unit @ K @ R ) ) ).
% subfieldE(1)
thf(fact_878_subfieldE_I1_J,axiom,
! [K: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subfie4546268998243038636t_unit @ K @ R )
=> ( subrin3541368690557094692t_unit @ K @ R ) ) ).
% subfieldE(1)
thf(fact_879_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_880_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_881_infinite__countable__subset,axiom,
! [S: set_set_int] :
( ~ ( finite6197958912794628473et_int @ S )
=> ? [F2: nat > set_int] :
( ( inj_on_nat_set_int @ F2 @ top_top_set_nat )
& ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F2 @ top_top_set_nat ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_882_infinite__countable__subset,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_883_infinite__countable__subset,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ? [F2: nat > a] :
( ( inj_on_nat_a @ F2 @ top_top_set_nat )
& ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ top_top_set_nat ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_884_infinite__countable__subset,axiom,
! [S: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ? [F2: nat > list_a] :
( ( inj_on_nat_list_a @ F2 @ top_top_set_nat )
& ( ord_le8861187494160871172list_a @ ( image_nat_list_a @ F2 @ top_top_set_nat ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_885_infinite__iff__countable__subset,axiom,
! [S: set_set_int] :
( ( ~ ( finite6197958912794628473et_int @ S ) )
= ( ? [F3: nat > set_int] :
( ( inj_on_nat_set_int @ F3 @ top_top_set_nat )
& ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F3 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_886_infinite__iff__countable__subset,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ? [F3: nat > nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_887_infinite__iff__countable__subset,axiom,
! [S: set_a] :
( ( ~ ( finite_finite_a @ S ) )
= ( ? [F3: nat > a] :
( ( inj_on_nat_a @ F3 @ top_top_set_nat )
& ( ord_less_eq_set_a @ ( image_nat_a @ F3 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_888_infinite__iff__countable__subset,axiom,
! [S: set_list_a] :
( ( ~ ( finite_finite_list_a @ S ) )
= ( ? [F3: nat > list_a] :
( ( inj_on_nat_list_a @ F3 @ top_top_set_nat )
& ( ord_le8861187494160871172list_a @ ( image_nat_list_a @ F3 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_889_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_890_subfieldE_I2_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( subcri7763218559781929323t_unit @ K @ R ) ) ).
% subfieldE(2)
thf(fact_891_subfieldE_I2_J,axiom,
! [K: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subfie4546268998243038636t_unit @ K @ R )
=> ( subcri8676831449680469861t_unit @ K @ R ) ) ).
% subfieldE(2)
thf(fact_892_subcringE_I3_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ H ) ) ).
% subcringE(3)
thf(fact_893_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_894_subfield_Oaxioms_I1_J,axiom,
! [K: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subfie4546268998243038636t_unit @ K @ R )
=> ( subdom561091866123308472t_unit @ K @ R ) ) ).
% subfield.axioms(1)
thf(fact_895_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_896_subfield_Oaxioms_I1_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( subdom7821232466298058046t_unit @ K @ R ) ) ).
% subfield.axioms(1)
thf(fact_897_ring__iso__memE_I4_J,axiom,
! [H2: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit] :
( ( member_list_a_list_a @ H2 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( H2 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_898_ring__iso__memE_I4_J,axiom,
! [H2: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_list_a_a @ H2 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( H2 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_899_ring__iso__memE_I4_J,axiom,
! [H2: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( H2 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_900_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_901_ring__iso__memE_I4_J,axiom,
! [H2: set_int > list_a,R: partia4934656038542163276t_unit,S: partia2670972154091845814t_unit] :
( ( member6804298160596253954list_a @ H2 @ ( ring_i8159631783994945698t_unit @ R @ S ) )
=> ( ( H2 @ ( one_se8065767436706823081t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_902_ring__iso__memE_I4_J,axiom,
! [H2: set_int > a,R: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_set_int_a @ H2 @ ( ring_i2213127016525301402it_a_b @ R @ S ) )
=> ( ( H2 @ ( one_se8065767436706823081t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_903_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_904_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_905_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_906_subdomainE_I3_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ H ) ) ).
% subdomainE(3)
thf(fact_907_subcringE_I4_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( H != bot_bot_set_list_a ) ) ).
% subcringE(4)
thf(fact_908_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_909_ring__hom__ring_Oimg__is__subfield_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a,K: set_a] :
( ( ring_h4024360765257340990_b_a_b @ R @ S @ H2 )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ( one_a_ring_ext_a_b @ S )
!= ( zero_a_b @ S ) )
=> ( subfield_a_b @ ( image_a_a @ H2 @ K ) @ S ) ) ) ) ).
% ring_hom_ring.img_is_subfield(2)
thf(fact_910_ring__hom__ring_Oimg__is__subfield_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,H2: a > list_list_a,K: set_a] :
( ( ring_h2621993495375454400t_unit @ R @ S @ H2 )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ( one_li8234411390022467901t_unit @ S )
!= ( zero_l347298301471573063t_unit @ S ) )
=> ( subfie4546268998243038636t_unit @ ( image_a_list_list_a @ H2 @ K ) @ S ) ) ) ) ).
% ring_hom_ring.img_is_subfield(2)
thf(fact_911_ring__hom__ring_Oimg__is__subfield_I2_J,axiom,
! [R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H2: list_a > list_a,K: set_list_a] :
( ( ring_h1334922693953046990t_unit @ R @ S @ H2 )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( ( one_li8328186300101108157t_unit @ S )
!= ( zero_l4142658623432671053t_unit @ S ) )
=> ( subfie1779122896746047282t_unit @ ( image_list_a_list_a @ H2 @ K ) @ S ) ) ) ) ).
% ring_hom_ring.img_is_subfield(2)
thf(fact_912_ring__hom__ring_Oimg__is__subfield_I2_J,axiom,
! [R: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,H2: list_a > list_list_a,K: set_list_a] :
( ( ring_h7560679385066609224t_unit @ R @ S @ H2 )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( ( one_li8234411390022467901t_unit @ S )
!= ( zero_l347298301471573063t_unit @ S ) )
=> ( subfie4546268998243038636t_unit @ ( image_8260866953997875467list_a @ H2 @ K ) @ S ) ) ) ) ).
% ring_hom_ring.img_is_subfield(2)
thf(fact_913_ring__hom__ring_Oimg__is__subfield_I2_J,axiom,
! [R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,H2: list_list_a > a,K: set_list_list_a] :
( ( ring_h3841606220870141376it_a_b @ R @ S @ H2 )
=> ( ( subfie4546268998243038636t_unit @ K @ R )
=> ( ( ( one_a_ring_ext_a_b @ S )
!= ( zero_a_b @ S ) )
=> ( subfield_a_b @ ( image_list_list_a_a @ H2 @ K ) @ S ) ) ) ) ).
% ring_hom_ring.img_is_subfield(2)
thf(fact_914_ring__hom__ring_Oimg__is__subfield_I2_J,axiom,
! [R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,H2: list_list_a > list_a,K: set_list_list_a] :
( ( ring_h4589914651911841480t_unit @ R @ S @ H2 )
=> ( ( subfie4546268998243038636t_unit @ K @ R )
=> ( ( ( one_li8328186300101108157t_unit @ S )
!= ( zero_l4142658623432671053t_unit @ S ) )
=> ( subfie1779122896746047282t_unit @ ( image_4177985004758493951list_a @ H2 @ K ) @ S ) ) ) ) ).
% ring_hom_ring.img_is_subfield(2)
thf(fact_915_ring__hom__ring_Oimg__is__subfield_I2_J,axiom,
! [R: partia2956882679547061052t_unit,S: partia2956882679547061052t_unit,H2: list_list_a > list_list_a,K: set_list_list_a] :
( ( ring_h3119652981342401090t_unit @ R @ S @ H2 )
=> ( ( subfie4546268998243038636t_unit @ K @ R )
=> ( ( ( one_li8234411390022467901t_unit @ S )
!= ( zero_l347298301471573063t_unit @ S ) )
=> ( subfie4546268998243038636t_unit @ ( image_5446147394702115205list_a @ H2 @ K ) @ S ) ) ) ) ).
% ring_hom_ring.img_is_subfield(2)
thf(fact_916_ring__hom__ring_Oimg__is__subfield_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,K: set_a] :
( ( ring_h5357930050666032198t_unit @ R @ S @ H2 )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ( one_li8328186300101108157t_unit @ S )
!= ( zero_l4142658623432671053t_unit @ S ) )
=> ( subfie1779122896746047282t_unit @ ( image_a_list_a @ H2 @ K ) @ S ) ) ) ) ).
% ring_hom_ring.img_is_subfield(2)
thf(fact_917_ring__hom__ring_Oimg__is__subfield_I2_J,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,K: set_list_a] :
( ( ring_h7848885096329822662it_a_b @ R @ S @ H2 )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( ( one_a_ring_ext_a_b @ S )
!= ( zero_a_b @ S ) )
=> ( subfield_a_b @ ( image_list_a_a @ H2 @ K ) @ S ) ) ) ) ).
% ring_hom_ring.img_is_subfield(2)
thf(fact_918_ring__hom__ring_Oimg__is__subfield_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a,K: set_a] :
( ( ring_h4024360765257340990_b_a_b @ R @ S @ H2 )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ( one_a_ring_ext_a_b @ S )
!= ( zero_a_b @ S ) )
=> ( inj_on_a_a @ H2 @ K ) ) ) ) ).
% ring_hom_ring.img_is_subfield(1)
thf(fact_919_ring__hom__ring_Oimg__is__subfield_I1_J,axiom,
! [R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H2: list_a > list_a,K: set_list_a] :
( ( ring_h1334922693953046990t_unit @ R @ S @ H2 )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( ( one_li8328186300101108157t_unit @ S )
!= ( zero_l4142658623432671053t_unit @ S ) )
=> ( inj_on_list_a_list_a @ H2 @ K ) ) ) ) ).
% ring_hom_ring.img_is_subfield(1)
thf(fact_920_ring__hom__ring_Oimg__is__subfield_I1_J,axiom,
! [R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,H2: list_list_a > list_a,K: set_list_list_a] :
( ( ring_h4589914651911841480t_unit @ R @ S @ H2 )
=> ( ( subfie4546268998243038636t_unit @ K @ R )
=> ( ( ( one_li8328186300101108157t_unit @ S )
!= ( zero_l4142658623432671053t_unit @ S ) )
=> ( inj_on2455837796590618731list_a @ H2 @ K ) ) ) ) ).
% ring_hom_ring.img_is_subfield(1)
thf(fact_921_ring__hom__ring_Oimg__is__subfield_I1_J,axiom,
! [R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,H2: list_list_a > a,K: set_list_list_a] :
( ( ring_h3841606220870141376it_a_b @ R @ S @ H2 )
=> ( ( subfie4546268998243038636t_unit @ K @ R )
=> ( ( ( one_a_ring_ext_a_b @ S )
!= ( zero_a_b @ S ) )
=> ( inj_on_list_list_a_a @ H2 @ K ) ) ) ) ).
% ring_hom_ring.img_is_subfield(1)
thf(fact_922_ring__hom__ring_Oimg__is__subfield_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,K: set_a] :
( ( ring_h5357930050666032198t_unit @ R @ S @ H2 )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ( one_li8328186300101108157t_unit @ S )
!= ( zero_l4142658623432671053t_unit @ S ) )
=> ( inj_on_a_list_a @ H2 @ K ) ) ) ) ).
% ring_hom_ring.img_is_subfield(1)
thf(fact_923_ring__hom__ring_Oimg__is__subfield_I1_J,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,K: set_list_a] :
( ( ring_h7848885096329822662it_a_b @ R @ S @ H2 )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( ( one_a_ring_ext_a_b @ S )
!= ( zero_a_b @ S ) )
=> ( inj_on_list_a_a @ H2 @ K ) ) ) ) ).
% ring_hom_ring.img_is_subfield(1)
thf(fact_924_subdomainE_I4_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( H != bot_bot_set_list_a ) ) ).
% subdomainE(4)
thf(fact_925_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_926_ring_Ouniv__poly__subfield__of__consts,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( subfie4546268998243038636t_unit @ K @ R )
=> ( subfie2726367901829656742t_unit @ ( image_1156962946714028939list_a @ ( poly_o1617770581224298896t_unit @ R ) @ K ) @ ( univ_p2250591967980070728t_unit @ R @ K ) ) ) ) ).
% ring.univ_poly_subfield_of_consts
thf(fact_927_ring_Ouniv__poly__subfield__of__consts,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ R ) @ K ) @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% ring.univ_poly_subfield_of_consts
thf(fact_928_ring_Ouniv__poly__subfield__of__consts,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( subfie4546268998243038636t_unit @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ R ) @ K ) @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% ring.univ_poly_subfield_of_consts
thf(fact_929_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_930_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_931_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_932_subfieldE_I3_J,axiom,
! [K: set_set_int,R: partia4934656038542163276t_unit] :
( ( subfie3888952257595785920t_unit @ K @ R )
=> ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ R ) ) ) ).
% subfieldE(3)
thf(fact_933_int__embed__def,axiom,
( ring_int_embed_a_b
= ( ^ [R4: partia2175431115845679010xt_a_b,K4: int] : ( add_pow_a_b_int @ R4 @ K4 @ ( one_a_ring_ext_a_b @ R4 ) ) ) ) ).
% int_embed_def
thf(fact_934_int__embed__def,axiom,
( ring_i7845631192508594762t_unit
= ( ^ [R4: partia2670972154091845814t_unit,K4: int] : ( add_po2638046716968164713it_int @ R4 @ K4 @ ( one_li8328186300101108157t_unit @ R4 ) ) ) ) ).
% int_embed_def
thf(fact_935_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_936_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_937_ring_Oint__embed__add,axiom,
! [R: partia2175431115845679010xt_a_b,X: int,Y3: int] :
( ( ring_a_b @ R )
=> ( ( ring_int_embed_a_b @ R @ ( plus_plus_int @ X @ Y3 ) )
= ( add_a_b @ R @ ( ring_int_embed_a_b @ R @ X ) @ ( ring_int_embed_a_b @ R @ Y3 ) ) ) ) ).
% ring.int_embed_add
thf(fact_938_ring_Oint__embed__add,axiom,
! [R: partia2670972154091845814t_unit,X: int,Y3: int] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ring_i7845631192508594762t_unit @ R @ ( plus_plus_int @ X @ Y3 ) )
= ( add_li7652885771158616974t_unit @ R @ ( ring_i7845631192508594762t_unit @ R @ X ) @ ( ring_i7845631192508594762t_unit @ R @ Y3 ) ) ) ) ).
% ring.int_embed_add
thf(fact_939_inj__on__iff__surj,axiom,
! [A3: set_list_a,A5: set_int] :
( ( A3 != bot_bot_set_list_a )
=> ( ( ? [F3: list_a > int] :
( ( inj_on_list_a_int @ F3 @ A3 )
& ( ord_less_eq_set_int @ ( image_list_a_int @ F3 @ A3 ) @ A5 ) ) )
= ( ? [G3: int > list_a] :
( ( image_int_list_a @ G3 @ A5 )
= A3 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_940_inj__on__iff__surj,axiom,
! [A3: set_list_a,A5: set_list_list_a] :
( ( A3 != bot_bot_set_list_a )
=> ( ( ? [F3: list_a > list_list_a] :
( ( inj_on6538719745830000247list_a @ F3 @ A3 )
& ( ord_le8488217952732425610list_a @ ( image_8260866953997875467list_a @ F3 @ A3 ) @ A5 ) ) )
= ( ? [G3: list_list_a > list_a] :
( ( image_4177985004758493951list_a @ G3 @ A5 )
= A3 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_941_inj__on__iff__surj,axiom,
! [A3: set_a,A5: set_int] :
( ( A3 != bot_bot_set_a )
=> ( ( ? [F3: a > int] :
( ( inj_on_a_int @ F3 @ A3 )
& ( ord_less_eq_set_int @ ( image_a_int @ F3 @ A3 ) @ A5 ) ) )
= ( ? [G3: int > a] :
( ( image_int_a @ G3 @ A5 )
= A3 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_942_inj__on__iff__surj,axiom,
! [A3: set_int,A5: set_a] :
( ( A3 != bot_bot_set_int )
=> ( ( ? [F3: int > a] :
( ( inj_on_int_a @ F3 @ A3 )
& ( ord_less_eq_set_a @ ( image_int_a @ F3 @ A3 ) @ A5 ) ) )
= ( ? [G3: a > int] :
( ( image_a_int @ G3 @ A5 )
= A3 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_943_inj__on__iff__surj,axiom,
! [A3: set_list_a,A5: set_a] :
( ( A3 != bot_bot_set_list_a )
=> ( ( ? [F3: list_a > a] :
( ( inj_on_list_a_a @ F3 @ A3 )
& ( ord_less_eq_set_a @ ( image_list_a_a @ F3 @ A3 ) @ A5 ) ) )
= ( ? [G3: a > list_a] :
( ( image_a_list_a @ G3 @ A5 )
= A3 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_944_inj__on__iff__surj,axiom,
! [A3: set_a,A5: set_a] :
( ( A3 != bot_bot_set_a )
=> ( ( ? [F3: a > a] :
( ( inj_on_a_a @ F3 @ A3 )
& ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A3 ) @ A5 ) ) )
= ( ? [G3: a > a] :
( ( image_a_a @ G3 @ A5 )
= A3 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_945_inj__on__iff__surj,axiom,
! [A3: set_int,A5: set_list_a] :
( ( A3 != bot_bot_set_int )
=> ( ( ? [F3: int > list_a] :
( ( inj_on_int_list_a @ F3 @ A3 )
& ( ord_le8861187494160871172list_a @ ( image_int_list_a @ F3 @ A3 ) @ A5 ) ) )
= ( ? [G3: list_a > int] :
( ( image_list_a_int @ G3 @ A5 )
= A3 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_946_inj__on__iff__surj,axiom,
! [A3: set_list_a,A5: set_list_a] :
( ( A3 != bot_bot_set_list_a )
=> ( ( ? [F3: list_a > list_a] :
( ( inj_on_list_a_list_a @ F3 @ A3 )
& ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ F3 @ A3 ) @ A5 ) ) )
= ( ? [G3: list_a > list_a] :
( ( image_list_a_list_a @ G3 @ A5 )
= A3 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_947_inj__on__iff__surj,axiom,
! [A3: set_a,A5: set_list_a] :
( ( A3 != bot_bot_set_a )
=> ( ( ? [F3: a > list_a] :
( ( inj_on_a_list_a @ F3 @ A3 )
& ( ord_le8861187494160871172list_a @ ( image_a_list_a @ F3 @ A3 ) @ A5 ) ) )
= ( ? [G3: list_a > a] :
( ( image_list_a_a @ G3 @ A5 )
= A3 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_948_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_949_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_950_ring_Oup__one__closed,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( member_nat_list_a
@ ^ [N4: nat] : ( if_list_a @ ( N4 = zero_zero_nat ) @ ( one_li8328186300101108157t_unit @ R ) @ ( zero_l4142658623432671053t_unit @ R ) )
@ ( up_lis8464167429055313730t_unit @ R ) ) ) ).
% ring.up_one_closed
thf(fact_951_ring_Oup__one__closed,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_nat_a
@ ^ [N4: nat] : ( if_a @ ( N4 = zero_zero_nat ) @ ( one_a_ring_ext_a_b @ R ) @ ( zero_a_b @ R ) )
@ ( up_a_b @ R ) ) ) ).
% ring.up_one_closed
thf(fact_952_ring_Ofinite__carr__imp__char__ge__0,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ R ) ) ) ) ).
% ring.finite_carr_imp_char_ge_0
thf(fact_953_ring_Ofinite__carr__imp__char__ge__0,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c500279861223467766t_unit @ R ) ) ) ) ).
% ring.finite_carr_imp_char_ge_0
thf(fact_954_ring_Ofinite__carr__imp__char__ge__0,axiom,
! [R: partia2956882679547061052t_unit] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( finite1660835950917165235list_a @ ( partia2464479390973590831t_unit @ R ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c667363890163664752t_unit @ R ) ) ) ) ).
% ring.finite_carr_imp_char_ge_0
thf(fact_955_ring_Ofinite__carr__imp__char__ge__0,axiom,
! [R: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R )
=> ( ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ R ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c6147214092195050492t_unit @ R ) ) ) ) ).
% ring.finite_carr_imp_char_ge_0
thf(fact_956_domain_Ochar__ring__is__subfield,axiom,
! [R: partia2956882679547061052t_unit] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ ( ring_c667363890163664752t_unit @ R ) )
=> ( subfie4546268998243038636t_unit @ ( image_8546930381880718615list_a @ ( ring_i3137826187831172292t_unit @ R ) @ top_top_set_int ) @ R ) ) ) ).
% domain.char_ring_is_subfield
thf(fact_957_domain_Ochar__ring__is__subfield,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ R ) )
=> ( subfield_a_b @ ( image_int_a @ ( ring_int_embed_a_b @ R ) @ top_top_set_int ) @ R ) ) ) ).
% domain.char_ring_is_subfield
thf(fact_958_domain_Ochar__ring__is__subfield,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ ( ring_c500279861223467766t_unit @ R ) )
=> ( subfie1779122896746047282t_unit @ ( image_int_list_a @ ( ring_i7845631192508594762t_unit @ R ) @ top_top_set_int ) @ R ) ) ) ).
% domain.char_ring_is_subfield
thf(fact_959_p_Oorder__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_3240872107759947550t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.order_gt_0_iff_finite
thf(fact_960_d_Ouniv__poly__is__principal,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ k2 )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ k2 @ K ) ) ) ).
% d.univ_poly_is_principal
thf(fact_961_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_962_p_Ohom__imp__img__subring,axiom,
! [H2: list_a > a,S: partia2175431115845679010xt_a_b,K: set_list_a] :
( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ ( univ_poly_a_b @ r @ k ) @ S ) )
=> ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
@ ( one_up5219539153869309446xt_a_b
@ ^ [Uu: a] : ( H2 @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_list_a_a @ H2 @ K )
@ S ) ) ) ) ) ) ).
% p.hom_imp_img_subring
thf(fact_963_p_Ohom__imp__img__subring,axiom,
! [H2: list_a > list_a,S: partia2670972154091845814t_unit,K: set_list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ ( univ_poly_a_b @ r @ k ) @ S ) )
=> ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
@ ( one_up8845488168893825172t_unit
@ ^ [Uu: list_a] : ( H2 @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_list_a_list_a @ H2 @ K )
@ S ) ) ) ) ) ) ).
% p.hom_imp_img_subring
thf(fact_964_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_965_d_Ocarrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ k2 )
!= bot_bot_set_a ) ).
% d.carrier_not_empty
thf(fact_966_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_967_d_Ozero__not__one,axiom,
( ( zero_a_b @ k2 )
!= ( one_a_ring_ext_a_b @ k2 ) ) ).
% d.zero_not_one
thf(fact_968_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_969_subring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( K != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_970_d_Osubring__props_I3_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ k2 )
=> ( member_a @ ( one_a_ring_ext_a_b @ k2 ) @ K ) ) ).
% d.subring_props(3)
thf(fact_971_d_Osubring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ k2 )
=> ( K != bot_bot_set_a ) ) ).
% d.subring_props(4)
thf(fact_972_p_Osubring__props_I2_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ K ) ) ).
% p.subring_props(2)
thf(fact_973_p_Osubring__props_I7_J,axiom,
! [K: set_list_a,H1: list_a,H22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( member_list_a @ H1 @ K )
=> ( ( member_list_a @ H22 @ K )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ H1 @ H22 ) @ K ) ) ) ) ).
% p.subring_props(7)
thf(fact_974_p_Osubring__props_I4_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( K != bot_bot_set_list_a ) ) ).
% p.subring_props(4)
thf(fact_975_p_Osubring__props_I5_J,axiom,
! [K: set_list_a,H2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( member_list_a @ H2 @ K )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ H2 ) @ K ) ) ) ).
% p.subring_props(5)
thf(fact_976_p_Osubring__props_I3_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ K ) ) ).
% p.subring_props(3)
thf(fact_977_up__one__closed,axiom,
( member_nat_a
@ ^ [N4: nat] : ( if_a @ ( N4 = zero_zero_nat ) @ ( one_a_ring_ext_a_b @ r ) @ ( zero_a_b @ r ) )
@ ( up_a_b @ r ) ) ).
% up_one_closed
thf(fact_978_d_Oup__one__closed,axiom,
( member_nat_a
@ ^ [N4: nat] : ( if_a @ ( N4 = zero_zero_nat ) @ ( one_a_ring_ext_a_b @ k2 ) @ ( zero_a_b @ k2 ) )
@ ( up_a_b @ k2 ) ) ).
% d.up_one_closed
thf(fact_979_finite__carr__imp__char__ge__0,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) ) ) ).
% finite_carr_imp_char_ge_0
thf(fact_980_d_Ofinite__carr__imp__char__ge__0,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ k2 ) ) ) ).
% d.finite_carr_imp_char_ge_0
thf(fact_981_univ__poly__subfield__of__consts,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_subfield_of_consts
thf(fact_982_d_Ouniv__poly__subfield__of__consts,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ k2 )
=> ( subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ k2 ) @ K ) @ ( univ_poly_a_b @ k2 @ K ) ) ) ).
% d.univ_poly_subfield_of_consts
thf(fact_983_p_Osubring__props_I1_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.subring_props(1)
thf(fact_984_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_985_d_Oadd_Oone__in__subset,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( H != bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ H )
=> ( member_a @ ( a_inv_a_b @ k2 @ X3 ) @ H ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ H )
=> ! [Xa: a] :
( ( member_a @ Xa @ H )
=> ( member_a @ ( add_a_b @ k2 @ X3 @ Xa ) @ H ) ) )
=> ( member_a @ ( zero_a_b @ k2 ) @ H ) ) ) ) ) ).
% d.add.one_in_subset
thf(fact_986_p_Ouniv__poly__subfield__of__consts,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( subfie4546268998243038636t_unit @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) @ K ) @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ).
% p.univ_poly_subfield_of_consts
thf(fact_987_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_988_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_989_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_990_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_991_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_992_d_Oone__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ k2 ) @ ( partia707051561876973205xt_a_b @ k2 ) ).
% d.one_closed
thf(fact_993_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_994_d_Onat__pow__one,axiom,
! [N: nat] :
( ( pow_a_1026414303147256608_b_nat @ k2 @ ( one_a_ring_ext_a_b @ k2 ) @ N )
= ( one_a_ring_ext_a_b @ k2 ) ) ).
% d.nat_pow_one
thf(fact_995_h_Oimg__is__subfield_I2_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ k2 )
=> ( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% h.img_is_subfield(2)
thf(fact_996_hom__imp__img__subring,axiom,
! [H2: a > a,S: partia2175431115845679010xt_a_b,K: set_a] :
( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ r @ S ) )
=> ( ( subring_a_b @ K @ r )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ r ) )
@ ( one_up5219539153869309446xt_a_b
@ ^ [Uu: a] : ( H2 @ ( one_a_ring_ext_a_b @ r ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_a_a @ H2 @ K )
@ S ) ) ) ) ) ) ).
% hom_imp_img_subring
thf(fact_997_hom__imp__img__subring,axiom,
! [H2: a > list_a,S: partia2670972154091845814t_unit,K: set_a] :
( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ r @ S ) )
=> ( ( subring_a_b @ K @ r )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ r ) )
@ ( one_up8845488168893825172t_unit
@ ^ [Uu: list_a] : ( H2 @ ( one_a_ring_ext_a_b @ r ) )
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_a_list_a @ H2 @ K )
@ S ) ) ) ) ) ) ).
% hom_imp_img_subring
thf(fact_998_d_Ohom__imp__img__subring,axiom,
! [H2: a > a,S: partia2175431115845679010xt_a_b,K: set_a] :
( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ k2 @ S ) )
=> ( ( subring_a_b @ K @ k2 )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ k2 ) )
@ ( one_up5219539153869309446xt_a_b
@ ^ [Uu: a] : ( H2 @ ( one_a_ring_ext_a_b @ k2 ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_a_a @ H2 @ K )
@ S ) ) ) ) ) ) ).
% d.hom_imp_img_subring
thf(fact_999_d_Ohom__imp__img__subring,axiom,
! [H2: a > list_a,S: partia2670972154091845814t_unit,K: set_a] :
( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ k2 @ S ) )
=> ( ( subring_a_b @ K @ k2 )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ k2 ) )
@ ( one_up8845488168893825172t_unit
@ ^ [Uu: list_a] : ( H2 @ ( one_a_ring_ext_a_b @ k2 ) )
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_a_list_a @ H2 @ K )
@ S ) ) ) ) ) ) ).
% d.hom_imp_img_subring
thf(fact_1000_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1001_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_1002_d_Onat__pow__0,axiom,
! [X: a] :
( ( pow_a_1026414303147256608_b_nat @ k2 @ X @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ k2 ) ) ).
% d.nat_pow_0
thf(fact_1003_h_Ohom__one,axiom,
( ( poly_of_const_a_b @ r @ ( one_a_ring_ext_a_b @ k2 ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% h.hom_one
thf(fact_1004_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1005_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1006_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_1007_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1008_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1009_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1010_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1011_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1012_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1013_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1014_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
? [K4: nat] :
( N4
= ( plus_plus_nat @ M4 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1015_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1016_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1017_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1018_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1019_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K2 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1020_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_1021_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1022_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1023_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1024_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_1025_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1026_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K2: nat] :
( ! [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K2 ) @ ( F @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1027_fin__zfact,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% fin_zfact
thf(fact_1028_domain_Ouniv__poly__is__principal,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( subfie4546268998243038636t_unit @ K @ R )
=> ( ring_p8404492108403472412t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_1029_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_1030_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_1031_ring_Ohom__imp__img__subring,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > a,S: partia2175431115845679010xt_a_b,K: set_a] :
( ( ring_a_b @ R )
=> ( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( subring_a_b @ K @ R )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_a_b @ R ) )
@ ( one_up5219539153869309446xt_a_b
@ ^ [Uu: a] : ( H2 @ ( one_a_ring_ext_a_b @ R ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_a_a @ H2 @ K )
@ S ) ) ) ) ) ) ) ).
% ring.hom_imp_img_subring
thf(fact_1032_ring_Ohom__imp__img__subring,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > a,S: partia2175431115845679010xt_a_b,K: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ring_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H2 @ ( zero_l4142658623432671053t_unit @ R ) )
@ ( one_up5219539153869309446xt_a_b
@ ^ [Uu: a] : ( H2 @ ( one_li8328186300101108157t_unit @ R ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : ( image_list_a_a @ H2 @ K )
@ S ) ) ) ) ) ) ) ).
% ring.hom_imp_img_subring
thf(fact_1033_ring_Ohom__imp__img__subring,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > list_a,S: partia2670972154091845814t_unit,K: set_a] :
( ( ring_a_b @ R )
=> ( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( subring_a_b @ K @ R )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ R ) )
@ ( one_up8845488168893825172t_unit
@ ^ [Uu: list_a] : ( H2 @ ( one_a_ring_ext_a_b @ R ) )
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_a_list_a @ H2 @ K )
@ S ) ) ) ) ) ) ) ).
% ring.hom_imp_img_subring
thf(fact_1034_ring_Ohom__imp__img__subring,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > list_a,S: partia2670972154091845814t_unit,K: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ring_l6212528067271185461t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H2 @ ( zero_l4142658623432671053t_unit @ R ) )
@ ( one_up8845488168893825172t_unit
@ ^ [Uu: list_a] : ( H2 @ ( one_li8328186300101108157t_unit @ R ) )
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( image_list_a_list_a @ H2 @ K )
@ S ) ) ) ) ) ) ) ).
% ring.hom_imp_img_subring
thf(fact_1035_d_Oorder__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_a_ring_ext_a_b @ k2 ) )
= ( finite_finite_a @ ( partia707051561876973205xt_a_b @ k2 ) ) ) ).
% d.order_gt_0_iff_finite
thf(fact_1036_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_1037_surj__plus__right,axiom,
! [A: int] :
( ( image_int_int
@ ^ [B5: int] : ( plus_plus_int @ B5 @ A )
@ top_top_set_int )
= top_top_set_int ) ).
% surj_plus_right
thf(fact_1038_inj__on__id2,axiom,
! [A3: set_a] :
( inj_on_a_a
@ ^ [X2: a] : X2
@ A3 ) ).
% inj_on_id2
thf(fact_1039_inj__on__id2,axiom,
! [A3: set_list_a] :
( inj_on_list_a_list_a
@ ^ [X2: list_a] : X2
@ A3 ) ).
% inj_on_id2
thf(fact_1040_surjD,axiom,
! [F: int > int,Y3: int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ? [X3: int] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1041_surjD,axiom,
! [F: int > a,Y3: a] :
( ( ( image_int_a @ F @ top_top_set_int )
= top_top_set_a )
=> ? [X3: int] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1042_surjD,axiom,
! [F: int > list_a,Y3: list_a] :
( ( ( image_int_list_a @ F @ top_top_set_int )
= top_top_set_list_a )
=> ? [X3: int] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1043_surjD,axiom,
! [F: a > int,Y3: int] :
( ( ( image_a_int @ F @ top_top_set_a )
= top_top_set_int )
=> ? [X3: a] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1044_surjD,axiom,
! [F: a > a,Y3: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ? [X3: a] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1045_surjD,axiom,
! [F: a > list_a,Y3: list_a] :
( ( ( image_a_list_a @ F @ top_top_set_a )
= top_top_set_list_a )
=> ? [X3: a] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1046_surjD,axiom,
! [F: list_a > int,Y3: int] :
( ( ( image_list_a_int @ F @ top_top_set_list_a )
= top_top_set_int )
=> ? [X3: list_a] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1047_surjD,axiom,
! [F: list_a > a,Y3: a] :
( ( ( image_list_a_a @ F @ top_top_set_list_a )
= top_top_set_a )
=> ? [X3: list_a] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1048_surjD,axiom,
! [F: list_a > list_a,Y3: list_a] :
( ( ( image_list_a_list_a @ F @ top_top_set_list_a )
= top_top_set_list_a )
=> ? [X3: list_a] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_1049_surjE,axiom,
! [F: int > int,Y3: int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ~ ! [X3: int] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1050_surjE,axiom,
! [F: int > a,Y3: a] :
( ( ( image_int_a @ F @ top_top_set_int )
= top_top_set_a )
=> ~ ! [X3: int] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1051_surjE,axiom,
! [F: int > list_a,Y3: list_a] :
( ( ( image_int_list_a @ F @ top_top_set_int )
= top_top_set_list_a )
=> ~ ! [X3: int] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1052_surjE,axiom,
! [F: a > int,Y3: int] :
( ( ( image_a_int @ F @ top_top_set_a )
= top_top_set_int )
=> ~ ! [X3: a] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1053_surjE,axiom,
! [F: a > a,Y3: a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
=> ~ ! [X3: a] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1054_surjE,axiom,
! [F: a > list_a,Y3: list_a] :
( ( ( image_a_list_a @ F @ top_top_set_a )
= top_top_set_list_a )
=> ~ ! [X3: a] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1055_surjE,axiom,
! [F: list_a > int,Y3: int] :
( ( ( image_list_a_int @ F @ top_top_set_list_a )
= top_top_set_int )
=> ~ ! [X3: list_a] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1056_surjE,axiom,
! [F: list_a > a,Y3: a] :
( ( ( image_list_a_a @ F @ top_top_set_list_a )
= top_top_set_a )
=> ~ ! [X3: list_a] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1057_surjE,axiom,
! [F: list_a > list_a,Y3: list_a] :
( ( ( image_list_a_list_a @ F @ top_top_set_list_a )
= top_top_set_list_a )
=> ~ ! [X3: list_a] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_1058_surjI,axiom,
! [G: int > int,F: int > int] :
( ! [X3: int] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_int_int @ G @ top_top_set_int )
= top_top_set_int ) ) ).
% surjI
thf(fact_1059_surjI,axiom,
! [G: int > a,F: a > int] :
( ! [X3: a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_int_a @ G @ top_top_set_int )
= top_top_set_a ) ) ).
% surjI
thf(fact_1060_surjI,axiom,
! [G: int > list_a,F: list_a > int] :
( ! [X3: list_a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_int_list_a @ G @ top_top_set_int )
= top_top_set_list_a ) ) ).
% surjI
thf(fact_1061_surjI,axiom,
! [G: a > int,F: int > a] :
( ! [X3: int] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_a_int @ G @ top_top_set_a )
= top_top_set_int ) ) ).
% surjI
thf(fact_1062_surjI,axiom,
! [G: a > a,F: a > a] :
( ! [X3: a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_a_a @ G @ top_top_set_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_1063_surjI,axiom,
! [G: a > list_a,F: list_a > a] :
( ! [X3: list_a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_a_list_a @ G @ top_top_set_a )
= top_top_set_list_a ) ) ).
% surjI
thf(fact_1064_surjI,axiom,
! [G: list_a > int,F: int > list_a] :
( ! [X3: int] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_list_a_int @ G @ top_top_set_list_a )
= top_top_set_int ) ) ).
% surjI
thf(fact_1065_surjI,axiom,
! [G: list_a > a,F: a > list_a] :
( ! [X3: a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_list_a_a @ G @ top_top_set_list_a )
= top_top_set_a ) ) ).
% surjI
thf(fact_1066_surjI,axiom,
! [G: list_a > list_a,F: list_a > list_a] :
( ! [X3: list_a] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image_list_a_list_a @ G @ top_top_set_list_a )
= top_top_set_list_a ) ) ).
% surjI
thf(fact_1067_surj__def,axiom,
! [F: int > int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
= ( ! [Y5: int] :
? [X2: int] :
( Y5
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_1068_surj__def,axiom,
! [F: int > a] :
( ( ( image_int_a @ F @ top_top_set_int )
= top_top_set_a )
= ( ! [Y5: a] :
? [X2: int] :
( Y5
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_1069_surj__def,axiom,
! [F: int > list_a] :
( ( ( image_int_list_a @ F @ top_top_set_int )
= top_top_set_list_a )
= ( ! [Y5: list_a] :
? [X2: int] :
( Y5
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_1070_surj__def,axiom,
! [F: a > int] :
( ( ( image_a_int @ F @ top_top_set_a )
= top_top_set_int )
= ( ! [Y5: int] :
? [X2: a] :
( Y5
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_1071_surj__def,axiom,
! [F: a > a] :
( ( ( image_a_a @ F @ top_top_set_a )
= top_top_set_a )
= ( ! [Y5: a] :
? [X2: a] :
( Y5
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_1072_surj__def,axiom,
! [F: a > list_a] :
( ( ( image_a_list_a @ F @ top_top_set_a )
= top_top_set_list_a )
= ( ! [Y5: list_a] :
? [X2: a] :
( Y5
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_1073_surj__def,axiom,
! [F: list_a > int] :
( ( ( image_list_a_int @ F @ top_top_set_list_a )
= top_top_set_int )
= ( ! [Y5: int] :
? [X2: list_a] :
( Y5
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_1074_surj__def,axiom,
! [F: list_a > a] :
( ( ( image_list_a_a @ F @ top_top_set_list_a )
= top_top_set_a )
= ( ! [Y5: a] :
? [X2: list_a] :
( Y5
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_1075_surj__def,axiom,
! [F: list_a > list_a] :
( ( ( image_list_a_list_a @ F @ top_top_set_list_a )
= top_top_set_list_a )
= ( ! [Y5: list_a] :
? [X2: list_a] :
( Y5
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_1076_subset__inj__on,axiom,
! [F: a > list_a,B3: set_a,A3: set_a] :
( ( inj_on_a_list_a @ F @ B3 )
=> ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( inj_on_a_list_a @ F @ A3 ) ) ) ).
% subset_inj_on
thf(fact_1077_subset__inj__on,axiom,
! [F: a > a,B3: set_a,A3: set_a] :
( ( inj_on_a_a @ F @ B3 )
=> ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( inj_on_a_a @ F @ A3 ) ) ) ).
% subset_inj_on
thf(fact_1078_subset__inj__on,axiom,
! [F: list_a > list_a,B3: set_list_a,A3: set_list_a] :
( ( inj_on_list_a_list_a @ F @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ( inj_on_list_a_list_a @ F @ A3 ) ) ) ).
% subset_inj_on
thf(fact_1079_subset__inj__on,axiom,
! [F: list_a > list_list_a,B3: set_list_a,A3: set_list_a] :
( ( inj_on6538719745830000247list_a @ F @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ( inj_on6538719745830000247list_a @ F @ A3 ) ) ) ).
% subset_inj_on
thf(fact_1080_inj__on__subset,axiom,
! [F: a > list_a,A3: set_a,B3: set_a] :
( ( inj_on_a_list_a @ F @ A3 )
=> ( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( inj_on_a_list_a @ F @ B3 ) ) ) ).
% inj_on_subset
thf(fact_1081_inj__on__subset,axiom,
! [F: a > a,A3: set_a,B3: set_a] :
( ( inj_on_a_a @ F @ A3 )
=> ( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( inj_on_a_a @ F @ B3 ) ) ) ).
% inj_on_subset
thf(fact_1082_inj__on__subset,axiom,
! [F: list_a > list_a,A3: set_list_a,B3: set_list_a] :
( ( inj_on_list_a_list_a @ F @ A3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ A3 )
=> ( inj_on_list_a_list_a @ F @ B3 ) ) ) ).
% inj_on_subset
thf(fact_1083_inj__on__subset,axiom,
! [F: list_a > list_list_a,A3: set_list_a,B3: set_list_a] :
( ( inj_on6538719745830000247list_a @ F @ A3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ A3 )
=> ( inj_on6538719745830000247list_a @ F @ B3 ) ) ) ).
% inj_on_subset
thf(fact_1084_image__strict__mono,axiom,
! [F: int > list_a,B3: set_int,A3: set_int] :
( ( inj_on_int_list_a @ F @ B3 )
=> ( ( ord_less_set_int @ A3 @ B3 )
=> ( ord_less_set_list_a @ ( image_int_list_a @ F @ A3 ) @ ( image_int_list_a @ F @ B3 ) ) ) ) ).
% image_strict_mono
thf(fact_1085_image__strict__mono,axiom,
! [F: int > a,B3: set_int,A3: set_int] :
( ( inj_on_int_a @ F @ B3 )
=> ( ( ord_less_set_int @ A3 @ B3 )
=> ( ord_less_set_a @ ( image_int_a @ F @ A3 ) @ ( image_int_a @ F @ B3 ) ) ) ) ).
% image_strict_mono
thf(fact_1086_image__strict__mono,axiom,
! [F: a > list_a,B3: set_a,A3: set_a] :
( ( inj_on_a_list_a @ F @ B3 )
=> ( ( ord_less_set_a @ A3 @ B3 )
=> ( ord_less_set_list_a @ ( image_a_list_a @ F @ A3 ) @ ( image_a_list_a @ F @ B3 ) ) ) ) ).
% image_strict_mono
thf(fact_1087_image__strict__mono,axiom,
! [F: a > a,B3: set_a,A3: set_a] :
( ( inj_on_a_a @ F @ B3 )
=> ( ( ord_less_set_a @ A3 @ B3 )
=> ( ord_less_set_a @ ( image_a_a @ F @ A3 ) @ ( image_a_a @ F @ B3 ) ) ) ) ).
% image_strict_mono
thf(fact_1088_image__strict__mono,axiom,
! [F: list_a > list_a,B3: set_list_a,A3: set_list_a] :
( ( inj_on_list_a_list_a @ F @ B3 )
=> ( ( ord_less_set_list_a @ A3 @ B3 )
=> ( ord_less_set_list_a @ ( image_list_a_list_a @ F @ A3 ) @ ( image_list_a_list_a @ F @ B3 ) ) ) ) ).
% image_strict_mono
thf(fact_1089_image__strict__mono,axiom,
! [F: list_a > list_list_a,B3: set_list_a,A3: set_list_a] :
( ( inj_on6538719745830000247list_a @ F @ B3 )
=> ( ( ord_less_set_list_a @ A3 @ B3 )
=> ( ord_le5338140678153942166list_a @ ( image_8260866953997875467list_a @ F @ A3 ) @ ( image_8260866953997875467list_a @ F @ B3 ) ) ) ) ).
% image_strict_mono
thf(fact_1090_inj__on__image__iff,axiom,
! [A3: set_a,G: a > list_a,F: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ! [Xa: a] :
( ( member_a @ Xa @ A3 )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X3 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_a_a @ F @ A3 )
=> ( ( inj_on_a_list_a @ G @ ( image_a_a @ F @ A3 ) )
= ( inj_on_a_list_a @ G @ A3 ) ) ) ) ).
% inj_on_image_iff
thf(fact_1091_inj__on__image__iff,axiom,
! [A3: set_a,G: a > a,F: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ! [Xa: a] :
( ( member_a @ Xa @ A3 )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X3 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_a_a @ F @ A3 )
=> ( ( inj_on_a_a @ G @ ( image_a_a @ F @ A3 ) )
= ( inj_on_a_a @ G @ A3 ) ) ) ) ).
% inj_on_image_iff
thf(fact_1092_inj__on__image__iff,axiom,
! [A3: set_list_a,G: list_a > list_a,F: list_a > list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A3 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ A3 )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X3 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_list_a_list_a @ F @ A3 )
=> ( ( inj_on_list_a_list_a @ G @ ( image_list_a_list_a @ F @ A3 ) )
= ( inj_on_list_a_list_a @ G @ A3 ) ) ) ) ).
% inj_on_image_iff
thf(fact_1093_inj__on__image__iff,axiom,
! [A3: set_list_a,G: list_a > list_list_a,F: list_a > list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A3 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ A3 )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X3 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_list_a_list_a @ F @ A3 )
=> ( ( inj_on6538719745830000247list_a @ G @ ( image_list_a_list_a @ F @ A3 ) )
= ( inj_on6538719745830000247list_a @ G @ A3 ) ) ) ) ).
% inj_on_image_iff
thf(fact_1094_inj__on__add_H,axiom,
! [A: int,A3: set_int] :
( inj_on_int_int
@ ^ [B5: int] : ( plus_plus_int @ B5 @ A )
@ A3 ) ).
% inj_on_add'
thf(fact_1095_inj__on__add_H,axiom,
! [A: nat,A3: set_nat] :
( inj_on_nat_nat
@ ^ [B5: nat] : ( plus_plus_nat @ B5 @ A )
@ A3 ) ).
% inj_on_add'
thf(fact_1096_inj__on__image__eq__iff,axiom,
! [F: int > list_a,C2: set_int,A3: set_int,B3: set_int] :
( ( inj_on_int_list_a @ F @ C2 )
=> ( ( ord_less_eq_set_int @ A3 @ C2 )
=> ( ( ord_less_eq_set_int @ B3 @ C2 )
=> ( ( ( image_int_list_a @ F @ A3 )
= ( image_int_list_a @ F @ B3 ) )
= ( A3 = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1097_inj__on__image__eq__iff,axiom,
! [F: int > a,C2: set_int,A3: set_int,B3: set_int] :
( ( inj_on_int_a @ F @ C2 )
=> ( ( ord_less_eq_set_int @ A3 @ C2 )
=> ( ( ord_less_eq_set_int @ B3 @ C2 )
=> ( ( ( image_int_a @ F @ A3 )
= ( image_int_a @ F @ B3 ) )
= ( A3 = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1098_inj__on__image__eq__iff,axiom,
! [F: a > list_a,C2: set_a,A3: set_a,B3: set_a] :
( ( inj_on_a_list_a @ F @ C2 )
=> ( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ( ( image_a_list_a @ F @ A3 )
= ( image_a_list_a @ F @ B3 ) )
= ( A3 = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1099_inj__on__image__eq__iff,axiom,
! [F: a > a,C2: set_a,A3: set_a,B3: set_a] :
( ( inj_on_a_a @ F @ C2 )
=> ( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ( ( image_a_a @ F @ A3 )
= ( image_a_a @ F @ B3 ) )
= ( A3 = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1100_inj__on__image__eq__iff,axiom,
! [F: list_a > list_a,C2: set_list_a,A3: set_list_a,B3: set_list_a] :
( ( inj_on_list_a_list_a @ F @ C2 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ C2 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C2 )
=> ( ( ( image_list_a_list_a @ F @ A3 )
= ( image_list_a_list_a @ F @ B3 ) )
= ( A3 = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1101_inj__on__image__eq__iff,axiom,
! [F: list_a > list_list_a,C2: set_list_a,A3: set_list_a,B3: set_list_a] :
( ( inj_on6538719745830000247list_a @ F @ C2 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ C2 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C2 )
=> ( ( ( image_8260866953997875467list_a @ F @ A3 )
= ( image_8260866953997875467list_a @ F @ B3 ) )
= ( A3 = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1102_inj__on__image__mem__iff,axiom,
! [F: list_a > list_list_a,B3: set_list_a,A: list_a,A3: set_list_a] :
( ( inj_on6538719745830000247list_a @ F @ B3 )
=> ( ( member_list_a @ A @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ( ( member_list_list_a @ ( F @ A ) @ ( image_8260866953997875467list_a @ F @ A3 ) )
= ( member_list_a @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1103_inj__on__image__mem__iff,axiom,
! [F: list_a > set_int > a,B3: set_list_a,A: list_a,A3: set_list_a] :
( ( inj_on4987226016813664192_int_a @ F @ B3 )
=> ( ( member_list_a @ A @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ( ( member_set_int_a @ ( F @ A ) @ ( image_751163810753716308_int_a @ F @ A3 ) )
= ( member_list_a @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1104_inj__on__image__mem__iff,axiom,
! [F: list_a > nat > list_a,B3: set_list_a,A: list_a,A3: set_list_a] :
( ( inj_on4494935751713688664list_a @ F @ B3 )
=> ( ( member_list_a @ A @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ( ( member_nat_list_a @ ( F @ A ) @ ( image_5238573146875157228list_a @ F @ A3 ) )
= ( member_list_a @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1105_inj__on__image__mem__iff,axiom,
! [F: list_a > nat > a,B3: set_list_a,A: list_a,A3: set_list_a] :
( ( inj_on_list_a_nat_a @ F @ B3 )
=> ( ( member_list_a @ A @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ( ( member_nat_a @ ( F @ A ) @ ( image_list_a_nat_a @ F @ A3 ) )
= ( member_list_a @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1106_inj__on__image__mem__iff,axiom,
! [F: list_a > a > list_a,B3: set_list_a,A: list_a,A3: set_list_a] :
( ( inj_on7824993887354538302list_a @ F @ B3 )
=> ( ( member_list_a @ A @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ( ( member_a_list_a @ ( F @ A ) @ ( image_6252139687081310122list_a @ F @ A3 ) )
= ( member_list_a @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1107_inj__on__image__mem__iff,axiom,
! [F: list_a > a,B3: set_list_a,A: list_a,A3: set_list_a] :
( ( inj_on_list_a_a @ F @ B3 )
=> ( ( member_list_a @ A @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
=> ( ( member_a @ ( F @ A ) @ ( image_list_a_a @ F @ A3 ) )
= ( member_list_a @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1108_p_OsubringI,axiom,
! [H: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ H )
=> ( ! [H3: list_a] :
( ( member_list_a @ H3 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ H3 ) @ H ) )
=> ( ! [H12: list_a,H23: list_a] :
( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ H12 @ H23 ) @ H ) ) )
=> ( ! [H12: list_a,H23: list_a] :
( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ H12 @ H23 ) @ H ) ) )
=> ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ) ).
% p.subringI
thf(fact_1109_p_Om__lcomm,axiom,
! [X: list_a,Y3: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ Z ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Z ) ) ) ) ) ) ).
% p.m_lcomm
thf(fact_1110_p_Om__comm,axiom,
! [X: list_a,Y3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ X ) ) ) ) ).
% p.m_comm
thf(fact_1111_p_Om__assoc,axiom,
! [X: list_a,Y3: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 ) @ Z )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ Z ) ) ) ) ) ) ).
% p.m_assoc
thf(fact_1112_p_Osubring__props_I6_J,axiom,
! [K: set_list_a,H1: list_a,H22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( member_list_a @ H1 @ K )
=> ( ( member_list_a @ H22 @ K )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ H1 @ H22 ) @ K ) ) ) ) ).
% p.subring_props(6)
thf(fact_1113_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_1114_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_1115_p_Or__distr,axiom,
! [X: list_a,Y3: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Z @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Z @ Y3 ) ) ) ) ) ) ).
% p.r_distr
thf(fact_1116_p_Ol__distr,axiom,
! [X: list_a,Y3: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ Z ) ) ) ) ) ) ).
% p.l_distr
thf(fact_1117_p_Or__minus,axiom,
! [X: list_a,Y3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 ) ) ) ) ) ).
% p.r_minus
thf(fact_1118_p_Ol__minus,axiom,
! [X: list_a,Y3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ Y3 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 ) ) ) ) ) ).
% p.l_minus
thf(fact_1119_p_Opow__mult__distrib,axiom,
! [X: list_a,Y3: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ X ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 ) @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ Y3 @ N ) ) ) ) ) ) ).
% p.pow_mult_distrib
thf(fact_1120_p_Onat__pow__distrib,axiom,
! [X: list_a,Y3: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 ) @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ Y3 @ N ) ) ) ) ) ).
% p.nat_pow_distrib
thf(fact_1121_p_Onat__pow__comm,axiom,
! [X: list_a,N: nat,M: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ M ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ M ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) ) ) ) ).
% p.nat_pow_comm
thf(fact_1122_p_Ogroup__commutes__pow,axiom,
! [X: list_a,Y3: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ X ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) @ Y3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) ) ) ) ) ) ).
% p.group_commutes_pow
thf(fact_1123_p_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ U @ X3 )
= X3 ) )
=> ( U
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.one_unique
thf(fact_1124_p_Oinv__unique,axiom,
! [Y3: list_a,X: list_a,Y4: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y3 @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y4 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( Y3 = Y4 ) ) ) ) ) ) ).
% p.inv_unique
thf(fact_1125_p_Oadd__pow__rdistr__int,axiom,
! [A: list_a,B: list_a,K2: int] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ K2 @ B ) )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ K2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B ) ) ) ) ) ).
% p.add_pow_rdistr_int
thf(fact_1126_p_Oadd__pow__ldistr__int,axiom,
! [A: list_a,B: list_a,K2: int] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ K2 @ A ) @ B )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ k ) @ K2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B ) ) ) ) ) ).
% p.add_pow_ldistr_int
thf(fact_1127_p_Oline__extension__mem__iff,axiom,
! [U: list_a,K: set_list_a,A: list_a,E: set_list_a] :
( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ A @ E ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ K )
& ? [Y5: list_a] :
( ( member_list_a @ Y5 @ E )
& ( U
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ A ) @ Y5 ) ) ) ) ) ) ).
% p.line_extension_mem_iff
thf(fact_1128_p_OsubcringI,axiom,
! [H: set_list_a] :
( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ k ) )
=> ( ! [H12: list_a,H23: list_a] :
( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ H12 @ H23 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ H23 @ H12 ) ) ) )
=> ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.subcringI
thf(fact_1129_p_Oup__smult__closed,axiom,
! [A: list_a,P2: nat > list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_nat_list_a @ P2 @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_nat_list_a
@ ^ [I3: nat] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ ( P2 @ I3 ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.up_smult_closed
thf(fact_1130_p_Onat__pow__mult,axiom,
! [X: list_a,N: nat,M: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ M ) )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ ( plus_plus_nat @ N @ M ) ) ) ) ).
% p.nat_pow_mult
thf(fact_1131_p_Oline__extension__smult__closed,axiom,
! [K: set_list_a,E: set_list_a,A: list_a,K2: list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ! [K3: list_a,V2: list_a] :
( ( member_list_a @ K3 @ K )
=> ( ( member_list_a @ V2 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ K3 @ V2 ) @ E ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ K2 @ K )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ A @ E ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ K2 @ U ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ A @ E ) ) ) ) ) ) ) ) ).
% p.line_extension_smult_closed
thf(fact_1132_p_OsubdomainI,axiom,
! [H: set_list_a] :
( ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ k ) )
=> ( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ! [H12: list_a,H23: list_a] :
( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ H12 @ H23 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( H12
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
| ( H23
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) )
=> ( subdom7821232466298058046t_unit @ H @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.subdomainI
thf(fact_1133_p_Om__closed,axiom,
! [X: list_a,Y3: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).
% p.m_closed
thf(fact_1134_p_Or__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.r_null
thf(fact_1135_p_Ol__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.l_null
thf(fact_1136_p_Or__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
= X ) ) ).
% p.r_one
thf(fact_1137_p_Ol__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ X )
= X ) ) ).
% p.l_one
thf(fact_1138_p_Omonoid__cancelI,axiom,
( ! [A2: list_a,B2: list_a,C3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ C3 @ A2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ C3 @ B2 ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: list_a,B2: list_a,C3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A2 @ C3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ B2 @ C3 ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.monoid_cancelI
thf(fact_1139_m__assoc,axiom,
! [X: a,Y3: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( 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 @ Y3 ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y3 @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_1140_m__comm,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y3 )
= ( mult_a_ring_ext_a_b @ r @ Y3 @ X ) ) ) ) ).
% m_comm
thf(fact_1141_m__lcomm,axiom,
! [X: a,Y3: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( 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 @ Y3 @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y3 @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_1142_d_Om__lcomm,axiom,
! [X: a,Y3: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ X @ ( mult_a_ring_ext_a_b @ k2 @ Y3 @ Z ) )
= ( mult_a_ring_ext_a_b @ k2 @ Y3 @ ( mult_a_ring_ext_a_b @ k2 @ X @ Z ) ) ) ) ) ) ).
% d.m_lcomm
thf(fact_1143_d_Om__comm,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ X @ Y3 )
= ( mult_a_ring_ext_a_b @ k2 @ Y3 @ X ) ) ) ) ).
% d.m_comm
thf(fact_1144_d_Om__assoc,axiom,
! [X: a,Y3: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ ( mult_a_ring_ext_a_b @ k2 @ X @ Y3 ) @ Z )
= ( mult_a_ring_ext_a_b @ k2 @ X @ ( mult_a_ring_ext_a_b @ k2 @ Y3 @ Z ) ) ) ) ) ) ).
% d.m_assoc
thf(fact_1145_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_1146_d_Osubring__props_I6_J,axiom,
! [K: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K @ k2 )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H22 @ K )
=> ( member_a @ ( mult_a_ring_ext_a_b @ k2 @ H1 @ H22 ) @ K ) ) ) ) ).
% d.subring_props(6)
thf(fact_1147_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_1148_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_1149_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_1150_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_1151_l__distr,axiom,
! [X: a,Y3: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y3 ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y3 @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_1152_r__distr,axiom,
! [X: a,Y3: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y3 ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y3 ) ) ) ) ) ) ).
% r_distr
thf(fact_1153_d_Om__rcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ k2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ( mult_a_ring_ext_a_b @ k2 @ B @ A )
= ( mult_a_ring_ext_a_b @ k2 @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% d.m_rcancel
thf(fact_1154_d_Om__lcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ k2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ( mult_a_ring_ext_a_b @ k2 @ A @ B )
= ( mult_a_ring_ext_a_b @ k2 @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% d.m_lcancel
thf(fact_1155_d_Ointegral__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ( mult_a_ring_ext_a_b @ k2 @ A @ B )
= ( zero_a_b @ k2 ) )
= ( ( A
= ( zero_a_b @ k2 ) )
| ( B
= ( zero_a_b @ k2 ) ) ) ) ) ) ).
% d.integral_iff
thf(fact_1156_d_Ointegral,axiom,
! [A: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ k2 @ A @ B )
= ( zero_a_b @ k2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( A
= ( zero_a_b @ k2 ) )
| ( B
= ( zero_a_b @ k2 ) ) ) ) ) ) ).
% d.integral
thf(fact_1157_d_Or__distr,axiom,
! [X: a,Y3: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ Z @ ( add_a_b @ k2 @ X @ Y3 ) )
= ( add_a_b @ k2 @ ( mult_a_ring_ext_a_b @ k2 @ Z @ X ) @ ( mult_a_ring_ext_a_b @ k2 @ Z @ Y3 ) ) ) ) ) ) ).
% d.r_distr
thf(fact_1158_d_Ol__distr,axiom,
! [X: a,Y3: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ ( add_a_b @ k2 @ X @ Y3 ) @ Z )
= ( add_a_b @ k2 @ ( mult_a_ring_ext_a_b @ k2 @ X @ Z ) @ ( mult_a_ring_ext_a_b @ k2 @ Y3 @ Z ) ) ) ) ) ) ).
% d.l_distr
thf(fact_1159_inv__unique,axiom,
! [Y3: a,X: a,Y4: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y3 @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y4 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y3 = Y4 ) ) ) ) ) ) ).
% inv_unique
thf(fact_1160_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_1161_d_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ U @ X3 )
= X3 ) )
=> ( U
= ( one_a_ring_ext_a_b @ k2 ) ) ) ) ).
% d.one_unique
thf(fact_1162_d_Oinv__unique,axiom,
! [Y3: a,X: a,Y4: a] :
( ( ( mult_a_ring_ext_a_b @ k2 @ Y3 @ X )
= ( one_a_ring_ext_a_b @ k2 ) )
=> ( ( ( mult_a_ring_ext_a_b @ k2 @ X @ Y4 )
= ( one_a_ring_ext_a_b @ k2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( Y3 = Y4 ) ) ) ) ) ) ).
% d.inv_unique
thf(fact_1163_l__minus,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y3 )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y3 ) ) ) ) ) ).
% l_minus
thf(fact_1164_r__minus,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y3 ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y3 ) ) ) ) ) ).
% r_minus
thf(fact_1165_d_Or__minus,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ X @ ( a_inv_a_b @ k2 @ Y3 ) )
= ( a_inv_a_b @ k2 @ ( mult_a_ring_ext_a_b @ k2 @ X @ Y3 ) ) ) ) ) ).
% d.r_minus
thf(fact_1166_d_Ol__minus,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ ( a_inv_a_b @ k2 @ X ) @ Y3 )
= ( a_inv_a_b @ k2 @ ( mult_a_ring_ext_a_b @ k2 @ X @ Y3 ) ) ) ) ) ).
% d.l_minus
thf(fact_1167_mult__cong__l,axiom,
! [A: a,A6: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ A6 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A6 @ B ) ) ) ) ) ) ).
% mult_cong_l
thf(fact_1168_mult__cong__r,axiom,
! [B: a,B6: a,A: a] :
( ( associ5860276527279195403xt_a_b @ r @ B @ B6 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B6 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_1169_group__commutes__pow,axiom,
! [X: a,Y3: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y3 )
= ( mult_a_ring_ext_a_b @ r @ Y3 @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ Y3 )
= ( mult_a_ring_ext_a_b @ r @ Y3 @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_1170_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_1171_nat__pow__distrib,axiom,
! [X: a,Y3: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y3 ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y3 @ N ) ) ) ) ) ).
% nat_pow_distrib
thf(fact_1172_pow__mult__distrib,axiom,
! [X: a,Y3: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y3 )
= ( mult_a_ring_ext_a_b @ r @ Y3 @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y3 ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y3 @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_1173_d_Omult__cong__r,axiom,
! [B: a,B6: a,A: a] :
( ( associ5860276527279195403xt_a_b @ k2 @ B @ B6 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( associ5860276527279195403xt_a_b @ k2 @ ( mult_a_ring_ext_a_b @ k2 @ A @ B ) @ ( mult_a_ring_ext_a_b @ k2 @ A @ B6 ) ) ) ) ) ) ).
% d.mult_cong_r
thf(fact_1174_d_Omult__cong__l,axiom,
! [A: a,A6: a,B: a] :
( ( associ5860276527279195403xt_a_b @ k2 @ A @ A6 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( associ5860276527279195403xt_a_b @ k2 @ ( mult_a_ring_ext_a_b @ k2 @ A @ B ) @ ( mult_a_ring_ext_a_b @ k2 @ A6 @ B ) ) ) ) ) ) ).
% d.mult_cong_l
thf(fact_1175_d_Opow__mult__distrib,axiom,
! [X: a,Y3: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ k2 @ X @ Y3 )
= ( mult_a_ring_ext_a_b @ k2 @ Y3 @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( pow_a_1026414303147256608_b_nat @ k2 @ ( mult_a_ring_ext_a_b @ k2 @ X @ Y3 ) @ N )
= ( mult_a_ring_ext_a_b @ k2 @ ( pow_a_1026414303147256608_b_nat @ k2 @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ k2 @ Y3 @ N ) ) ) ) ) ) ).
% d.pow_mult_distrib
thf(fact_1176_d_Onat__pow__distrib,axiom,
! [X: a,Y3: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( pow_a_1026414303147256608_b_nat @ k2 @ ( mult_a_ring_ext_a_b @ k2 @ X @ Y3 ) @ N )
= ( mult_a_ring_ext_a_b @ k2 @ ( pow_a_1026414303147256608_b_nat @ k2 @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ k2 @ Y3 @ N ) ) ) ) ) ).
% d.nat_pow_distrib
thf(fact_1177_d_Onat__pow__comm,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ ( pow_a_1026414303147256608_b_nat @ k2 @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ k2 @ X @ M ) )
= ( mult_a_ring_ext_a_b @ k2 @ ( pow_a_1026414303147256608_b_nat @ k2 @ X @ M ) @ ( pow_a_1026414303147256608_b_nat @ k2 @ X @ N ) ) ) ) ).
% d.nat_pow_comm
thf(fact_1178_d_Ogroup__commutes__pow,axiom,
! [X: a,Y3: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ k2 @ X @ Y3 )
= ( mult_a_ring_ext_a_b @ k2 @ Y3 @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ ( pow_a_1026414303147256608_b_nat @ k2 @ X @ N ) @ Y3 )
= ( mult_a_ring_ext_a_b @ k2 @ Y3 @ ( pow_a_1026414303147256608_b_nat @ k2 @ X @ N ) ) ) ) ) ) ).
% d.group_commutes_pow
thf(fact_1179_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_1180_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_1181_d_Oadd__pow__rdistr__int,axiom,
! [A: a,B: a,K2: int] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ A @ ( add_pow_a_b_int @ k2 @ K2 @ B ) )
= ( add_pow_a_b_int @ k2 @ K2 @ ( mult_a_ring_ext_a_b @ k2 @ A @ B ) ) ) ) ) ).
% d.add_pow_rdistr_int
thf(fact_1182_d_Oadd__pow__ldistr__int,axiom,
! [A: a,B: a,K2: int] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ ( add_pow_a_b_int @ k2 @ K2 @ A ) @ B )
= ( add_pow_a_b_int @ k2 @ K2 @ ( mult_a_ring_ext_a_b @ k2 @ A @ B ) ) ) ) ) ).
% d.add_pow_ldistr_int
thf(fact_1183_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 )
& ? [Y5: a] :
( ( member_a @ Y5 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ A ) @ Y5 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_1184_d_Oline__extension__mem__iff,axiom,
! [U: a,K: set_a,A: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ k2 @ K @ A @ E ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ K )
& ? [Y5: a] :
( ( member_a @ Y5 @ E )
& ( U
= ( add_a_b @ k2 @ ( mult_a_ring_ext_a_b @ k2 @ X2 @ A ) @ Y5 ) ) ) ) ) ) ).
% d.line_extension_mem_iff
thf(fact_1185_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_1186_d_OsubcringI,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ k2 )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( ( mult_a_ring_ext_a_b @ k2 @ H12 @ H23 )
= ( mult_a_ring_ext_a_b @ k2 @ H23 @ H12 ) ) ) )
=> ( subcring_a_b @ H @ k2 ) ) ) ).
% d.subcringI
thf(fact_1187_up__smult__closed,axiom,
! [A: a,P2: nat > a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_nat_a @ P2 @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( mult_a_ring_ext_a_b @ r @ A @ ( P2 @ I3 ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_smult_closed
thf(fact_1188_d_Oup__smult__closed,axiom,
! [A: a,P2: nat > a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_nat_a @ P2 @ ( up_a_b @ k2 ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( mult_a_ring_ext_a_b @ k2 @ A @ ( P2 @ I3 ) )
@ ( up_a_b @ k2 ) ) ) ) ).
% d.up_smult_closed
thf(fact_1189_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_1190_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_1191_d_Osquare__eq__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( ( mult_a_ring_ext_a_b @ k2 @ X @ X )
= ( one_a_ring_ext_a_b @ k2 ) )
=> ( ( X
= ( one_a_ring_ext_a_b @ k2 ) )
| ( X
= ( a_inv_a_b @ k2 @ ( one_a_ring_ext_a_b @ k2 ) ) ) ) ) ) ).
% d.square_eq_one
thf(fact_1192_d_Onat__pow__mult,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ ( pow_a_1026414303147256608_b_nat @ k2 @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ k2 @ X @ M ) )
= ( pow_a_1026414303147256608_b_nat @ k2 @ X @ ( plus_plus_nat @ N @ M ) ) ) ) ).
% d.nat_pow_mult
thf(fact_1193_line__extension__smult__closed,axiom,
! [K: set_a,E: set_a,A: a,K2: a,U: a] :
( ( subfield_a_b @ K @ r )
=> ( ! [K3: a,V2: a] :
( ( member_a @ K3 @ K )
=> ( ( member_a @ V2 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ V2 ) @ 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_1194_d_Oline__extension__smult__closed,axiom,
! [K: set_a,E: set_a,A: a,K2: a,U: a] :
( ( subfield_a_b @ K @ k2 )
=> ( ! [K3: a,V2: a] :
( ( member_a @ K3 @ K )
=> ( ( member_a @ V2 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ k2 @ K3 @ V2 ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ k2 @ K @ A @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ k2 @ K2 @ U ) @ ( embedd971793762689825387on_a_b @ k2 @ K @ A @ E ) ) ) ) ) ) ) ) ).
% d.line_extension_smult_closed
thf(fact_1195_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_1196_d_OsubdomainI,axiom,
! [H: set_a] :
( ( subcring_a_b @ H @ k2 )
=> ( ( ( one_a_ring_ext_a_b @ k2 )
!= ( zero_a_b @ k2 ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ k2 @ H12 @ H23 )
= ( zero_a_b @ k2 ) )
=> ( ( H12
= ( zero_a_b @ k2 ) )
| ( H23
= ( zero_a_b @ k2 ) ) ) ) ) )
=> ( subdomain_a_b @ H @ k2 ) ) ) ) ).
% d.subdomainI
thf(fact_1197_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_1198_d_OsubringI,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ k2 ) @ H )
=> ( ! [H3: a] :
( ( member_a @ H3 @ H )
=> ( member_a @ ( a_inv_a_b @ k2 @ H3 ) @ H ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ k2 @ H12 @ H23 ) @ H ) ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( add_a_b @ k2 @ H12 @ H23 ) @ H ) ) )
=> ( subring_a_b @ H @ k2 ) ) ) ) ) ) ).
% d.subringI
thf(fact_1199_m__closed,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_1200_d_Om__closed,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ k2 @ X @ Y3 ) @ ( partia707051561876973205xt_a_b @ k2 ) ) ) ) ).
% d.m_closed
thf(fact_1201_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_1202_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_1203_d_Or__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ X @ ( zero_a_b @ k2 ) )
= ( zero_a_b @ k2 ) ) ) ).
% d.r_null
thf(fact_1204_d_Ol__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ ( zero_a_b @ k2 ) @ X )
= ( zero_a_b @ k2 ) ) ) ).
% d.l_null
thf(fact_1205_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_1206_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_1207_d_Ol__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ ( one_a_ring_ext_a_b @ k2 ) @ X )
= X ) ) ).
% d.l_one
thf(fact_1208_d_Or__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( mult_a_ring_ext_a_b @ k2 @ X @ ( one_a_ring_ext_a_b @ k2 ) )
= X ) ) ).
% d.r_one
thf(fact_1209_h_Ohom__mult,axiom,
! [X: a,Y3: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( poly_of_const_a_b @ r @ ( mult_a_ring_ext_a_b @ k2 @ X @ Y3 ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( poly_of_const_a_b @ r @ X ) @ ( poly_of_const_a_b @ r @ Y3 ) ) ) ) ) ).
% h.hom_mult
thf(fact_1210_p_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K: set_list_a,E: set_list_a,V: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E )
=> ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ k ) )
=> ( ( ord_le8861187494160871172list_a @ V @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ V ) ) ) ) ) ).
% p.subalbegra_incl_imp_finite_dimension
thf(fact_1211_p_Ogenideal__one,axiom,
( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) )
= ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.genideal_one
thf(fact_1212_p_Otelescopic__base__dim_I1_J,axiom,
! [K: set_list_a,F4: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( subfie1779122896746047282t_unit @ F4 @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ F4 )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ F4 @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E ) ) ) ) ) ).
% p.telescopic_base_dim(1)
thf(fact_1213_p_Ofinite__dimension__imp__subalgebra,axiom,
! [K: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E )
=> ( embedd1768981623711841426t_unit @ K @ E @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.finite_dimension_imp_subalgebra
thf(fact_1214_univ__poly__infinite__dimension,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ~ ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ K ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).
% univ_poly_infinite_dimension
thf(fact_1215_d_Ouniv__poly__infinite__dimension,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ k2 )
=> ~ ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ k2 @ K ) @ ( image_a_list_a @ ( poly_of_const_a_b @ k2 ) @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ k2 @ K ) ) ) ) ).
% d.univ_poly_infinite_dimension
thf(fact_1216_p_Ogenideal__self_H,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( member_list_a @ I @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ I @ bot_bot_set_list_a ) ) ) ) ).
% p.genideal_self'
thf(fact_1217_p_Ogenideal__zero,axiom,
( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) ).
% p.genideal_zero
thf(fact_1218_p_Ozeropideal,axiom,
princi8786919440553033881t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ k ) ).
% p.zeropideal
thf(fact_1219_p_Ocarrier__one__not__zero,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.carrier_one_not_zero
thf(fact_1220_p_Ocarrier__one__zero,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.carrier_one_zero
thf(fact_1221_p_Oone__zeroD,axiom,
( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) ) ).
% p.one_zeroD
thf(fact_1222_p_Oone__zeroI,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) )
=> ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.one_zeroI
thf(fact_1223_p_OIdl__subset__ideal_H,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) )
= ( member_list_a @ A @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ) ) ).
% p.Idl_subset_ideal'
thf(fact_1224_p_Odomain__eq__zeroprimeideal,axiom,
( ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ k ) )
= ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.domain_eq_zeroprimeideal
thf(fact_1225_p_Ozeroprimeideal__domainI,axiom,
( ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ k ) )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.zeroprimeideal_domainI
thf(fact_1226_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_1227_d_Ozeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ k2 ) @ bot_bot_set_a ) @ k2 ).
% d.zeropideal
thf(fact_1228_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_1229_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_1230_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_1231_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_1232_d_Oone__zeroI,axiom,
( ( ( partia707051561876973205xt_a_b @ k2 )
= ( insert_a @ ( zero_a_b @ k2 ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ k2 )
= ( zero_a_b @ k2 ) ) ) ).
% d.one_zeroI
thf(fact_1233_d_Oone__zeroD,axiom,
( ( ( one_a_ring_ext_a_b @ k2 )
= ( zero_a_b @ k2 ) )
=> ( ( partia707051561876973205xt_a_b @ k2 )
= ( insert_a @ ( zero_a_b @ k2 ) @ bot_bot_set_a ) ) ) ).
% d.one_zeroD
thf(fact_1234_d_Ocarrier__one__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ k2 )
= ( insert_a @ ( zero_a_b @ k2 ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ k2 )
= ( zero_a_b @ k2 ) ) ) ).
% d.carrier_one_zero
thf(fact_1235_d_Ocarrier__one__not__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ k2 )
!= ( insert_a @ ( zero_a_b @ k2 ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ k2 )
!= ( zero_a_b @ k2 ) ) ) ).
% d.carrier_one_not_zero
thf(fact_1236_p_Omaximalideal__prime,axiom,
! [I2: set_list_a] :
( ( maxima6585700282301356660t_unit @ I2 @ ( univ_poly_a_b @ r @ k ) )
=> ( primei6309817859076077608t_unit @ I2 @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.maximalideal_prime
thf(fact_1237_h_Oprimeideal__vimage,axiom,
! [P: set_list_a] :
( ( cring_a_b @ k2 )
=> ( ( primei6309817859076077608t_unit @ P @ ( univ_poly_a_b @ r @ k ) )
=> ( primeideal_a_b
@ ( collect_a
@ ^ [R5: a] :
( ( member_a @ R5 @ ( partia707051561876973205xt_a_b @ k2 ) )
& ( member_list_a @ ( poly_of_const_a_b @ r @ R5 ) @ P ) ) )
@ k2 ) ) ) ).
% h.primeideal_vimage
thf(fact_1238_zeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroprimeideal
thf(fact_1239_d_Ozeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ k2 ) @ bot_bot_set_a ) @ k2 ).
% d.zeroprimeideal
thf(fact_1240_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_1241_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_1242_d_Ozeroprimeideal__domainI,axiom,
( ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ k2 ) @ bot_bot_set_a ) @ k2 )
=> ( domain_a_b @ k2 ) ) ).
% d.zeroprimeideal_domainI
thf(fact_1243_d_Odomain__eq__zeroprimeideal,axiom,
( ( domain_a_b @ k2 )
= ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ k2 ) @ bot_bot_set_a ) @ k2 ) ) ).
% d.domain_eq_zeroprimeideal
thf(fact_1244_h_Oinfinite__dimension__hom,axiom,
! [K: set_a,E: set_a] :
( ( subfield_a_b @ K @ k2 )
=> ( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( inj_on_a_list_a @ ( poly_of_const_a_b @ r ) @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K @ E @ k2 )
=> ( ~ ( embedd8708762675212832759on_a_b @ k2 @ K @ E )
=> ~ ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ E ) ) ) ) ) ) ) ).
% h.infinite_dimension_hom
thf(fact_1245_h_Onon__trivial__field__hom__imp__inj,axiom,
( ( field_a_b @ k2 )
=> ( ( ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ ( partia707051561876973205xt_a_b @ k2 ) )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) )
=> ( inj_on_a_list_a @ ( poly_of_const_a_b @ r ) @ ( partia707051561876973205xt_a_b @ k2 ) ) ) ) ).
% h.non_trivial_field_hom_imp_inj
thf(fact_1246_telescopic__base__dim_I1_J,axiom,
! [K: set_a,F4: set_a,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( subfield_a_b @ F4 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ F4 )
=> ( ( embedd8708762675212832759on_a_b @ r @ F4 @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K @ E ) ) ) ) ) ).
% telescopic_base_dim(1)
thf(fact_1247_d_Otelescopic__base__dim_I1_J,axiom,
! [K: set_a,F4: set_a,E: set_a] :
( ( subfield_a_b @ K @ k2 )
=> ( ( subfield_a_b @ F4 @ k2 )
=> ( ( embedd8708762675212832759on_a_b @ k2 @ K @ F4 )
=> ( ( embedd8708762675212832759on_a_b @ k2 @ F4 @ E )
=> ( embedd8708762675212832759on_a_b @ k2 @ K @ E ) ) ) ) ) ).
% d.telescopic_base_dim(1)
thf(fact_1248_finite__dimension__imp__subalgebra,axiom,
! [K: set_a,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ( embedd9027525575939734154ra_a_b @ K @ E @ r ) ) ) ).
% finite_dimension_imp_subalgebra
thf(fact_1249_d_Ofinite__dimension__imp__subalgebra,axiom,
! [K: set_a,E: set_a] :
( ( subfield_a_b @ K @ k2 )
=> ( ( embedd8708762675212832759on_a_b @ k2 @ K @ E )
=> ( embedd9027525575939734154ra_a_b @ K @ E @ k2 ) ) ) ).
% d.finite_dimension_imp_subalgebra
thf(fact_1250_subfield__iff_I2_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( field_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) ) ) ).
% subfield_iff(2)
thf(fact_1251_d_Osubfield__iff_I2_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ k2 )
=> ( field_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ k2 ) ) ) ).
% d.subfield_iff(2)
thf(fact_1252_subalbegra__incl__imp__finite__dimension,axiom,
! [K: set_a,E: set_a,V: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ( ord_less_eq_set_a @ V @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K @ V ) ) ) ) ) ).
% subalbegra_incl_imp_finite_dimension
thf(fact_1253_d_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K: set_a,E: set_a,V: set_a] :
( ( subfield_a_b @ K @ k2 )
=> ( ( embedd8708762675212832759on_a_b @ k2 @ K @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V @ k2 )
=> ( ( ord_less_eq_set_a @ V @ E )
=> ( embedd8708762675212832759on_a_b @ k2 @ K @ V ) ) ) ) ) ).
% d.subalbegra_incl_imp_finite_dimension
thf(fact_1254_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A2
!= ( zero_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A2 @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_1255_d_Ocring__fieldI2,axiom,
( ( ( zero_a_b @ k2 )
!= ( one_a_ring_ext_a_b @ k2 ) )
=> ( ! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( ( A2
!= ( zero_a_b @ k2 ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ k2 ) )
& ( ( mult_a_ring_ext_a_b @ k2 @ A2 @ X4 )
= ( one_a_ring_ext_a_b @ k2 ) ) ) ) )
=> ( field_a_b @ k2 ) ) ) ).
% d.cring_fieldI2
thf(fact_1256_subfield__iff_I1_J,axiom,
! [K: set_a] :
( ( field_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( subfield_a_b @ K @ r ) ) ) ).
% subfield_iff(1)
thf(fact_1257_d_Osubfield__iff_I1_J,axiom,
! [K: set_a] :
( ( field_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ k2 ) )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ k2 ) )
=> ( subfield_a_b @ K @ k2 ) ) ) ).
% d.subfield_iff(1)
thf(fact_1258_d_Ozeromaximalideal__fieldI,axiom,
( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ k2 ) @ bot_bot_set_a ) @ k2 )
=> ( field_a_b @ k2 ) ) ).
% d.zeromaximalideal_fieldI
thf(fact_1259_d_Ozeromaximalideal__eq__field,axiom,
( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ k2 ) @ bot_bot_set_a ) @ k2 )
= ( field_a_b @ k2 ) ) ).
% d.zeromaximalideal_eq_field
thf(fact_1260_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_1261_d_Ouniv__poly__not__field,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ k2 )
=> ~ ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ k2 @ K ) ) ) ).
% d.univ_poly_not_field
thf(fact_1262_maximalideal__prime,axiom,
! [I2: set_a] :
( ( maximalideal_a_b @ I2 @ r )
=> ( primeideal_a_b @ I2 @ r ) ) ).
% maximalideal_prime
thf(fact_1263_d_Omaximalideal__prime,axiom,
! [I2: set_a] :
( ( maximalideal_a_b @ I2 @ k2 )
=> ( primeideal_a_b @ I2 @ k2 ) ) ).
% d.maximalideal_prime
thf(fact_1264_p_Osubfield__iff_I2_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( field_6388047844668329575t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.subfield_iff(2)
thf(fact_1265_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_1266_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_1267_p_Ocring__fieldI2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ! [A2: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( A2
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A2 @ X4 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.cring_fieldI2
thf(fact_1268_p_Osubfield__iff_I1_J,axiom,
! [K: set_list_a] :
( ( field_6388047844668329575t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) ) ) ) ).
% p.subfield_iff(1)
thf(fact_1269_p_Ozeromaximalideal__fieldI,axiom,
( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ k ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.zeromaximalideal_fieldI
thf(fact_1270_p_Ozeromaximalideal__eq__field,axiom,
( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ k ) )
= ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).
% p.zeromaximalideal_eq_field
thf(fact_1271_p_Osubfield__m__inv__simprule,axiom,
! [K: set_list_a,K2: list_a,A: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ K2 @ A ) @ K )
=> ( member_list_a @ A @ K ) ) ) ) ) ).
% p.subfield_m_inv_simprule
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y3: a] :
( ( if_a @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y3: a] :
( ( if_a @ $true @ X @ Y3 )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y3: list_a] :
( ( if_list_a @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y3: list_a] :
( ( if_list_a @ $true @ X @ Y3 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( poly_of_const_a_b @ r
@ ( ring_int_embed_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : k
@ r )
@ ( semiri1314217659103216013at_int @ n ) ) )
= ( poly_of_const_a_b @ r @ ( ring_int_embed_a_b @ r @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
%------------------------------------------------------------------------------