TPTP Problem File: SLH0648^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Universal_Hash_Families/0028_Field/prob_00218_007806__18339162_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1415 ( 473 unt; 139 typ; 0 def)
% Number of atoms : 3684 (1254 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 13452 ( 259 ~; 27 |; 145 &;11253 @)
% ( 0 <=>;1768 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 402 ( 402 >; 0 *; 0 +; 0 <<)
% Number of symbols : 129 ( 126 usr; 13 con; 0-4 aty)
% Number of variables : 3274 ( 82 ^;3131 !; 61 ?;3274 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:42:26.904
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
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__Nat__Onat_Mt__Group__Omonoid__Omonoid____ext_It__Nat__Onat_Mt__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J_J_J,type,
partia4692342223508353374t_unit: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
set_set_int_set_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J_J,type,
set_set_int_nat: $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__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $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_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__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (126)
thf(sy_c_AbelCoset_OA__RCOSETS_001t__Nat__Onat_001t__Product____Type__Ounit,type,
a_RCOS6328597822339572043t_unit: partia4692342223508353374t_unit > set_nat > set_set_nat ).
thf(sy_c_AbelCoset_Oa__l__coset_001t__Nat__Onat_001t__Product____Type__Ounit,type,
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thf(sy_c_AbelCoset_Oadditive__subgroup_001t__Nat__Onat_001t__Product____Type__Ounit,type,
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thf(sy_c_AbelCoset_Oset__add_001t__Nat__Onat_001t__Product____Type__Ounit,type,
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thf(sy_c_AbelCoset_Oset__add_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Nat__Onat_001t__Group__Omonoid__Omonoid____ext_It__Nat__Onat_Mt__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
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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,
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thf(sy_c_Divisibility_Omonoid__cancel_001t__Nat__Onat_001t__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Embedded__Algebras_Oring_Odimension_001t__Nat__Onat_001t__Product____Type__Ounit,type,
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thf(sy_c_Embedded__Algebras_Oring_Odimension_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Embedded__Algebras_Oring_Ofinite__dimension_001t__Nat__Onat_001t__Product____Type__Ounit,type,
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thf(sy_c_Embedded__Algebras_Oring_Ofinite__dimension_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001t__Nat__Onat_001t__Product____Type__Ounit,type,
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thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Embedded__Algebras_Osubalgebra_001t__Nat__Onat_001t__Product____Type__Ounit,type,
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thf(sy_c_Embedded__Algebras_Osubalgebra_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Field_Omod__ring,type,
mod_ring: nat > partia4692342223508353374t_unit ).
thf(sy_c_Field_Ozfact__iso__inv,type,
zfact_iso_inv: nat > set_int > nat ).
thf(sy_c_Group_Ogroup_001t__Nat__Onat_001t__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Group_Omonoid_Omult_001t__Nat__Onat_001t__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Group_Omonoid_Omult_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,
mult_s3864001451298473021t_unit: partia4934656038542163276t_unit > set_int > set_int > set_int ).
thf(sy_c_Group_Omonoid_Oone_001t__Nat__Onat_001t__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
minus_3247115583872269408et_int: set_nat_set_int > set_nat_set_int > set_nat_set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
minus_8897228262479074673et_int: set_set_int > set_set_int > set_set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
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_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
uminus4718767861138198480et_int: set_nat_set_int > set_nat_set_int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
uminus613421341184616069et_nat: set_set_nat > set_set_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_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
hilber3159750049796175616et_int: set_nat > ( nat > set_int ) > set_int > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Set__Oset_It__Int__Oint_J_001t__Nat__Onat,type,
hilber5958887739343024896nt_nat: set_set_int > ( set_int > nat ) > nat > set_int ).
thf(sy_c_Ideal_Ocgenideal_001t__Nat__Onat_001t__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Ideal_Ocgenideal_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,
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thf(sy_c_Ideal_Ogenideal_001t__Nat__Onat_001t__Product____Type__Ounit,type,
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thf(sy_c_Ideal_Ogenideal_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Ideal_Oprincipalideal_001t__Nat__Onat_001t__Product____Type__Ounit,type,
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thf(sy_c_Ideal_Oprincipalideal_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_IntRing_OZFact,type,
zFact: int > partia4934656038542163276t_unit ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
inf_inf_int: int > int > int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
inf_in1752217752563533465et_int: set_nat_set_int > set_nat_set_int > set_nat_set_int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
inf_inf_set_set_int: set_set_int > set_set_int > set_set_int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
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_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
bot_bo8417611410066262939et_int: set_nat_set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
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thf(sy_c_QuotRing_Oring__iso_001t__Nat__Onat_001t__Product____Type__Ounit_001t__Nat__Onat_001t__Product____Type__Ounit,type,
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add_nat_Product_unit: partia4692342223508353374t_unit > nat > nat > nat ).
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_Ozero_001t__Nat__Onat_001t__Product____Type__Ounit,type,
zero_n5149899317435570679t_unit: partia4692342223508353374t_unit > nat ).
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__hom_001t__Nat__Onat_001t__Product____Type__Ounit_001t__Nat__Onat_001t__Product____Type__Ounit,type,
ring_h4412161176302437050t_unit: partia4692342223508353374t_unit > partia4692342223508353374t_unit > set_nat_nat ).
thf(sy_c_Ring_Oring__hom_001t__Nat__Onat_001t__Product____Type__Ounit_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_h4752909569380436264t_unit: partia4692342223508353374t_unit > partia4934656038542163276t_unit > set_nat_set_int ).
thf(sy_c_Ring_Oring__hom_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Nat__Onat_001t__Product____Type__Ounit,type,
ring_h9051218956466089420t_unit: partia4934656038542163276t_unit > partia4692342223508353374t_unit > set_set_int_nat ).
thf(sy_c_Ring_Oring__hom_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_h3404898052528352314t_unit: partia4934656038542163276t_unit > partia4934656038542163276t_unit > set_set_int_set_int ).
thf(sy_c_Ring_Osemiring_001t__Nat__Onat_001t__Product____Type__Ounit,type,
semiri3921172975686117281t_unit: partia4692342223508353374t_unit > $o ).
thf(sy_c_Ring_Osemiring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
semiri8708897239777792527t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring__Characteristic_Ozfact__iso,type,
ring_zfact_iso: nat > nat > set_int ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
collect_nat_set_int: ( ( nat > set_int ) > $o ) > set_nat_set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Nat__Onat_J,type,
pow_set_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
insert_nat_set_int: ( nat > set_int ) > set_nat_set_int > set_nat_set_int ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Int__Oint_J,type,
insert_set_int: set_int > set_set_int > set_set_int ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Subrings_Osubcring_001t__Nat__Onat_001t__Product____Type__Ounit,type,
subcri1627753237249443161t_unit: set_nat > partia4692342223508353374t_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_Osubdomain_001t__Nat__Onat_001t__Product____Type__Ounit,type,
subdom2148668005855505734t_unit: set_nat > partia4692342223508353374t_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_Osubfield_001t__Nat__Onat_001t__Product____Type__Ounit,type,
subfie4892355163478727762t_unit: set_nat > partia4692342223508353374t_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_Osubring_001t__Nat__Onat_001t__Product____Type__Ounit,type,
subrin2893992908230074586t_unit: set_nat > partia4692342223508353374t_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_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_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__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J,type,
member_set_int_nat: ( set_int > nat ) > set_set_int_nat > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
member5205197933313416826et_int: ( set_int > set_int ) > set_set_int_set_int > $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_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_x,type,
x: nat ).
% Relevant facts (1271)
thf(fact_0_s_Oonepideal,axiom,
princi4652470909602072491t_unit @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) @ ( mod_ring @ n ) ).
% s.onepideal
thf(fact_1_calculation,axiom,
member_nat_set_int @ ( hilber5958887739343024896nt_nat @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zfact_iso_inv @ n ) ) @ ( ring_i1863809825068120638t_unit @ ( mod_ring @ n ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% calculation
thf(fact_2_s_Ocgenideal__self,axiom,
! [I: nat] :
( ( member_nat @ I @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ I @ ( cgenid8693976350862946099t_unit @ ( mod_ring @ n ) @ I ) ) ) ).
% s.cgenideal_self
thf(fact_3_s_Oring__iso__restrict,axiom,
! [F: nat > set_int,S: partia4934656038542163276t_unit,G: nat > set_int] :
( ( member_nat_set_int @ F @ ( ring_i1863809825068120638t_unit @ ( mod_ring @ n ) @ S ) )
=> ( ! [R: nat] :
( ( member_nat @ R @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( F @ R )
= ( G @ R ) ) )
=> ( member_nat_set_int @ G @ ( ring_i1863809825068120638t_unit @ ( mod_ring @ n ) @ S ) ) ) ) ).
% s.ring_iso_restrict
thf(fact_4_s_Osemiring__axioms,axiom,
semiri3921172975686117281t_unit @ ( mod_ring @ n ) ).
% s.semiring_axioms
thf(fact_5_s_Ocarrier__not__empty,axiom,
( ( partia3499330772048238685t_unit @ ( mod_ring @ n ) )
!= bot_bot_set_nat ) ).
% s.carrier_not_empty
thf(fact_6_r,axiom,
ring_s5316885176909347197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r
thf(fact_7_s_Oadd_Ol__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ C @ A )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ C @ B ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( A = B ) ) ) ) ) ).
% s.add.l_cancel
thf(fact_8_s_Oadd_Om__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ Z ) ) ) ) ) ) ).
% s.add.m_assoc
thf(fact_9_s_Oadd_Om__comm,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ X ) ) ) ) ).
% s.add.m_comm
thf(fact_10_s_Oadd_Om__lcomm,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ Z ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Z ) ) ) ) ) ) ).
% s.add.m_lcomm
thf(fact_11_s_Oadd_Or__cancel,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ A @ C )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ B @ C ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( A = B ) ) ) ) ) ).
% s.add.r_cancel
thf(fact_12_s_Oadd_Oright__cancel,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ X )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% s.add.right_cancel
thf(fact_13_s_Oadd_Om__closed,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.add.m_closed
thf(fact_14_ring__iso__set__sym,axiom,
! [R2: partia4692342223508353374t_unit,H: nat > set_int,S: partia4934656038542163276t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R2 @ S ) )
=> ( member_set_int_nat @ ( hilber3159750049796175616et_int @ ( partia3499330772048238685t_unit @ R2 ) @ H ) @ ( ring_i6162119212153773794t_unit @ S @ R2 ) ) ) ) ).
% ring_iso_set_sym
thf(fact_15_ring__iso__set__sym,axiom,
! [R2: partia4934656038542163276t_unit,H: set_int > nat,S: partia4692342223508353374t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ R2 @ S ) )
=> ( member_nat_set_int @ ( hilber5958887739343024896nt_nat @ ( partia966996272515721803t_unit @ R2 ) @ H ) @ ( ring_i1863809825068120638t_unit @ S @ R2 ) ) ) ) ).
% ring_iso_set_sym
thf(fact_16_s_Oadd_Oint__pow__mult__distrib,axiom,
! [X: nat,Y: nat,I: int] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ X ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ X ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ Y ) ) ) ) ) ) ).
% s.add.int_pow_mult_distrib
thf(fact_17_s_Oadd_Oint__pow__distrib,axiom,
! [X: nat,Y: nat,I: int] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ X ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ Y ) ) ) ) ) ).
% s.add.int_pow_distrib
thf(fact_18_s_Ominus__unique,axiom,
! [Y: nat,X: nat,Y2: nat] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y2 )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% s.minus_unique
thf(fact_19_s_Oadd_Or__inv__ex,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
& ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ X2 )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.add.r_inv_ex
thf(fact_20_s_Oadd_Oone__unique,axiom,
! [U: nat] :
( ( member_nat @ U @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.add.one_unique
thf(fact_21_s_Oadd_Ol__inv__ex,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
& ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X2 @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.add.l_inv_ex
thf(fact_22_s_Oadd_Oinv__comm,axiom,
! [X: nat,Y: nat] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ) ).
% s.add.inv_comm
thf(fact_23_s_Or__distr,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Z @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Z @ X ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Z @ Y ) ) ) ) ) ) ).
% s.r_distr
thf(fact_24_s_Ol__distr,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Z ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Y @ Z ) ) ) ) ) ) ).
% s.l_distr
thf(fact_25_ring_Oonepideal,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( princi4652470909602072491t_unit @ ( partia3499330772048238685t_unit @ R2 ) @ R2 ) ) ).
% ring.onepideal
thf(fact_26_ring_Oonepideal,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( princi8860937869964495385t_unit @ ( partia966996272515721803t_unit @ R2 ) @ R2 ) ) ).
% ring.onepideal
thf(fact_27_ring_Oring__iso__restrict,axiom,
! [R2: partia4692342223508353374t_unit,F: nat > set_int,S: partia4934656038542163276t_unit,G: nat > set_int] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat_set_int @ F @ ( ring_i1863809825068120638t_unit @ R2 @ S ) )
=> ( ! [R: nat] :
( ( member_nat @ R @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( F @ R )
= ( G @ R ) ) )
=> ( member_nat_set_int @ G @ ( ring_i1863809825068120638t_unit @ R2 @ S ) ) ) ) ) ).
% ring.ring_iso_restrict
thf(fact_28_s_Oring__axioms,axiom,
ring_n9194430563101542159t_unit @ ( mod_ring @ n ) ).
% s.ring_axioms
thf(fact_29_s_Om__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y ) @ Z )
= ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Y @ Z ) ) ) ) ) ) ).
% s.m_assoc
thf(fact_30_s_Oadd__pow__ldistr__int,axiom,
! [A: nat,B: nat,K: int] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ K @ A ) @ B )
= ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ K @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ B ) ) ) ) ) ).
% s.add_pow_ldistr_int
thf(fact_31_s_Oadd__pow__rdistr__int,axiom,
! [A: nat,B: nat,K: int] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ K @ B ) )
= ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ K @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ B ) ) ) ) ) ).
% s.add_pow_rdistr_int
thf(fact_32_s_Ozero__closed,axiom,
member_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ).
% s.zero_closed
thf(fact_33_s_Om__closed,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.m_closed
thf(fact_34_s_Oadd_Oint__pow__closed,axiom,
! [X: nat,I: int] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ X ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.add.int_pow_closed
thf(fact_35_s_Oadd_Oint__pow__one,axiom,
! [Z: int] :
( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ Z @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ).
% s.add.int_pow_one
thf(fact_36_s_Oadd_Ol__cancel__one,axiom,
! [X: nat,A: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ A )
= X )
= ( A
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ) ).
% s.add.l_cancel_one
thf(fact_37_s_Oadd_Ol__cancel__one_H,axiom,
! [X: nat,A: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( X
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ A ) )
= ( A
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ) ).
% s.add.l_cancel_one'
thf(fact_38_s_Oadd_Or__cancel__one,axiom,
! [X: nat,A: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ A @ X )
= X )
= ( A
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ) ).
% s.add.r_cancel_one
thf(fact_39_s_Oadd_Or__cancel__one_H,axiom,
! [X: nat,A: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( X
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ A @ X ) )
= ( A
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ) ).
% s.add.r_cancel_one'
thf(fact_40_s_Ol__zero,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ X )
= X ) ) ).
% s.l_zero
thf(fact_41_s_Or__zero,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
= X ) ) ).
% s.r_zero
thf(fact_42_s_Ol__null,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.l_null
thf(fact_43_s_Or__null,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.r_null
thf(fact_44_ring__iso__memE_I2_J,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( member_nat_nat @ H @ ( ring_i3628375019190340560t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) )
= ( mult_n6028127365542633569t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_45_ring__iso__memE_I2_J,axiom,
! [H: nat > set_int,R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,X: nat,Y: nat] :
( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) )
= ( mult_s3864001451298473021t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_46_ring__iso__memE_I2_J,axiom,
! [H: set_int > nat,R2: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,X: set_int,Y: set_int] :
( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ R2 @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( H @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Y ) )
= ( mult_n6028127365542633569t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_47_principalideal_Ois__principalideal,axiom,
! [I2: set_nat,R2: partia4692342223508353374t_unit] :
( ( princi4652470909602072491t_unit @ I2 @ R2 )
=> ( princi4652470909602072491t_unit @ I2 @ R2 ) ) ).
% principalideal.is_principalideal
thf(fact_48_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
! [A: nat > set_int,P: ( nat > set_int ) > $o] :
( ( member_nat_set_int @ A @ ( collect_nat_set_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_50_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A2: set_nat_set_int] :
( ( collect_nat_set_int
@ ^ [X3: nat > set_int] : ( member_nat_set_int @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_54_ring__iso__memE_I1_J,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit,X: nat] :
( ( member_nat_nat @ H @ ( ring_i3628375019190340560t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( H @ X ) @ ( partia3499330772048238685t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_55_ring__iso__memE_I1_J,axiom,
! [H: nat > set_int,R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,X: nat] :
( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_set_int @ ( H @ X ) @ ( partia966996272515721803t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_56_ring__iso__memE_I1_J,axiom,
! [H: set_int > nat,R2: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,X: set_int] :
( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ R2 @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_nat @ ( H @ X ) @ ( partia3499330772048238685t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_57_ring__iso__memE_I1_J,axiom,
! [H: set_int > set_int,R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,X: set_int] :
( ( member5205197933313416826et_int @ H @ ( ring_i7857697894474155344t_unit @ R2 @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( H @ X ) @ ( partia966996272515721803t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_58_ring_Ocgenideal__self,axiom,
! [R2: partia4692342223508353374t_unit,I: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ I @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ I @ ( cgenid8693976350862946099t_unit @ R2 @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_59_ring_Ocgenideal__self,axiom,
! [R2: partia4934656038542163276t_unit,I: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ I @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ I @ ( cgenid8502489213727343375t_unit @ R2 @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_60_ring__iso__memE_I3_J,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( member_nat_nat @ H @ ( ring_i3628375019190340560t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( add_nat_Product_unit @ R2 @ X @ Y ) )
= ( add_nat_Product_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_61_ring__iso__memE_I3_J,axiom,
! [H: nat > set_int,R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,X: nat,Y: nat] :
( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( add_nat_Product_unit @ R2 @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_62_ring__iso__memE_I3_J,axiom,
! [H: set_int > nat,R2: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,X: set_int,Y: set_int] :
( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ R2 @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( H @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) )
= ( add_nat_Product_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_63_s_Omonoid__cancelI,axiom,
( ! [A3: nat,B2: nat,C2: nat] :
( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ C2 @ A3 )
= ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ C2 @ B2 ) )
=> ( ( member_nat @ A3 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: nat,B2: nat,C2: nat] :
( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A3 @ C2 )
= ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ B2 @ C2 ) )
=> ( ( member_nat @ A3 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid2713784563552164055t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.monoid_cancelI
thf(fact_64_s_Ozeropideal,axiom,
princi4652470909602072491t_unit @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) @ ( mod_ring @ n ) ).
% s.zeropideal
thf(fact_65_s_Oline__extension__mem__iff,axiom,
! [U: nat,K2: set_nat,A: nat,E: set_nat] :
( ( member_nat @ U @ ( embedd838748496991043025t_unit @ ( mod_ring @ n ) @ K2 @ A @ E ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ K2 )
& ? [Y3: nat] :
( ( member_nat @ Y3 @ E )
& ( U
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X3 @ A ) @ Y3 ) ) ) ) ) ) ).
% s.line_extension_mem_iff
thf(fact_66_s_Oadd_Oint__pow__mult,axiom,
! [X: nat,I: int,J: int] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ ( plus_plus_int @ I @ J ) @ X )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ X ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ J @ X ) ) ) ) ).
% s.add.int_pow_mult
thf(fact_67_semiring_Or__distr,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ Z @ ( add_nat_Product_unit @ R2 @ X @ Y ) )
= ( add_nat_Product_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ Z @ X ) @ ( mult_n6028127365542633569t_unit @ R2 @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_68_semiring_Or__distr,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ Z @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ Z @ X ) @ ( mult_s3864001451298473021t_unit @ R2 @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_69_semiring_Ol__distr,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( add_nat_Product_unit @ R2 @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Z ) @ ( mult_n6028127365542633569t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_70_semiring_Ol__distr,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Z ) @ ( mult_s3864001451298473021t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_71_semiring_Or__null,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ X @ ( zero_n5149899317435570679t_unit @ R2 ) )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% semiring.r_null
thf(fact_72_semiring_Or__null,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ X @ ( zero_s6269048424454532197t_unit @ R2 ) )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% semiring.r_null
thf(fact_73_semiring_Ol__null,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( zero_n5149899317435570679t_unit @ R2 ) @ X )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% semiring.l_null
thf(fact_74_semiring_Ol__null,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( zero_s6269048424454532197t_unit @ R2 ) @ X )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% semiring.l_null
thf(fact_75_semiring_Osemiring__simprules_I6_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( zero_n5149899317435570679t_unit @ R2 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_76_semiring_Osemiring__simprules_I6_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( zero_s6269048424454532197t_unit @ R2 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_77_semiring_Osemiring__simprules_I11_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ ( zero_n5149899317435570679t_unit @ R2 ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_78_semiring_Osemiring__simprules_I11_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ ( zero_s6269048424454532197t_unit @ R2 ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_79_ring_Oadd__pow__rdistr__int,axiom,
! [R2: partia4692342223508353374t_unit,A: nat,B: nat,K: int] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ A @ ( add_po2420080144553951285it_int @ R2 @ K @ B ) )
= ( add_po2420080144553951285it_int @ R2 @ K @ ( mult_n6028127365542633569t_unit @ R2 @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_rdistr_int
thf(fact_80_ring_Oadd__pow__rdistr__int,axiom,
! [R2: partia4934656038542163276t_unit,A: set_int,B: set_int,K: int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ A @ ( add_po7581009264371422883it_int @ R2 @ K @ B ) )
= ( add_po7581009264371422883it_int @ R2 @ K @ ( mult_s3864001451298473021t_unit @ R2 @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_rdistr_int
thf(fact_81_ring_Ozeropideal,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( princi8860937869964495385t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) @ R2 ) ) ).
% ring.zeropideal
thf(fact_82_ring_Ozeropideal,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( princi4652470909602072491t_unit @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) @ R2 ) ) ).
% ring.zeropideal
thf(fact_83_ring__hom__closed,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit,X: nat] :
( ( member_nat_nat @ H @ ( ring_h4412161176302437050t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( H @ X ) @ ( partia3499330772048238685t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_84_ring__hom__closed,axiom,
! [H: nat > set_int,R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,X: nat] :
( ( member_nat_set_int @ H @ ( ring_h4752909569380436264t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_set_int @ ( H @ X ) @ ( partia966996272515721803t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_85_ring__hom__closed,axiom,
! [H: set_int > nat,R2: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,X: set_int] :
( ( member_set_int_nat @ H @ ( ring_h9051218956466089420t_unit @ R2 @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_nat @ ( H @ X ) @ ( partia3499330772048238685t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_86_ring__hom__closed,axiom,
! [H: set_int > set_int,R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,X: set_int] :
( ( member5205197933313416826et_int @ H @ ( ring_h3404898052528352314t_unit @ R2 @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( H @ X ) @ ( partia966996272515721803t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_87_ring_Oring__simprules_I2_J,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% ring.ring_simprules(2)
thf(fact_88_ring_Oring__simprules_I2_J,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% ring.ring_simprules(2)
thf(fact_89_ring_Oring__simprules_I22_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ ( add_nat_Product_unit @ R2 @ Y @ Z ) )
= ( add_nat_Product_unit @ R2 @ Y @ ( add_nat_Product_unit @ R2 @ X @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_90_ring_Oring__simprules_I22_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ ( add_se5859248395121729892t_unit @ R2 @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ R2 @ Y @ ( add_se5859248395121729892t_unit @ R2 @ X @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_91_ring_Oring__simprules_I10_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ Y )
= ( add_nat_Product_unit @ R2 @ Y @ X ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_92_ring_Oring__simprules_I10_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ Y )
= ( add_se5859248395121729892t_unit @ R2 @ Y @ X ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_93_ring_Oring__simprules_I7_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( add_nat_Product_unit @ R2 @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ R2 @ X @ ( add_nat_Product_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_94_ring_Oring__simprules_I7_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R2 @ X @ ( add_se5859248395121729892t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_95_ring_Oring__simprules_I1_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ X @ Y ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_96_ring_Oring__simprules_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_97_ring_Oring__simprules_I11_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) @ Z )
= ( mult_n6028127365542633569t_unit @ R2 @ X @ ( mult_n6028127365542633569t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_98_ring_Oring__simprules_I11_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ R2 @ X @ ( mult_s3864001451298473021t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_99_ring_Oring__simprules_I5_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_100_ring_Oring__simprules_I5_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Y ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_101_ring__hom__add,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( member_nat_nat @ H @ ( ring_h4412161176302437050t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( add_nat_Product_unit @ R2 @ X @ Y ) )
= ( add_nat_Product_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_102_ring__hom__add,axiom,
! [H: set_int > nat,R2: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,X: set_int,Y: set_int] :
( ( member_set_int_nat @ H @ ( ring_h9051218956466089420t_unit @ R2 @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( H @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) )
= ( add_nat_Product_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_103_ring__hom__mult,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( member_nat_nat @ H @ ( ring_h4412161176302437050t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) )
= ( mult_n6028127365542633569t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_104_ring__hom__mult,axiom,
! [H: set_int > nat,R2: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,X: set_int,Y: set_int] :
( ( member_set_int_nat @ H @ ( ring_h9051218956466089420t_unit @ R2 @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( H @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Y ) )
= ( mult_n6028127365542633569t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_105_ring__hom__zero,axiom,
! [H: set_int > set_int,R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit] :
( ( member5205197933313416826et_int @ H @ ( ring_h3404898052528352314t_unit @ R2 @ S ) )
=> ( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ring_s5316885176909347197t_unit @ S )
=> ( ( H @ ( zero_s6269048424454532197t_unit @ R2 ) )
= ( zero_s6269048424454532197t_unit @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_106_ring__hom__zero,axiom,
! [H: set_int > nat,R2: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit] :
( ( member_set_int_nat @ H @ ( ring_h9051218956466089420t_unit @ R2 @ S ) )
=> ( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ring_n9194430563101542159t_unit @ S )
=> ( ( H @ ( zero_s6269048424454532197t_unit @ R2 ) )
= ( zero_n5149899317435570679t_unit @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_107_ring__hom__zero,axiom,
! [H: nat > set_int,R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit] :
( ( member_nat_set_int @ H @ ( ring_h4752909569380436264t_unit @ R2 @ S ) )
=> ( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ring_s5316885176909347197t_unit @ S )
=> ( ( H @ ( zero_n5149899317435570679t_unit @ R2 ) )
= ( zero_s6269048424454532197t_unit @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_108_ring__hom__zero,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit] :
( ( member_nat_nat @ H @ ( ring_h4412161176302437050t_unit @ R2 @ S ) )
=> ( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ring_n9194430563101542159t_unit @ S )
=> ( ( H @ ( zero_n5149899317435570679t_unit @ R2 ) )
= ( zero_n5149899317435570679t_unit @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_109_semiring_Osemiring__simprules_I2_J,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_110_semiring_Osemiring__simprules_I2_J,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_111_semiring_Osemiring__simprules_I12_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ ( add_nat_Product_unit @ R2 @ Y @ Z ) )
= ( add_nat_Product_unit @ R2 @ Y @ ( add_nat_Product_unit @ R2 @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_112_semiring_Osemiring__simprules_I12_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ ( add_se5859248395121729892t_unit @ R2 @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ R2 @ Y @ ( add_se5859248395121729892t_unit @ R2 @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_113_semiring_Osemiring__simprules_I7_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ Y )
= ( add_nat_Product_unit @ R2 @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_114_semiring_Osemiring__simprules_I7_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ Y )
= ( add_se5859248395121729892t_unit @ R2 @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_115_semiring_Osemiring__simprules_I5_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( add_nat_Product_unit @ R2 @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ R2 @ X @ ( add_nat_Product_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_116_semiring_Osemiring__simprules_I5_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R2 @ X @ ( add_se5859248395121729892t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_117_semiring_Osemiring__simprules_I1_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ X @ Y ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_118_semiring_Osemiring__simprules_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_119_semiring_Osemiring__simprules_I8_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) @ Z )
= ( mult_n6028127365542633569t_unit @ R2 @ X @ ( mult_n6028127365542633569t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_120_semiring_Osemiring__simprules_I8_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ R2 @ X @ ( mult_s3864001451298473021t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_121_semiring_Osemiring__simprules_I3_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_122_semiring_Osemiring__simprules_I3_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Y ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_123_ring_Oring__simprules_I15_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ ( zero_n5149899317435570679t_unit @ R2 ) )
= X ) ) ) ).
% ring.ring_simprules(15)
thf(fact_124_ring_Oring__simprules_I15_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ ( zero_s6269048424454532197t_unit @ R2 ) )
= X ) ) ) ).
% ring.ring_simprules(15)
thf(fact_125_ring_Oring__simprules_I8_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( zero_n5149899317435570679t_unit @ R2 ) @ X )
= X ) ) ) ).
% ring.ring_simprules(8)
thf(fact_126_ring_Oring__simprules_I8_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( zero_s6269048424454532197t_unit @ R2 ) @ X )
= X ) ) ) ).
% ring.ring_simprules(8)
thf(fact_127_ring_Oring__simprules_I25_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ X @ ( zero_n5149899317435570679t_unit @ R2 ) )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_128_ring_Oring__simprules_I25_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ X @ ( zero_s6269048424454532197t_unit @ R2 ) )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_129_ring_Oring__simprules_I24_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( zero_n5149899317435570679t_unit @ R2 ) @ X )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_130_ring_Oring__simprules_I24_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( zero_s6269048424454532197t_unit @ R2 ) @ X )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_131_ring_Oring__simprules_I23_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ Z @ ( add_nat_Product_unit @ R2 @ X @ Y ) )
= ( add_nat_Product_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ Z @ X ) @ ( mult_n6028127365542633569t_unit @ R2 @ Z @ Y ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_132_ring_Oring__simprules_I23_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ Z @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ Z @ X ) @ ( mult_s3864001451298473021t_unit @ R2 @ Z @ Y ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_133_ring_Oring__simprules_I13_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( add_nat_Product_unit @ R2 @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Z ) @ ( mult_n6028127365542633569t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_134_ring_Oring__simprules_I13_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Z ) @ ( mult_s3864001451298473021t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_135_ring_Oadd__pow__ldistr__int,axiom,
! [R2: partia4692342223508353374t_unit,A: nat,B: nat,K: int] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( add_po2420080144553951285it_int @ R2 @ K @ A ) @ B )
= ( add_po2420080144553951285it_int @ R2 @ K @ ( mult_n6028127365542633569t_unit @ R2 @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_ldistr_int
thf(fact_136_ring_Oadd__pow__ldistr__int,axiom,
! [R2: partia4934656038542163276t_unit,A: set_int,B: set_int,K: int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( add_po7581009264371422883it_int @ R2 @ K @ A ) @ B )
= ( add_po7581009264371422883it_int @ R2 @ K @ ( mult_s3864001451298473021t_unit @ R2 @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_ldistr_int
thf(fact_137_s_Ogenideal__zero,axiom,
( ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) ).
% s.genideal_zero
thf(fact_138_s_Ogenideal__self_H,axiom,
! [I: nat] :
( ( member_nat @ I @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ I @ ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) ) ) ).
% s.genideal_self'
thf(fact_139_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_140_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_141_singletonI,axiom,
! [A: nat > set_int] : ( member_nat_set_int @ A @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ) ).
% singletonI
thf(fact_142_singletonI,axiom,
! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_143_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_144_s_Oone__zeroI,axiom,
( ( ( partia3499330772048238685t_unit @ ( mod_ring @ n ) )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) )
=> ( ( one_na902338870878123981t_unit @ ( mod_ring @ n ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.one_zeroI
thf(fact_145_s_Oone__zeroD,axiom,
( ( ( one_na902338870878123981t_unit @ ( mod_ring @ n ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
=> ( ( partia3499330772048238685t_unit @ ( mod_ring @ n ) )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) ) ).
% s.one_zeroD
thf(fact_146_s_Ocarrier__one__zero,axiom,
( ( ( partia3499330772048238685t_unit @ ( mod_ring @ n ) )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) )
= ( ( one_na902338870878123981t_unit @ ( mod_ring @ n ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.carrier_one_zero
thf(fact_147_s_Ocarrier__one__not__zero,axiom,
( ( ( partia3499330772048238685t_unit @ ( mod_ring @ n ) )
!= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) )
= ( ( one_na902338870878123981t_unit @ ( mod_ring @ n ) )
!= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.carrier_one_not_zero
thf(fact_148_ring_Oline__extension__mem__iff,axiom,
! [R2: partia4934656038542163276t_unit,U: set_int,K2: set_set_int,A: set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ U @ ( embedd4283282269743769663t_unit @ R2 @ K2 @ A @ E ) )
= ( ? [X3: set_int] :
( ( member_set_int @ X3 @ K2 )
& ? [Y3: set_int] :
( ( member_set_int @ Y3 @ E )
& ( U
= ( add_se5859248395121729892t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ X3 @ A ) @ Y3 ) ) ) ) ) ) ) ).
% ring.line_extension_mem_iff
thf(fact_149_ring_Oline__extension__mem__iff,axiom,
! [R2: partia4692342223508353374t_unit,U: nat,K2: set_nat,A: nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ U @ ( embedd838748496991043025t_unit @ R2 @ K2 @ A @ E ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ K2 )
& ? [Y3: nat] :
( ( member_nat @ Y3 @ E )
& ( U
= ( add_nat_Product_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ X3 @ A ) @ Y3 ) ) ) ) ) ) ) ).
% ring.line_extension_mem_iff
thf(fact_150_s_Oline__extension__in__carrier,axiom,
! [K2: set_nat,A: nat,E: set_nat] :
( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ E @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ord_less_eq_set_nat @ ( embedd838748496991043025t_unit @ ( mod_ring @ n ) @ K2 @ A @ E ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ) ).
% s.line_extension_in_carrier
thf(fact_151_subset__antisym,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_152_subset__antisym,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_153_subsetI,axiom,
! [A2: set_nat_set_int,B3: set_nat_set_int] :
( ! [X2: nat > set_int] :
( ( member_nat_set_int @ X2 @ A2 )
=> ( member_nat_set_int @ X2 @ B3 ) )
=> ( ord_le5995675665013768039et_int @ A2 @ B3 ) ) ).
% subsetI
thf(fact_154_subsetI,axiom,
! [A2: set_nat,B3: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B3 ) )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_155_subsetI,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ X2 @ B3 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_156_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_157_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_158_all__not__in__conv,axiom,
! [A2: set_nat_set_int] :
( ( ! [X3: nat > set_int] :
~ ( member_nat_set_int @ X3 @ A2 ) )
= ( A2 = bot_bo8417611410066262939et_int ) ) ).
% all_not_in_conv
thf(fact_159_all__not__in__conv,axiom,
! [A2: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_160_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_161_empty__iff,axiom,
! [C: nat > set_int] :
~ ( member_nat_set_int @ C @ bot_bo8417611410066262939et_int ) ).
% empty_iff
thf(fact_162_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_163_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_164_insert__absorb2,axiom,
! [X: nat,A2: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ X @ A2 ) )
= ( insert_nat @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_165_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_166_insert__iff,axiom,
! [A: nat > set_int,B: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_set_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_167_insert__iff,axiom,
! [A: set_nat,B: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_set_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_168_insertCI,axiom,
! [A: nat,B3: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B3 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B3 ) ) ) ).
% insertCI
thf(fact_169_insertCI,axiom,
! [A: nat > set_int,B3: set_nat_set_int,B: nat > set_int] :
( ( ~ ( member_nat_set_int @ A @ B3 )
=> ( A = B ) )
=> ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ B3 ) ) ) ).
% insertCI
thf(fact_170_insertCI,axiom,
! [A: set_nat,B3: set_set_nat,B: set_nat] :
( ( ~ ( member_set_nat @ A @ B3 )
=> ( A = B ) )
=> ( member_set_nat @ A @ ( insert_set_nat @ B @ B3 ) ) ) ).
% insertCI
thf(fact_171_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_172_s_Oinv__unique,axiom,
! [Y: nat,X: nat,Y2: nat] :
( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Y @ X )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y2 )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% s.inv_unique
thf(fact_173_s_Oone__unique,axiom,
! [U: nat] :
( ( member_nat @ U @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ U @ X2 )
= X2 ) )
=> ( U
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.one_unique
thf(fact_174_s_Ogenideal__self,axiom,
! [S: set_nat] :
( ( ord_less_eq_set_nat @ S @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ord_less_eq_set_nat @ S @ ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ S ) ) ) ).
% s.genideal_self
thf(fact_175_s_Osubset__Idl__subset,axiom,
! [I2: set_nat,H2: set_nat] :
( ( ord_less_eq_set_nat @ I2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ H2 @ I2 )
=> ( ord_less_eq_set_nat @ ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ H2 ) @ ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ I2 ) ) ) ) ).
% s.subset_Idl_subset
thf(fact_176_s_OIdl__subset__ideal_H,axiom,
! [A: nat,B: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( member_nat @ A @ ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ) ).
% s.Idl_subset_ideal'
thf(fact_177_s_Ogenideal__one,axiom,
( ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ ( insert_nat @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) )
= ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ).
% s.genideal_one
thf(fact_178_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_179_empty__subsetI,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_180_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_181_subset__empty,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% subset_empty
thf(fact_182_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_183_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_184_insert__subset,axiom,
! [X: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ ( insert_nat_set_int @ X @ A2 ) @ B3 )
= ( ( member_nat_set_int @ X @ B3 )
& ( ord_le5995675665013768039et_int @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_185_insert__subset,axiom,
! [X: nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B3 )
= ( ( member_nat @ X @ B3 )
& ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_186_insert__subset,axiom,
! [X: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A2 ) @ B3 )
= ( ( member_set_nat @ X @ B3 )
& ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_187_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_188_singleton__insert__inj__eq,axiom,
! [B: set_nat,A: set_nat,A2: set_set_nat] :
( ( ( insert_set_nat @ B @ bot_bot_set_set_nat )
= ( insert_set_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_189_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_190_singleton__insert__inj__eq_H,axiom,
! [A: set_nat,A2: set_set_nat,B: set_nat] :
( ( ( insert_set_nat @ A @ A2 )
= ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
= ( ( A = B )
& ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_191_s_Oone__closed,axiom,
member_nat @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ).
% s.one_closed
thf(fact_192_s_Ol__one,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) @ X )
= X ) ) ).
% s.l_one
thf(fact_193_s_Or__one,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
= X ) ) ).
% s.r_one
thf(fact_194_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_195_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_196_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z2: set_nat] : ( Y4 = Z2 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_197_set__eq__subset,axiom,
( ( ^ [Y4: set_set_nat,Z2: set_set_nat] : ( Y4 = Z2 ) )
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_198_subset__trans,axiom,
! [A2: set_nat,B3: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C3 )
=> ( ord_less_eq_set_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_199_subset__trans,axiom,
! [A2: set_set_nat,B3: set_set_nat,C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( ord_le6893508408891458716et_nat @ B3 @ C3 )
=> ( ord_le6893508408891458716et_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_200_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_201_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X2: set_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_202_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_203_subset__refl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_204_subset__iff,axiom,
( ord_le5995675665013768039et_int
= ( ^ [A4: set_nat_set_int,B4: set_nat_set_int] :
! [T: nat > set_int] :
( ( member_nat_set_int @ T @ A4 )
=> ( member_nat_set_int @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_205_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A4 )
=> ( member_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_206_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A4 )
=> ( member_set_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_207_equalityD2,axiom,
! [A2: set_nat,B3: set_nat] :
( ( A2 = B3 )
=> ( ord_less_eq_set_nat @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_208_equalityD2,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( A2 = B3 )
=> ( ord_le6893508408891458716et_nat @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_209_equalityD1,axiom,
! [A2: set_nat,B3: set_nat] :
( ( A2 = B3 )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_210_equalityD1,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( A2 = B3 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_211_subset__eq,axiom,
( ord_le5995675665013768039et_int
= ( ^ [A4: set_nat_set_int,B4: set_nat_set_int] :
! [X3: nat > set_int] :
( ( member_nat_set_int @ X3 @ A4 )
=> ( member_nat_set_int @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_212_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_213_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
=> ( member_set_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_214_equalityE,axiom,
! [A2: set_nat,B3: set_nat] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ~ ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_215_equalityE,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( A2 = B3 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ~ ( ord_le6893508408891458716et_nat @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_216_subsetD,axiom,
! [A2: set_nat_set_int,B3: set_nat_set_int,C: nat > set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ B3 )
=> ( ( member_nat_set_int @ C @ A2 )
=> ( member_nat_set_int @ C @ B3 ) ) ) ).
% subsetD
thf(fact_217_subsetD,axiom,
! [A2: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_218_subsetD,axiom,
! [A2: set_set_nat,B3: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_219_in__mono,axiom,
! [A2: set_nat_set_int,B3: set_nat_set_int,X: nat > set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ B3 )
=> ( ( member_nat_set_int @ X @ A2 )
=> ( member_nat_set_int @ X @ B3 ) ) ) ).
% in_mono
thf(fact_220_in__mono,axiom,
! [A2: set_nat,B3: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_221_in__mono,axiom,
! [A2: set_set_nat,B3: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_222_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_223_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_224_subset__insertI2,axiom,
! [A2: set_nat,B3: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_225_subset__insertI2,axiom,
! [A2: set_set_nat,B3: set_set_nat,B: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_226_subset__insertI,axiom,
! [B3: set_nat,A: nat] : ( ord_less_eq_set_nat @ B3 @ ( insert_nat @ A @ B3 ) ) ).
% subset_insertI
thf(fact_227_subset__insertI,axiom,
! [B3: set_set_nat,A: set_nat] : ( ord_le6893508408891458716et_nat @ B3 @ ( insert_set_nat @ A @ B3 ) ) ).
% subset_insertI
thf(fact_228_subset__insert,axiom,
! [X: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ~ ( member_nat_set_int @ X @ A2 )
=> ( ( ord_le5995675665013768039et_int @ A2 @ ( insert_nat_set_int @ X @ B3 ) )
= ( ord_le5995675665013768039et_int @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_229_subset__insert,axiom,
! [X: nat,A2: set_nat,B3: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B3 ) )
= ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_230_subset__insert,axiom,
! [X: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ~ ( member_set_nat @ X @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B3 ) )
= ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_231_insert__mono,axiom,
! [C3: set_nat,D: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C3 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C3 ) @ ( insert_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_232_insert__mono,axiom,
! [C3: set_set_nat,D: set_set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ D )
=> ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ A @ C3 ) @ ( insert_set_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_233_ring_Osubset__Idl__subset,axiom,
! [R2: partia4692342223508353374t_unit,I2: set_nat,H2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ord_less_eq_set_nat @ I2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( ord_less_eq_set_nat @ H2 @ I2 )
=> ( ord_less_eq_set_nat @ ( genide4496968333291595629t_unit @ R2 @ H2 ) @ ( genide4496968333291595629t_unit @ R2 @ I2 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_234_ring_Osubset__Idl__subset,axiom,
! [R2: partia4934656038542163276t_unit,I2: set_set_int,H2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( ord_le4403425263959731960et_int @ H2 @ I2 )
=> ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ R2 @ H2 ) @ ( genide1545711809618862555t_unit @ R2 @ I2 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_235_ring_Ogenideal__self,axiom,
! [R2: partia4692342223508353374t_unit,S: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ord_less_eq_set_nat @ S @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ord_less_eq_set_nat @ S @ ( genide4496968333291595629t_unit @ R2 @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_236_ring_Ogenideal__self,axiom,
! [R2: partia4934656038542163276t_unit,S: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ord_le4403425263959731960et_int @ S @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ord_le4403425263959731960et_int @ S @ ( genide1545711809618862555t_unit @ R2 @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_237_ring__hom__one,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit] :
( ( member_nat_nat @ H @ ( ring_h4412161176302437050t_unit @ R2 @ S ) )
=> ( ( H @ ( one_na902338870878123981t_unit @ R2 ) )
= ( one_na902338870878123981t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_238_ring__iso__memE_I4_J,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit] :
( ( member_nat_nat @ H @ ( ring_i3628375019190340560t_unit @ R2 @ S ) )
=> ( ( H @ ( one_na902338870878123981t_unit @ R2 ) )
= ( one_na902338870878123981t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_239_ring__iso__memE_I4_J,axiom,
! [H: nat > set_int,R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit] :
( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R2 @ S ) )
=> ( ( H @ ( one_na902338870878123981t_unit @ R2 ) )
= ( one_se8065767436706823081t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_240_subset__singletonD,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_241_subset__singletonD,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
=> ( ( A2 = bot_bot_set_set_nat )
| ( A2
= ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_242_subset__singleton__iff,axiom,
! [X4: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X4 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X4 = bot_bot_set_nat )
| ( X4
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_243_subset__singleton__iff,axiom,
! [X4: set_set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
= ( ( X4 = bot_bot_set_set_nat )
| ( X4
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_244_ring_Oring__simprules_I6_J,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ R2 ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% ring.ring_simprules(6)
thf(fact_245_ring_Oring__simprules_I6_J,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ R2 ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% ring.ring_simprules(6)
thf(fact_246_ring_Ogenideal__one,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( genide4496968333291595629t_unit @ R2 @ ( insert_nat @ ( one_na902338870878123981t_unit @ R2 ) @ bot_bot_set_nat ) )
= ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% ring.genideal_one
thf(fact_247_ring_Ogenideal__one,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ ( one_se8065767436706823081t_unit @ R2 ) @ bot_bot_set_set_int ) )
= ( partia966996272515721803t_unit @ R2 ) ) ) ).
% ring.genideal_one
thf(fact_248_semiring_Osemiring__simprules_I4_J,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ R2 ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_249_semiring_Osemiring__simprules_I4_J,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ R2 ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_250_ring_OIdl__subset__ideal_H,axiom,
! [R2: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( ord_less_eq_set_nat @ ( genide4496968333291595629t_unit @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( genide4496968333291595629t_unit @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( member_nat @ A @ ( genide4496968333291595629t_unit @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_251_ring_OIdl__subset__ideal_H,axiom,
! [R2: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) @ ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) )
= ( member_set_int @ A @ ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_252_ring_Oline__extension__in__carrier,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,A: set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ord_le4403425263959731960et_int @ K2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ord_le4403425263959731960et_int @ ( embedd4283282269743769663t_unit @ R2 @ K2 @ A @ E ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_253_ring_Oline__extension__in__carrier,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,A: nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( ord_less_eq_set_nat @ E @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ord_less_eq_set_nat @ ( embedd838748496991043025t_unit @ R2 @ K2 @ A @ E ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_254_ring_Oring__simprules_I12_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( one_na902338870878123981t_unit @ R2 ) @ X )
= X ) ) ) ).
% ring.ring_simprules(12)
thf(fact_255_ring_Oring__simprules_I12_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( one_se8065767436706823081t_unit @ R2 ) @ X )
= X ) ) ) ).
% ring.ring_simprules(12)
thf(fact_256_semiring_Osemiring__simprules_I9_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( one_na902338870878123981t_unit @ R2 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_257_semiring_Osemiring__simprules_I9_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( one_se8065767436706823081t_unit @ R2 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_258_ex__in__conv,axiom,
! [A2: set_nat_set_int] :
( ( ? [X3: nat > set_int] : ( member_nat_set_int @ X3 @ A2 ) )
= ( A2 != bot_bo8417611410066262939et_int ) ) ).
% ex_in_conv
thf(fact_259_ex__in__conv,axiom,
! [A2: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_260_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_261_equals0I,axiom,
! [A2: set_nat_set_int] :
( ! [Y5: nat > set_int] :
~ ( member_nat_set_int @ Y5 @ A2 )
=> ( A2 = bot_bo8417611410066262939et_int ) ) ).
% equals0I
thf(fact_262_equals0I,axiom,
! [A2: set_set_nat] :
( ! [Y5: set_nat] :
~ ( member_set_nat @ Y5 @ A2 )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_263_equals0I,axiom,
! [A2: set_nat] :
( ! [Y5: nat] :
~ ( member_nat @ Y5 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_264_equals0D,axiom,
! [A2: set_nat_set_int,A: nat > set_int] :
( ( A2 = bot_bo8417611410066262939et_int )
=> ~ ( member_nat_set_int @ A @ A2 ) ) ).
% equals0D
thf(fact_265_equals0D,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( A2 = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_266_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_267_emptyE,axiom,
! [A: nat > set_int] :
~ ( member_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ).
% emptyE
thf(fact_268_emptyE,axiom,
! [A: set_nat] :
~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_269_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_270_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B5: set_nat] :
( ( A2
= ( insert_nat @ A @ B5 ) )
& ~ ( member_nat @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_271_mk__disjoint__insert,axiom,
! [A: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ A2 )
=> ? [B5: set_nat_set_int] :
( ( A2
= ( insert_nat_set_int @ A @ B5 ) )
& ~ ( member_nat_set_int @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_272_mk__disjoint__insert,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ? [B5: set_set_nat] :
( ( A2
= ( insert_set_nat @ A @ B5 ) )
& ~ ( member_set_nat @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_273_insert__commute,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ Y @ A2 ) )
= ( insert_nat @ Y @ ( insert_nat @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_274_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B3: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B3 )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_nat] :
( ( A2
= ( insert_nat @ B @ C4 ) )
& ~ ( member_nat @ B @ C4 )
& ( B3
= ( insert_nat @ A @ C4 ) )
& ~ ( member_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_275_insert__eq__iff,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B: nat > set_int,B3: set_nat_set_int] :
( ~ ( member_nat_set_int @ A @ A2 )
=> ( ~ ( member_nat_set_int @ B @ B3 )
=> ( ( ( insert_nat_set_int @ A @ A2 )
= ( insert_nat_set_int @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_nat_set_int] :
( ( A2
= ( insert_nat_set_int @ B @ C4 ) )
& ~ ( member_nat_set_int @ B @ C4 )
& ( B3
= ( insert_nat_set_int @ A @ C4 ) )
& ~ ( member_nat_set_int @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_276_insert__eq__iff,axiom,
! [A: set_nat,A2: set_set_nat,B: set_nat,B3: set_set_nat] :
( ~ ( member_set_nat @ A @ A2 )
=> ( ~ ( member_set_nat @ B @ B3 )
=> ( ( ( insert_set_nat @ A @ A2 )
= ( insert_set_nat @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_set_nat] :
( ( A2
= ( insert_set_nat @ B @ C4 ) )
& ~ ( member_set_nat @ B @ C4 )
& ( B3
= ( insert_set_nat @ A @ C4 ) )
& ~ ( member_set_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_277_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_278_insert__absorb,axiom,
! [A: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ A2 )
=> ( ( insert_nat_set_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_279_insert__absorb,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( insert_set_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_280_insert__ident,axiom,
! [X: nat,A2: set_nat,B3: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B3 )
=> ( ( ( insert_nat @ X @ A2 )
= ( insert_nat @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_281_insert__ident,axiom,
! [X: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ~ ( member_nat_set_int @ X @ A2 )
=> ( ~ ( member_nat_set_int @ X @ B3 )
=> ( ( ( insert_nat_set_int @ X @ A2 )
= ( insert_nat_set_int @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_282_insert__ident,axiom,
! [X: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ~ ( member_set_nat @ X @ A2 )
=> ( ~ ( member_set_nat @ X @ B3 )
=> ( ( ( insert_set_nat @ X @ A2 )
= ( insert_set_nat @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_283_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B5: set_nat] :
( ( A2
= ( insert_nat @ X @ B5 ) )
=> ( member_nat @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_284_Set_Oset__insert,axiom,
! [X: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ X @ A2 )
=> ~ ! [B5: set_nat_set_int] :
( ( A2
= ( insert_nat_set_int @ X @ B5 ) )
=> ( member_nat_set_int @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_285_Set_Oset__insert,axiom,
! [X: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X @ A2 )
=> ~ ! [B5: set_set_nat] :
( ( A2
= ( insert_set_nat @ X @ B5 ) )
=> ( member_set_nat @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_286_insertI2,axiom,
! [A: nat,B3: set_nat,B: nat] :
( ( member_nat @ A @ B3 )
=> ( member_nat @ A @ ( insert_nat @ B @ B3 ) ) ) ).
% insertI2
thf(fact_287_insertI2,axiom,
! [A: nat > set_int,B3: set_nat_set_int,B: nat > set_int] :
( ( member_nat_set_int @ A @ B3 )
=> ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ B3 ) ) ) ).
% insertI2
thf(fact_288_insertI2,axiom,
! [A: set_nat,B3: set_set_nat,B: set_nat] :
( ( member_set_nat @ A @ B3 )
=> ( member_set_nat @ A @ ( insert_set_nat @ B @ B3 ) ) ) ).
% insertI2
thf(fact_289_insertI1,axiom,
! [A: nat,B3: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B3 ) ) ).
% insertI1
thf(fact_290_insertI1,axiom,
! [A: nat > set_int,B3: set_nat_set_int] : ( member_nat_set_int @ A @ ( insert_nat_set_int @ A @ B3 ) ) ).
% insertI1
thf(fact_291_insertI1,axiom,
! [A: set_nat,B3: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B3 ) ) ).
% insertI1
thf(fact_292_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_293_insertE,axiom,
! [A: nat > set_int,B: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat_set_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_294_insertE,axiom,
! [A: set_nat,B: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_295_ring_Oline__extension_Ocong,axiom,
embedd838748496991043025t_unit = embedd838748496991043025t_unit ).
% ring.line_extension.cong
thf(fact_296_ring__hom__memI,axiom,
! [R2: partia4692342223508353374t_unit,H: nat > nat,S: partia4692342223508353374t_unit] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( H @ X2 ) @ ( partia3499330772048238685t_unit @ S ) ) )
=> ( ! [X2: nat,Y5: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y5 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( mult_n6028127365542633569t_unit @ R2 @ X2 @ Y5 ) )
= ( mult_n6028127365542633569t_unit @ S @ ( H @ X2 ) @ ( H @ Y5 ) ) ) ) )
=> ( ! [X2: nat,Y5: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y5 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( add_nat_Product_unit @ R2 @ X2 @ Y5 ) )
= ( add_nat_Product_unit @ S @ ( H @ X2 ) @ ( H @ Y5 ) ) ) ) )
=> ( ( ( H @ ( one_na902338870878123981t_unit @ R2 ) )
= ( one_na902338870878123981t_unit @ S ) )
=> ( member_nat_nat @ H @ ( ring_h4412161176302437050t_unit @ R2 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_297_ring__hom__memI,axiom,
! [R2: partia4692342223508353374t_unit,H: nat > set_int,S: partia4934656038542163276t_unit] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_set_int @ ( H @ X2 ) @ ( partia966996272515721803t_unit @ S ) ) )
=> ( ! [X2: nat,Y5: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y5 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( mult_n6028127365542633569t_unit @ R2 @ X2 @ Y5 ) )
= ( mult_s3864001451298473021t_unit @ S @ ( H @ X2 ) @ ( H @ Y5 ) ) ) ) )
=> ( ! [X2: nat,Y5: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y5 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( add_nat_Product_unit @ R2 @ X2 @ Y5 ) )
= ( add_se5859248395121729892t_unit @ S @ ( H @ X2 ) @ ( H @ Y5 ) ) ) ) )
=> ( ( ( H @ ( one_na902338870878123981t_unit @ R2 ) )
= ( one_se8065767436706823081t_unit @ S ) )
=> ( member_nat_set_int @ H @ ( ring_h4752909569380436264t_unit @ R2 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_298_ring__hom__memI,axiom,
! [R2: partia4934656038542163276t_unit,H: set_int > nat,S: partia4692342223508353374t_unit] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_nat @ ( H @ X2 ) @ ( partia3499330772048238685t_unit @ S ) ) )
=> ( ! [X2: set_int,Y5: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y5 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( H @ ( mult_s3864001451298473021t_unit @ R2 @ X2 @ Y5 ) )
= ( mult_n6028127365542633569t_unit @ S @ ( H @ X2 ) @ ( H @ Y5 ) ) ) ) )
=> ( ! [X2: set_int,Y5: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y5 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( H @ ( add_se5859248395121729892t_unit @ R2 @ X2 @ Y5 ) )
= ( add_nat_Product_unit @ S @ ( H @ X2 ) @ ( H @ Y5 ) ) ) ) )
=> ( ( ( H @ ( one_se8065767436706823081t_unit @ R2 ) )
= ( one_na902338870878123981t_unit @ S ) )
=> ( member_set_int_nat @ H @ ( ring_h9051218956466089420t_unit @ R2 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_299_ring__hom__memI,axiom,
! [R2: partia4934656038542163276t_unit,H: set_int > set_int,S: partia4934656038542163276t_unit] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( H @ X2 ) @ ( partia966996272515721803t_unit @ S ) ) )
=> ( ! [X2: set_int,Y5: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y5 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( H @ ( mult_s3864001451298473021t_unit @ R2 @ X2 @ Y5 ) )
= ( mult_s3864001451298473021t_unit @ S @ ( H @ X2 ) @ ( H @ Y5 ) ) ) ) )
=> ( ! [X2: set_int,Y5: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y5 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( H @ ( add_se5859248395121729892t_unit @ R2 @ X2 @ Y5 ) )
= ( add_se5859248395121729892t_unit @ S @ ( H @ X2 ) @ ( H @ Y5 ) ) ) ) )
=> ( ( ( H @ ( one_se8065767436706823081t_unit @ R2 ) )
= ( one_se8065767436706823081t_unit @ S ) )
=> ( member5205197933313416826et_int @ H @ ( ring_h3404898052528352314t_unit @ R2 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_300_semiring_Ocarrier__one__not__zero,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( ( partia3499330772048238685t_unit @ R2 )
!= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) )
= ( ( one_na902338870878123981t_unit @ R2 )
!= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_301_semiring_Ocarrier__one__not__zero,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( ( partia966996272515721803t_unit @ R2 )
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) )
= ( ( one_se8065767436706823081t_unit @ R2 )
!= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_302_semiring_Ocarrier__one__zero,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( ( partia3499330772048238685t_unit @ R2 )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) )
= ( ( one_na902338870878123981t_unit @ R2 )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_303_semiring_Ocarrier__one__zero,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( ( partia966996272515721803t_unit @ R2 )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) )
= ( ( one_se8065767436706823081t_unit @ R2 )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_304_semiring_Oone__zeroI,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( ( partia3499330772048238685t_unit @ R2 )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) )
=> ( ( one_na902338870878123981t_unit @ R2 )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% semiring.one_zeroI
thf(fact_305_semiring_Oone__zeroI,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( ( partia966996272515721803t_unit @ R2 )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) )
=> ( ( one_se8065767436706823081t_unit @ R2 )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% semiring.one_zeroI
thf(fact_306_semiring_Oone__zeroD,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( ( one_na902338870878123981t_unit @ R2 )
= ( zero_n5149899317435570679t_unit @ R2 ) )
=> ( ( partia3499330772048238685t_unit @ R2 )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) ) ) ).
% semiring.one_zeroD
thf(fact_307_semiring_Oone__zeroD,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( ( one_se8065767436706823081t_unit @ R2 )
= ( zero_s6269048424454532197t_unit @ R2 ) )
=> ( ( partia966996272515721803t_unit @ R2 )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) ) ) ).
% semiring.one_zeroD
thf(fact_308_ring_Ogenideal__self_H,axiom,
! [R2: partia4692342223508353374t_unit,I: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ I @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ I @ ( genide4496968333291595629t_unit @ R2 @ ( insert_nat @ I @ bot_bot_set_nat ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_309_ring_Ogenideal__self_H,axiom,
! [R2: partia4934656038542163276t_unit,I: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ I @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ I @ ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ I @ bot_bot_set_set_int ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_310_ring_Ogenideal__zero,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) ) ).
% ring.genideal_zero
thf(fact_311_ring_Ogenideal__zero,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( genide4496968333291595629t_unit @ R2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) ) ).
% ring.genideal_zero
thf(fact_312_principalideal_Ogenerate,axiom,
! [I2: set_nat,R2: partia4692342223508353374t_unit] :
( ( princi4652470909602072491t_unit @ I2 @ R2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R2 ) )
& ( I2
= ( genide4496968333291595629t_unit @ R2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% principalideal.generate
thf(fact_313_principalideal_Ogenerate,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( princi8860937869964495385t_unit @ I2 @ R2 )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R2 ) )
& ( I2
= ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ X2 @ bot_bot_set_set_int ) ) ) ) ) ).
% principalideal.generate
thf(fact_314_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_315_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_316_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_317_singleton__iff,axiom,
! [B: nat > set_int,A: nat > set_int] :
( ( member_nat_set_int @ B @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_318_singleton__iff,axiom,
! [B: set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_319_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_320_singletonD,axiom,
! [B: nat > set_int,A: nat > set_int] :
( ( member_nat_set_int @ B @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_321_singletonD,axiom,
! [B: set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_322_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_323_s_Oa__lcos__mult__one,axiom,
! [M2: set_nat] :
( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_l_co3340896127261556338t_unit @ ( mod_ring @ n ) @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ M2 )
= M2 ) ) ).
% s.a_lcos_mult_one
thf(fact_324_s_Oa__lcos__m__assoc,axiom,
! [M2: set_nat,G: nat,H: nat] :
( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ G @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ H @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_l_co3340896127261556338t_unit @ ( mod_ring @ n ) @ G @ ( a_l_co3340896127261556338t_unit @ ( mod_ring @ n ) @ H @ M2 ) )
= ( a_l_co3340896127261556338t_unit @ ( mod_ring @ n ) @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ G @ H ) @ M2 ) ) ) ) ) ).
% s.a_lcos_m_assoc
thf(fact_325_s_Oset__add__zero,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) @ A2 )
= A2 ) ) ).
% s.set_add_zero
thf(fact_326_s_Ocarrier__is__subalgebra,axiom,
! [K2: set_nat] :
( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( embedd2609395410403458802t_unit @ K2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) @ ( mod_ring @ n ) ) ) ).
% s.carrier_is_subalgebra
thf(fact_327_s_Osubalgebra__in__carrier,axiom,
! [K2: set_nat,V: set_nat] :
( ( embedd2609395410403458802t_unit @ K2 @ V @ ( mod_ring @ n ) )
=> ( ord_less_eq_set_nat @ V @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.subalgebra_in_carrier
thf(fact_328_s_Oa__l__coset__subset__G,axiom,
! [H2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ord_less_eq_set_nat @ ( a_l_co3340896127261556338t_unit @ ( mod_ring @ n ) @ X @ H2 ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.a_l_coset_subset_G
thf(fact_329_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_330_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_331_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_332_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_333_s_Oadd_Oone__in__subset,axiom,
! [H2: set_nat] :
( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( H2 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ H2 )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X2 ) @ H2 ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ H2 )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ H2 )
=> ( member_nat @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X2 @ Xa ) @ H2 ) ) )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ H2 ) ) ) ) ) ).
% s.add.one_in_subset
thf(fact_334_s_Oline__extension__smult__closed,axiom,
! [K2: set_nat,E: set_nat,A: nat,K: nat,U: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ! [K3: nat,V2: nat] :
( ( member_nat @ K3 @ K2 )
=> ( ( member_nat @ V2 @ E )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ K3 @ V2 ) @ E ) ) )
=> ( ( ord_less_eq_set_nat @ E @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ K @ K2 )
=> ( ( member_nat @ U @ ( embedd838748496991043025t_unit @ ( mod_ring @ n ) @ K2 @ A @ E ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ K @ U ) @ ( embedd838748496991043025t_unit @ ( mod_ring @ n ) @ K2 @ A @ E ) ) ) ) ) ) ) ) ).
% s.line_extension_smult_closed
thf(fact_335_s_Osubring__props_I7_J,axiom,
! [K2: set_nat,H1: nat,H22: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( member_nat @ H1 @ K2 )
=> ( ( member_nat @ H22 @ K2 )
=> ( member_nat @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ H1 @ H22 ) @ K2 ) ) ) ) ).
% s.subring_props(7)
thf(fact_336_s_Osubring__props_I2_J,axiom,
! [K2: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ K2 ) ) ).
% s.subring_props(2)
thf(fact_337_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_338_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_339_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_340_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_341_s_Osubring__props_I4_J,axiom,
! [K2: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( K2 != bot_bot_set_nat ) ) ).
% s.subring_props(4)
thf(fact_342_s_Osubring__props_I6_J,axiom,
! [K2: set_nat,H1: nat,H22: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( member_nat @ H1 @ K2 )
=> ( ( member_nat @ H22 @ K2 )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ H1 @ H22 ) @ K2 ) ) ) ) ).
% s.subring_props(6)
thf(fact_343_s_Osubring__props_I5_J,axiom,
! [K2: set_nat,H: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( member_nat @ H @ K2 )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ H ) @ K2 ) ) ) ).
% s.subring_props(5)
thf(fact_344_s_Osubring__props_I3_J,axiom,
! [K2: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( member_nat @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) @ K2 ) ) ).
% s.subring_props(3)
thf(fact_345_s_Or__neg2,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) @ Y ) )
= Y ) ) ) ).
% s.r_neg2
thf(fact_346_s_Or__neg1,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) )
= Y ) ) ) ).
% s.r_neg1
thf(fact_347_s_Ominus__add,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ Y ) ) ) ) ) ).
% s.minus_add
thf(fact_348_s_Oadd_Oinv__solve__right_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ B @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ C ) )
= A )
= ( B
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ A @ C ) ) ) ) ) ) ).
% s.add.inv_solve_right'
thf(fact_349_s_Oadd_Oinv__solve__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( A
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ B @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ C ) ) )
= ( B
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ A @ C ) ) ) ) ) ) ).
% s.add.inv_solve_right
thf(fact_350_s_Oadd_Oinv__solve__left_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ B ) @ C )
= A )
= ( C
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ B @ A ) ) ) ) ) ) ).
% s.add.inv_solve_left'
thf(fact_351_s_Oadd_Oinv__solve__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( A
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ B ) @ C ) )
= ( C
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ B @ A ) ) ) ) ) ) ).
% s.add.inv_solve_left
thf(fact_352_s_Oadd_Oinv__mult__group,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ Y ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) ) ) ) ) ).
% s.add.inv_mult_group
thf(fact_353_s_Oa__transpose__inv,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y )
= Z )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) @ Z )
= Y ) ) ) ) ) ).
% s.a_transpose_inv
thf(fact_354_s_Osubring__props_I1_J,axiom,
! [K2: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.subring_props(1)
thf(fact_355_s_Or__minus,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ Y ) )
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y ) ) ) ) ) ).
% s.r_minus
thf(fact_356_s_Ol__minus,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) @ Y )
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y ) ) ) ) ) ).
% s.l_minus
thf(fact_357_s_Osetadd__subset__G,axiom,
! [H2: set_nat,K2: set_nat] :
( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ord_less_eq_set_nat @ ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ H2 @ K2 ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.setadd_subset_G
thf(fact_358_s_Oset__add__comm,axiom,
! [I2: set_nat,J2: set_nat] :
( ( ord_less_eq_set_nat @ I2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ J2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ I2 @ J2 )
= ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ J2 @ I2 ) ) ) ) ).
% s.set_add_comm
thf(fact_359_s_Oset__add__closed,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ B3 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ord_less_eq_set_nat @ ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ A2 @ B3 ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.set_add_closed
thf(fact_360_s_Oadd_Oint__pow__inv,axiom,
! [X: nat,I: int] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) )
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ X ) ) ) ) ).
% s.add.int_pow_inv
thf(fact_361_s_Or__neg,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.r_neg
thf(fact_362_s_Ominus__equality,axiom,
! [Y: nat,X: nat] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X )
= Y ) ) ) ) ).
% s.minus_equality
thf(fact_363_s_Ol__neg,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.l_neg
thf(fact_364_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_365_s_Ominus__minus,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) )
= X ) ) ).
% s.minus_minus
thf(fact_366_s_Oadd_Oinv__closed,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.add.inv_closed
thf(fact_367_s_Ominus__zero,axiom,
( ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ).
% s.minus_zero
thf(fact_368_s_Oadd_Oinv__eq__1__iff,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
= ( X
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.add.inv_eq_1_iff
thf(fact_369_ring_Osubring__props_I5_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,H: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( member_set_int @ H @ K2 )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R2 @ H ) @ K2 ) ) ) ) ).
% ring.subring_props(5)
thf(fact_370_ring_Osubring__props_I5_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,H: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ H @ K2 )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ R2 @ H ) @ K2 ) ) ) ) ).
% ring.subring_props(5)
thf(fact_371_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_372_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_373_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_374_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_375_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_376_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_377_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K4: nat] :
( N2
= ( plus_plus_nat @ M3 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_378_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_379_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_380_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_381_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_382_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_383_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_384_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_385_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_386_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_387_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_388_subalgebra_Osmult__closed,axiom,
! [K2: set_nat,V: set_nat,R2: partia4692342223508353374t_unit,K: nat,V3: nat] :
( ( embedd2609395410403458802t_unit @ K2 @ V @ R2 )
=> ( ( member_nat @ K @ K2 )
=> ( ( member_nat @ V3 @ V )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ K @ V3 ) @ V ) ) ) ) ).
% subalgebra.smult_closed
thf(fact_389_ring_Oring__simprules_I20_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( a_inv_2472168910397739247t_unit @ R2 @ ( a_inv_2472168910397739247t_unit @ R2 @ X ) )
= X ) ) ) ).
% ring.ring_simprules(20)
thf(fact_390_ring_Oring__simprules_I20_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( a_inv_5951419416477254493t_unit @ R2 @ ( a_inv_5951419416477254493t_unit @ R2 @ X ) )
= X ) ) ) ).
% ring.ring_simprules(20)
thf(fact_391_ring_Oring__simprules_I3_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ R2 @ X ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_392_ring_Oring__simprules_I3_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R2 @ X ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_393_ring_Ominus__zero,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( a_inv_5951419416477254493t_unit @ R2 @ ( zero_s6269048424454532197t_unit @ R2 ) )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ).
% ring.minus_zero
thf(fact_394_ring_Ominus__zero,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( a_inv_2472168910397739247t_unit @ R2 @ ( zero_n5149899317435570679t_unit @ R2 ) )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ).
% ring.minus_zero
thf(fact_395_ring_Osubring__props_I2_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ K2 ) ) ) ).
% ring.subring_props(2)
thf(fact_396_ring_Osubring__props_I2_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ K2 ) ) ) ).
% ring.subring_props(2)
thf(fact_397_ring_Osubring__props_I7_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,H1: set_int,H22: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( member_set_int @ H1 @ K2 )
=> ( ( member_set_int @ H22 @ K2 )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R2 @ H1 @ H22 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(7)
thf(fact_398_ring_Osubring__props_I7_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,H1: nat,H22: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ H1 @ K2 )
=> ( ( member_nat @ H22 @ K2 )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ H1 @ H22 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(7)
thf(fact_399_ring_Osubring__props_I4_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( K2 != bot_bot_set_set_int ) ) ) ).
% ring.subring_props(4)
thf(fact_400_ring_Osubring__props_I4_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( K2 != bot_bot_set_nat ) ) ) ).
% ring.subring_props(4)
thf(fact_401_ring_Osubring__props_I6_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,H1: set_int,H22: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( member_set_int @ H1 @ K2 )
=> ( ( member_set_int @ H22 @ K2 )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ H1 @ H22 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(6)
thf(fact_402_ring_Osubring__props_I6_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,H1: nat,H22: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ H1 @ K2 )
=> ( ( member_nat @ H22 @ K2 )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ H1 @ H22 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(6)
thf(fact_403_ring_Osubring__props_I3_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ R2 ) @ K2 ) ) ) ).
% ring.subring_props(3)
thf(fact_404_ring_Osubring__props_I3_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ R2 ) @ K2 ) ) ) ).
% ring.subring_props(3)
thf(fact_405_ring_Oring__simprules_I19_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( a_inv_2472168910397739247t_unit @ R2 @ ( add_nat_Product_unit @ R2 @ X @ Y ) )
= ( add_nat_Product_unit @ R2 @ ( a_inv_2472168910397739247t_unit @ R2 @ X ) @ ( a_inv_2472168910397739247t_unit @ R2 @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_406_ring_Oring__simprules_I19_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( a_inv_5951419416477254493t_unit @ R2 @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ R2 @ ( a_inv_5951419416477254493t_unit @ R2 @ X ) @ ( a_inv_5951419416477254493t_unit @ R2 @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_407_ring_Oring__simprules_I18_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( a_inv_2472168910397739247t_unit @ R2 @ X ) @ ( add_nat_Product_unit @ R2 @ X @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_408_ring_Oring__simprules_I18_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( a_inv_5951419416477254493t_unit @ R2 @ X ) @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_409_ring_Oring__simprules_I17_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ ( add_nat_Product_unit @ R2 @ ( a_inv_2472168910397739247t_unit @ R2 @ X ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_410_ring_Oring__simprules_I17_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ ( add_se5859248395121729892t_unit @ R2 @ ( a_inv_5951419416477254493t_unit @ R2 @ X ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_411_ring_Ol__minus,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( a_inv_2472168910397739247t_unit @ R2 @ X ) @ Y )
= ( a_inv_2472168910397739247t_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_412_ring_Ol__minus,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( a_inv_5951419416477254493t_unit @ R2 @ X ) @ Y )
= ( a_inv_5951419416477254493t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Y ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_413_ring_Or__minus,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ X @ ( a_inv_2472168910397739247t_unit @ R2 @ Y ) )
= ( a_inv_2472168910397739247t_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_414_ring_Or__minus,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ X @ ( a_inv_5951419416477254493t_unit @ R2 @ Y ) )
= ( a_inv_5951419416477254493t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Y ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_415_ring_Osubring__props_I1_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ).
% ring.subring_props(1)
thf(fact_416_ring_Osubring__props_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ord_le4403425263959731960et_int @ K2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ).
% ring.subring_props(1)
thf(fact_417_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_418_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_419_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_420_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_421_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_422_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_423_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_424_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_425_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_426_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_427_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_428_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_429_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A5: int,B6: int] : ( plus_plus_int @ B6 @ A5 ) ) ) ).
% add.commute
thf(fact_430_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B6: nat] : ( plus_plus_nat @ B6 @ A5 ) ) ) ).
% add.commute
thf(fact_431_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_432_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_433_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_434_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_435_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_436_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_437_ring_Oset__add__comm,axiom,
! [R2: partia4692342223508353374t_unit,I2: set_nat,J2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ord_less_eq_set_nat @ I2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( ord_less_eq_set_nat @ J2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( set_ad320919470248169786t_unit @ R2 @ I2 @ J2 )
= ( set_ad320919470248169786t_unit @ R2 @ J2 @ I2 ) ) ) ) ) ).
% ring.set_add_comm
thf(fact_438_ring_Oset__add__comm,axiom,
! [R2: partia4934656038542163276t_unit,I2: set_set_int,J2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( ord_le4403425263959731960et_int @ J2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( set_ad273131178244904872t_unit @ R2 @ I2 @ J2 )
= ( set_ad273131178244904872t_unit @ R2 @ J2 @ I2 ) ) ) ) ) ).
% ring.set_add_comm
thf(fact_439_ring_Oring__simprules_I16_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ ( a_inv_2472168910397739247t_unit @ R2 @ X ) )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_440_ring_Oring__simprules_I16_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ ( a_inv_5951419416477254493t_unit @ R2 @ X ) )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_441_ring_Oring__simprules_I9_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( a_inv_2472168910397739247t_unit @ R2 @ X ) @ X )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_442_ring_Oring__simprules_I9_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( a_inv_5951419416477254493t_unit @ R2 @ X ) @ X )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_443_ring_Ocarrier__is__subalgebra,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( embedd2609395410403458802t_unit @ K2 @ ( partia3499330772048238685t_unit @ R2 ) @ R2 ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_444_ring_Ocarrier__is__subalgebra,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ord_le4403425263959731960et_int @ K2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( embedd2743979684206749024t_unit @ K2 @ ( partia966996272515721803t_unit @ R2 ) @ R2 ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_445_ring_Osubalgebra__in__carrier,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,V: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( embedd2609395410403458802t_unit @ K2 @ V @ R2 )
=> ( ord_less_eq_set_nat @ V @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_446_ring_Osubalgebra__in__carrier,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,V: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( embedd2743979684206749024t_unit @ K2 @ V @ R2 )
=> ( ord_le4403425263959731960et_int @ V @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_447_ring_Oline__extension__smult__closed,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,E: set_set_int,A: set_int,K: set_int,U: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ! [K3: set_int,V2: set_int] :
( ( member_set_int @ K3 @ K2 )
=> ( ( member_set_int @ V2 @ E )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ K3 @ V2 ) @ E ) ) )
=> ( ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ K @ K2 )
=> ( ( member_set_int @ U @ ( embedd4283282269743769663t_unit @ R2 @ K2 @ A @ E ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ K @ U ) @ ( embedd4283282269743769663t_unit @ R2 @ K2 @ A @ E ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_448_ring_Oline__extension__smult__closed,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,E: set_nat,A: nat,K: nat,U: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ! [K3: nat,V2: nat] :
( ( member_nat @ K3 @ K2 )
=> ( ( member_nat @ V2 @ E )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ K3 @ V2 ) @ E ) ) )
=> ( ( ord_less_eq_set_nat @ E @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ K @ K2 )
=> ( ( member_nat @ U @ ( embedd838748496991043025t_unit @ R2 @ K2 @ A @ E ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ K @ U ) @ ( embedd838748496991043025t_unit @ R2 @ K2 @ A @ E ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_449_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_450_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_451_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_452_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_453_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_454_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_455_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_456_add__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_457_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_458_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_459_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_460_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_461_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_462_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B6: nat] :
? [C5: nat] :
( B6
= ( plus_plus_nat @ A5 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_463_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_464_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_465_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_466_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_467_s_Ominus__eq,axiom,
! [X: nat,Y: nat] :
( ( a_minu1751788497103602224t_unit @ ( mod_ring @ n ) @ X @ Y )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ Y ) ) ) ).
% s.minus_eq
thf(fact_468_ring_Oset__add__zero,axiom,
! [R2: partia4692342223508353374t_unit,A2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( set_ad320919470248169786t_unit @ R2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) @ A2 )
= A2 ) ) ) ).
% ring.set_add_zero
thf(fact_469_ring_Oset__add__zero,axiom,
! [R2: partia4934656038542163276t_unit,A2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ord_le4403425263959731960et_int @ A2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( set_ad273131178244904872t_unit @ R2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) @ A2 )
= A2 ) ) ) ).
% ring.set_add_zero
thf(fact_470_s_Osubfield__m__inv__simprule,axiom,
! [K2: set_nat,K: nat,A: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ K @ A ) @ K2 )
=> ( member_nat @ A @ K2 ) ) ) ) ) ).
% s.subfield_m_inv_simprule
thf(fact_471_s_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K2: set_nat,E: set_nat,V: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ E )
=> ( ( embedd2609395410403458802t_unit @ K2 @ V @ ( mod_ring @ n ) )
=> ( ( ord_less_eq_set_nat @ V @ E )
=> ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ V ) ) ) ) ) ).
% s.subalbegra_incl_imp_finite_dimension
thf(fact_472_s_OsubringI,axiom,
! [H2: set_nat] :
( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) @ H2 )
=> ( ! [H3: nat] :
( ( member_nat @ H3 @ H2 )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ H3 ) @ H2 ) )
=> ( ! [H12: nat,H23: nat] :
( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ H12 @ H23 ) @ H2 ) ) )
=> ( ! [H12: nat,H23: nat] :
( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( member_nat @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ H12 @ H23 ) @ H2 ) ) )
=> ( subrin2893992908230074586t_unit @ H2 @ ( mod_ring @ n ) ) ) ) ) ) ) ).
% s.subringI
thf(fact_473_s_Oadd__additive__subgroups,axiom,
! [H2: set_nat,K2: set_nat] :
( ( additi4297497278381856430t_unit @ H2 @ ( mod_ring @ n ) )
=> ( ( additi4297497278381856430t_unit @ K2 @ ( mod_ring @ n ) )
=> ( additi4297497278381856430t_unit @ ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ H2 @ K2 ) @ ( mod_ring @ n ) ) ) ) ).
% s.add_additive_subgroups
thf(fact_474_s_Oadd_Oint__pow__diff,axiom,
! [X: nat,N: int,M: int] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ ( minus_minus_int @ N @ M ) @ X )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ N @ X ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ M @ X ) ) ) ) ) ).
% s.add.int_pow_diff
thf(fact_475_s_Ofinite__dimension__imp__subalgebra,axiom,
! [K2: set_nat,E: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ E )
=> ( embedd2609395410403458802t_unit @ K2 @ E @ ( mod_ring @ n ) ) ) ) ).
% s.finite_dimension_imp_subalgebra
thf(fact_476_s_Ocarrier__is__subring,axiom,
subrin2893992908230074586t_unit @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) @ ( mod_ring @ n ) ).
% s.carrier_is_subring
thf(fact_477_DiffI,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ C @ A2 )
=> ( ~ ( member_nat_set_int @ C @ B3 )
=> ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_478_DiffI,axiom,
! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ A2 )
=> ( ~ ( member_set_nat @ C @ B3 )
=> ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_479_DiffI,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_480_Diff__iff,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B3 ) )
= ( ( member_nat_set_int @ C @ A2 )
& ~ ( member_nat_set_int @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_481_Diff__iff,axiom,
! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
= ( ( member_set_nat @ C @ A2 )
& ~ ( member_set_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_482_Diff__iff,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_483_Diff__idemp,axiom,
! [A2: set_nat,B3: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ B3 )
= ( minus_minus_set_nat @ A2 @ B3 ) ) ).
% Diff_idemp
thf(fact_484_s_Otelescopic__base__dim_I1_J,axiom,
! [K2: set_nat,F2: set_nat,E: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( subfie4892355163478727762t_unit @ F2 @ ( mod_ring @ n ) )
=> ( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ F2 )
=> ( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ F2 @ E )
=> ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ E ) ) ) ) ) ).
% s.telescopic_base_dim(1)
thf(fact_485_s_Osum__space__dim_I1_J,axiom,
! [K2: set_nat,E: set_nat,F2: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ E )
=> ( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ F2 )
=> ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ E @ F2 ) ) ) ) ) ).
% s.sum_space_dim(1)
thf(fact_486_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_487_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_488_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_489_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_490_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_491_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_492_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_493_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_494_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_495_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_496_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_497_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_498_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_499_insert__Diff1,axiom,
! [X: nat > set_int,B3: set_nat_set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ X @ B3 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X @ A2 ) @ B3 )
= ( minus_3247115583872269408et_int @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_500_insert__Diff1,axiom,
! [X: set_nat,B3: set_set_nat,A2: set_set_nat] :
( ( member_set_nat @ X @ B3 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B3 )
= ( minus_2163939370556025621et_nat @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_501_insert__Diff1,axiom,
! [X: nat,B3: set_nat,A2: set_nat] :
( ( member_nat @ X @ B3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B3 )
= ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_502_Diff__insert0,axiom,
! [X: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ~ ( member_nat_set_int @ X @ A2 )
=> ( ( minus_3247115583872269408et_int @ A2 @ ( insert_nat_set_int @ X @ B3 ) )
= ( minus_3247115583872269408et_int @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_503_Diff__insert0,axiom,
! [X: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ~ ( member_set_nat @ X @ A2 )
=> ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ B3 ) )
= ( minus_2163939370556025621et_nat @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_504_Diff__insert0,axiom,
! [X: nat,A2: set_nat,B3: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B3 ) )
= ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_505_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_506_Diff__eq__empty__iff,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ( minus_2163939370556025621et_nat @ A2 @ B3 )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_507_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_508_s_Ominus__closed,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( a_minu1751788497103602224t_unit @ ( mod_ring @ n ) @ X @ Y ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.minus_closed
thf(fact_509_s_Or__right__minus__eq,axiom,
! [A: nat,B: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( a_minu1751788497103602224t_unit @ ( mod_ring @ n ) @ A @ B )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
= ( A = B ) ) ) ) ).
% s.r_right_minus_eq
thf(fact_510_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_511_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_512_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_513_ring_Ofinite__dimension_Ocong,axiom,
embedd6096501799845681477t_unit = embedd6096501799845681477t_unit ).
% ring.finite_dimension.cong
thf(fact_514_DiffE,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B3 ) )
=> ~ ( ( member_nat_set_int @ C @ A2 )
=> ( member_nat_set_int @ C @ B3 ) ) ) ).
% DiffE
thf(fact_515_DiffE,axiom,
! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
=> ~ ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_516_DiffE,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_517_DiffD1,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B3 ) )
=> ( member_nat_set_int @ C @ A2 ) ) ).
% DiffD1
thf(fact_518_DiffD1,axiom,
! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
=> ( member_set_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_519_DiffD1,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_520_DiffD2,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B3 ) )
=> ~ ( member_nat_set_int @ C @ B3 ) ) ).
% DiffD2
thf(fact_521_DiffD2,axiom,
! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
=> ~ ( member_set_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_522_DiffD2,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
=> ~ ( member_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_523_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_524_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_525_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_526_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_527_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_528_diff__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D2 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_529_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_530_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_531_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_532_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_533_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_534_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_535_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_536_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_537_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_538_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_539_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_540_double__diff,axiom,
! [A2: set_nat,B3: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C3 )
=> ( ( minus_minus_set_nat @ B3 @ ( minus_minus_set_nat @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_541_double__diff,axiom,
! [A2: set_set_nat,B3: set_set_nat,C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( ord_le6893508408891458716et_nat @ B3 @ C3 )
=> ( ( minus_2163939370556025621et_nat @ B3 @ ( minus_2163939370556025621et_nat @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_542_Diff__subset,axiom,
! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_543_Diff__subset,axiom,
! [A2: set_set_nat,B3: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_544_Diff__mono,axiom,
! [A2: set_nat,C3: set_nat,D: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ D @ B3 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ ( minus_minus_set_nat @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_545_Diff__mono,axiom,
! [A2: set_set_nat,C3: set_set_nat,D: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C3 )
=> ( ( ord_le6893508408891458716et_nat @ D @ B3 )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) @ ( minus_2163939370556025621et_nat @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_546_insert__Diff__if,axiom,
! [X: nat > set_int,B3: set_nat_set_int,A2: set_nat_set_int] :
( ( ( member_nat_set_int @ X @ B3 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X @ A2 ) @ B3 )
= ( minus_3247115583872269408et_int @ A2 @ B3 ) ) )
& ( ~ ( member_nat_set_int @ X @ B3 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X @ A2 ) @ B3 )
= ( insert_nat_set_int @ X @ ( minus_3247115583872269408et_int @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_547_insert__Diff__if,axiom,
! [X: set_nat,B3: set_set_nat,A2: set_set_nat] :
( ( ( member_set_nat @ X @ B3 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B3 )
= ( minus_2163939370556025621et_nat @ A2 @ B3 ) ) )
& ( ~ ( member_set_nat @ X @ B3 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B3 )
= ( insert_set_nat @ X @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_548_insert__Diff__if,axiom,
! [X: nat,B3: set_nat,A2: set_nat] :
( ( ( member_nat @ X @ B3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B3 )
= ( minus_minus_set_nat @ A2 @ B3 ) ) )
& ( ~ ( member_nat @ X @ B3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B3 )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_549_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_550_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_551_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_552_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_553_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_554_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_555_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_556_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_557_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_558_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_559_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_560_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_561_subset__Diff__insert,axiom,
! [A2: set_nat_set_int,B3: set_nat_set_int,X: nat > set_int,C3: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ ( minus_3247115583872269408et_int @ B3 @ ( insert_nat_set_int @ X @ C3 ) ) )
= ( ( ord_le5995675665013768039et_int @ A2 @ ( minus_3247115583872269408et_int @ B3 @ C3 ) )
& ~ ( member_nat_set_int @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_562_subset__Diff__insert,axiom,
! [A2: set_nat,B3: set_nat,X: nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B3 @ ( insert_nat @ X @ C3 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B3 @ C3 ) )
& ~ ( member_nat @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_563_subset__Diff__insert,axiom,
! [A2: set_set_nat,B3: set_set_nat,X: set_nat,C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B3 @ ( insert_set_nat @ X @ C3 ) ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B3 @ C3 ) )
& ~ ( member_set_nat @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_564_Diff__insert,axiom,
! [A2: set_nat,A: nat,B3: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B3 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_565_insert__Diff,axiom,
! [A: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ A2 )
=> ( ( insert_nat_set_int @ A @ ( minus_3247115583872269408et_int @ A2 @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_566_insert__Diff,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_567_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_568_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B3: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B3 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_569_Diff__insert__absorb,axiom,
! [X: nat > set_int,A2: set_nat_set_int] :
( ~ ( member_nat_set_int @ X @ A2 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X @ A2 ) @ ( insert_nat_set_int @ X @ bot_bo8417611410066262939et_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_570_Diff__insert__absorb,axiom,
! [X: set_nat,A2: set_set_nat] :
( ~ ( member_set_nat @ X @ A2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_571_Diff__insert__absorb,axiom,
! [X: nat,A2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_572_ring_Otelescopic__base__dim_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,F2: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( subfie3888952257595785920t_unit @ F2 @ R2 )
=> ( ( embedd8246663962306818995t_unit @ R2 @ K2 @ F2 )
=> ( ( embedd8246663962306818995t_unit @ R2 @ F2 @ E )
=> ( embedd8246663962306818995t_unit @ R2 @ K2 @ E ) ) ) ) ) ) ).
% ring.telescopic_base_dim(1)
thf(fact_573_ring_Otelescopic__base__dim_I1_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,F2: set_nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( subfie4892355163478727762t_unit @ F2 @ R2 )
=> ( ( embedd6096501799845681477t_unit @ R2 @ K2 @ F2 )
=> ( ( embedd6096501799845681477t_unit @ R2 @ F2 @ E )
=> ( embedd6096501799845681477t_unit @ R2 @ K2 @ E ) ) ) ) ) ) ).
% ring.telescopic_base_dim(1)
thf(fact_574_ring_Oring__simprules_I4_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( a_minu1751788497103602224t_unit @ R2 @ X @ Y ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_575_ring_Oring__simprules_I4_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( a_minu5974516859897376926t_unit @ R2 @ X @ Y ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_576_subset__insert__iff,axiom,
! [A2: set_nat_set_int,X: nat > set_int,B3: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ ( insert_nat_set_int @ X @ B3 ) )
= ( ( ( member_nat_set_int @ X @ A2 )
=> ( ord_le5995675665013768039et_int @ ( minus_3247115583872269408et_int @ A2 @ ( insert_nat_set_int @ X @ bot_bo8417611410066262939et_int ) ) @ B3 ) )
& ( ~ ( member_nat_set_int @ X @ A2 )
=> ( ord_le5995675665013768039et_int @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_577_subset__insert__iff,axiom,
! [A2: set_nat,X: nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B3 ) )
= ( ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B3 ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_578_subset__insert__iff,axiom,
! [A2: set_set_nat,X: set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B3 ) )
= ( ( ( member_set_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B3 ) )
& ( ~ ( member_set_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_579_Diff__single__insert,axiom,
! [A2: set_nat,X: nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B3 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_580_Diff__single__insert,axiom,
! [A2: set_set_nat,X: set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B3 )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_581_a__minus__def,axiom,
( a_minu1751788497103602224t_unit
= ( ^ [R3: partia4692342223508353374t_unit,X3: nat,Y3: nat] : ( add_nat_Product_unit @ R3 @ X3 @ ( a_inv_2472168910397739247t_unit @ R3 @ Y3 ) ) ) ) ).
% a_minus_def
thf(fact_582_ring_Osum__space__dim_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,E: set_set_int,F2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd8246663962306818995t_unit @ R2 @ K2 @ E )
=> ( ( embedd8246663962306818995t_unit @ R2 @ K2 @ F2 )
=> ( embedd8246663962306818995t_unit @ R2 @ K2 @ ( set_ad273131178244904872t_unit @ R2 @ E @ F2 ) ) ) ) ) ) ).
% ring.sum_space_dim(1)
thf(fact_583_ring_Osum__space__dim_I1_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,E: set_nat,F2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd6096501799845681477t_unit @ R2 @ K2 @ E )
=> ( ( embedd6096501799845681477t_unit @ R2 @ K2 @ F2 )
=> ( embedd6096501799845681477t_unit @ R2 @ K2 @ ( set_ad320919470248169786t_unit @ R2 @ E @ F2 ) ) ) ) ) ) ).
% ring.sum_space_dim(1)
thf(fact_584_ring_Ofinite__dimension__imp__subalgebra,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd8246663962306818995t_unit @ R2 @ K2 @ E )
=> ( embedd2743979684206749024t_unit @ K2 @ E @ R2 ) ) ) ) ).
% ring.finite_dimension_imp_subalgebra
thf(fact_585_ring_Ofinite__dimension__imp__subalgebra,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd6096501799845681477t_unit @ R2 @ K2 @ E )
=> ( embedd2609395410403458802t_unit @ K2 @ E @ R2 ) ) ) ) ).
% ring.finite_dimension_imp_subalgebra
thf(fact_586_ring_Oring__simprules_I14_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( a_minu5974516859897376926t_unit @ R2 @ X @ Y )
= ( add_se5859248395121729892t_unit @ R2 @ X @ ( a_inv_5951419416477254493t_unit @ R2 @ Y ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_587_ring_Oring__simprules_I14_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( a_minu1751788497103602224t_unit @ R2 @ X @ Y )
= ( add_nat_Product_unit @ R2 @ X @ ( a_inv_2472168910397739247t_unit @ R2 @ Y ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_588_ring_Osubalbegra__incl__imp__finite__dimension,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,E: set_set_int,V: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd8246663962306818995t_unit @ R2 @ K2 @ E )
=> ( ( embedd2743979684206749024t_unit @ K2 @ V @ R2 )
=> ( ( ord_le4403425263959731960et_int @ V @ E )
=> ( embedd8246663962306818995t_unit @ R2 @ K2 @ V ) ) ) ) ) ) ).
% ring.subalbegra_incl_imp_finite_dimension
thf(fact_589_ring_Osubalbegra__incl__imp__finite__dimension,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,E: set_nat,V: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd6096501799845681477t_unit @ R2 @ K2 @ E )
=> ( ( embedd2609395410403458802t_unit @ K2 @ V @ R2 )
=> ( ( ord_less_eq_set_nat @ V @ E )
=> ( embedd6096501799845681477t_unit @ R2 @ K2 @ V ) ) ) ) ) ) ).
% ring.subalbegra_incl_imp_finite_dimension
thf(fact_590_ring_OsubringI,axiom,
! [R2: partia4692342223508353374t_unit,H2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ ( one_na902338870878123981t_unit @ R2 ) @ H2 )
=> ( ! [H3: nat] :
( ( member_nat @ H3 @ H2 )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ R2 @ H3 ) @ H2 ) )
=> ( ! [H12: nat,H23: nat] :
( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ H12 @ H23 ) @ H2 ) ) )
=> ( ! [H12: nat,H23: nat] :
( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ H12 @ H23 ) @ H2 ) ) )
=> ( subrin2893992908230074586t_unit @ H2 @ R2 ) ) ) ) ) ) ) ).
% ring.subringI
thf(fact_591_ring_OsubringI,axiom,
! [R2: partia4934656038542163276t_unit,H2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ ( one_se8065767436706823081t_unit @ R2 ) @ H2 )
=> ( ! [H3: set_int] :
( ( member_set_int @ H3 @ H2 )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R2 @ H3 ) @ H2 ) )
=> ( ! [H12: set_int,H23: set_int] :
( ( member_set_int @ H12 @ H2 )
=> ( ( member_set_int @ H23 @ H2 )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ H12 @ H23 ) @ H2 ) ) )
=> ( ! [H12: set_int,H23: set_int] :
( ( member_set_int @ H12 @ H2 )
=> ( ( member_set_int @ H23 @ H2 )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R2 @ H12 @ H23 ) @ H2 ) ) )
=> ( subrin7689096310803670856t_unit @ H2 @ R2 ) ) ) ) ) ) ) ).
% ring.subringI
thf(fact_592_ring_Osubfield__m__inv__simprule,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,K: nat,A: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ K @ A ) @ K2 )
=> ( member_nat @ A @ K2 ) ) ) ) ) ) ).
% ring.subfield_m_inv_simprule
thf(fact_593_ring_Osubfield__m__inv__simprule,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,K: set_int,A: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( member_set_int @ K @ ( minus_8897228262479074673et_int @ K2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ K @ A ) @ K2 )
=> ( member_set_int @ A @ K2 ) ) ) ) ) ) ).
% ring.subfield_m_inv_simprule
thf(fact_594_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_595_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_596_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_597_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_598_s_Osubfield__m__inv_I2_J,axiom,
! [K2: set_nat,K: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ K @ ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ K ) )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.subfield_m_inv(2)
thf(fact_599_s_Osubfield__m__inv_I3_J,axiom,
! [K2: set_nat,K: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ K ) @ K )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.subfield_m_inv(3)
thf(fact_600_s_Oinv__unique_H,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Y @ X )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
=> ( Y
= ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ X ) ) ) ) ) ) ).
% s.inv_unique'
thf(fact_601_s_Oinv__char,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Y @ X )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
=> ( ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ X )
= Y ) ) ) ) ) ).
% s.inv_char
thf(fact_602_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_603_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_604_s_Osubfield__m__inv_I1_J,axiom,
! [K2: set_nat,K: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ K ) @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) ) ) ) ).
% s.subfield_m_inv(1)
thf(fact_605_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_606_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_607_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_608_s_Oinv__one,axiom,
( ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ).
% s.inv_one
thf(fact_609_s_Oinv__neg__one,axiom,
( ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) )
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.inv_neg_one
thf(fact_610_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_611_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_612_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_613_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_614_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_615_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_616_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_617_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_618_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_619_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_620_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_621_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_622_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_623_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_624_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_625_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_626_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_627_ring_Oinv__neg__one,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( m_inv_4894562657074299959t_unit @ R2 @ ( a_inv_5951419416477254493t_unit @ R2 @ ( one_se8065767436706823081t_unit @ R2 ) ) )
= ( a_inv_5951419416477254493t_unit @ R2 @ ( one_se8065767436706823081t_unit @ R2 ) ) ) ) ).
% ring.inv_neg_one
thf(fact_628_ring_Oinv__neg__one,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( m_inv_3931797133766013019t_unit @ R2 @ ( a_inv_2472168910397739247t_unit @ R2 @ ( one_na902338870878123981t_unit @ R2 ) ) )
= ( a_inv_2472168910397739247t_unit @ R2 @ ( one_na902338870878123981t_unit @ R2 ) ) ) ) ).
% ring.inv_neg_one
thf(fact_629_ring_Osubfield__m__inv_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,K: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( member_set_int @ K @ ( minus_8897228262479074673et_int @ K2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) )
=> ( member_set_int @ ( m_inv_4894562657074299959t_unit @ R2 @ K ) @ ( minus_8897228262479074673et_int @ K2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) ) ) ) ) ).
% ring.subfield_m_inv(1)
thf(fact_630_ring_Osubfield__m__inv_I1_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,K: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( m_inv_3931797133766013019t_unit @ R2 @ K ) @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) ) ) ) ) ).
% ring.subfield_m_inv(1)
thf(fact_631_ring_Osubfield__m__inv_I2_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,K: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( member_set_int @ K @ ( minus_8897228262479074673et_int @ K2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ K @ ( m_inv_4894562657074299959t_unit @ R2 @ K ) )
= ( one_se8065767436706823081t_unit @ R2 ) ) ) ) ) ).
% ring.subfield_m_inv(2)
thf(fact_632_ring_Osubfield__m__inv_I2_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,K: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ K @ ( m_inv_3931797133766013019t_unit @ R2 @ K ) )
= ( one_na902338870878123981t_unit @ R2 ) ) ) ) ) ).
% ring.subfield_m_inv(2)
thf(fact_633_ring_Osubfield__m__inv_I3_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,K: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( member_set_int @ K @ ( minus_8897228262479074673et_int @ K2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( m_inv_4894562657074299959t_unit @ R2 @ K ) @ K )
= ( one_se8065767436706823081t_unit @ R2 ) ) ) ) ) ).
% ring.subfield_m_inv(3)
thf(fact_634_ring_Osubfield__m__inv_I3_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,K: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( m_inv_3931797133766013019t_unit @ R2 @ K ) @ K )
= ( one_na902338870878123981t_unit @ R2 ) ) ) ) ) ).
% ring.subfield_m_inv(3)
thf(fact_635_subringE_I2_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subrin2893992908230074586t_unit @ H2 @ R2 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ H2 ) ) ).
% subringE(2)
thf(fact_636_subringE_I7_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H1: nat,H22: nat] :
( ( subrin2893992908230074586t_unit @ H2 @ R2 )
=> ( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ H1 @ H22 ) @ H2 ) ) ) ) ).
% subringE(7)
thf(fact_637_subringE_I4_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subrin2893992908230074586t_unit @ H2 @ R2 )
=> ( H2 != bot_bot_set_nat ) ) ).
% subringE(4)
thf(fact_638_subfieldE_I4_J,axiom,
! [K2: set_nat,R2: partia4692342223508353374t_unit,K1: nat,K22: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ K1 @ K2 )
=> ( ( member_nat @ K22 @ K2 )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ K1 @ K22 )
= ( mult_n6028127365542633569t_unit @ R2 @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_639_subringE_I6_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H1: nat,H22: nat] :
( ( subrin2893992908230074586t_unit @ H2 @ R2 )
=> ( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ H1 @ H22 ) @ H2 ) ) ) ) ).
% subringE(6)
thf(fact_640_subfieldE_I1_J,axiom,
! [K2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( subrin2893992908230074586t_unit @ K2 @ R2 ) ) ).
% subfieldE(1)
thf(fact_641_subringE_I3_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subrin2893992908230074586t_unit @ H2 @ R2 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ R2 ) @ H2 ) ) ).
% subringE(3)
thf(fact_642_subringE_I5_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H: nat] :
( ( subrin2893992908230074586t_unit @ H2 @ R2 )
=> ( ( member_nat @ H @ H2 )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ R2 @ H ) @ H2 ) ) ) ).
% subringE(5)
thf(fact_643_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_644_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_645_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_646_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_647_subfieldE_I3_J,axiom,
! [K2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% subfieldE(3)
thf(fact_648_subfieldE_I3_J,axiom,
! [K2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ord_le4403425263959731960et_int @ K2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% subfieldE(3)
thf(fact_649_subringE_I1_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subrin2893992908230074586t_unit @ H2 @ R2 )
=> ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% subringE(1)
thf(fact_650_subringE_I1_J,axiom,
! [H2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( subrin7689096310803670856t_unit @ H2 @ R2 )
=> ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% subringE(1)
thf(fact_651_subfieldE_I5_J,axiom,
! [K2: set_nat,R2: partia4692342223508353374t_unit,K1: nat,K22: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ K1 @ K2 )
=> ( ( member_nat @ K22 @ K2 )
=> ( ( ( mult_n6028127365542633569t_unit @ R2 @ K1 @ K22 )
= ( zero_n5149899317435570679t_unit @ R2 ) )
=> ( ( K1
= ( zero_n5149899317435570679t_unit @ R2 ) )
| ( K22
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_652_ring_Ocarrier__is__subring,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( subrin2893992908230074586t_unit @ ( partia3499330772048238685t_unit @ R2 ) @ R2 ) ) ).
% ring.carrier_is_subring
thf(fact_653_ring_Ocarrier__is__subring,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( subrin7689096310803670856t_unit @ ( partia966996272515721803t_unit @ R2 ) @ R2 ) ) ).
% ring.carrier_is_subring
thf(fact_654_subfieldE_I6_J,axiom,
! [K2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( one_na902338870878123981t_unit @ R2 )
!= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ).
% subfieldE(6)
thf(fact_655_s_Ospace__subgroup__props_I6_J,axiom,
! [K2: set_nat,N: nat,E: set_nat,K: nat,A: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ K @ A ) @ E )
=> ( member_nat @ A @ E ) ) ) ) ) ) ).
% s.space_subgroup_props(6)
thf(fact_656_s_OsubcringI,axiom,
! [H2: set_nat] :
( ( subrin2893992908230074586t_unit @ H2 @ ( mod_ring @ n ) )
=> ( ! [H12: nat,H23: nat] :
( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ H12 @ H23 )
= ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ H23 @ H12 ) ) ) )
=> ( subcri1627753237249443161t_unit @ H2 @ ( mod_ring @ n ) ) ) ) ).
% s.subcringI
thf(fact_657_s_Oadd_Oint__pow__neg,axiom,
! [X: nat,I: int] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ ( uminus_uminus_int @ I ) @ X )
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ X ) ) ) ) ).
% s.add.int_pow_neg
thf(fact_658_s_Oa__rcos__assoc__lcos,axiom,
! [H2: set_nat,K2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ H2 @ X ) @ K2 )
= ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ H2 @ ( a_l_co3340896127261556338t_unit @ ( mod_ring @ n ) @ X @ K2 ) ) ) ) ) ) ).
% s.a_rcos_assoc_lcos
thf(fact_659_ring_Or__right__minus__eq,axiom,
! [R2: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( ( a_minu1751788497103602224t_unit @ R2 @ A @ B )
= ( zero_n5149899317435570679t_unit @ R2 ) )
= ( A = B ) ) ) ) ) ).
% ring.r_right_minus_eq
thf(fact_660_ring_Or__right__minus__eq,axiom,
! [R2: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( ( a_minu5974516859897376926t_unit @ R2 @ A @ B )
= ( zero_s6269048424454532197t_unit @ R2 ) )
= ( A = B ) ) ) ) ) ).
% ring.r_right_minus_eq
thf(fact_661_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_662_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_663_s_Odimension__is__inj,axiom,
! [K2: set_nat,N: nat,E: set_nat,M: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ M @ K2 @ E )
=> ( N = M ) ) ) ) ).
% s.dimension_is_inj
thf(fact_664_s_Ofinite__dimensionE_H,axiom,
! [K2: set_nat,E: set_nat] :
( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ E )
=> ~ ! [N3: nat] :
~ ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N3 @ K2 @ E ) ) ).
% s.finite_dimensionE'
thf(fact_665_s_Ofinite__dimensionI,axiom,
! [N: nat,K2: set_nat,E: set_nat] :
( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ E ) ) ).
% s.finite_dimensionI
thf(fact_666_s_Ofinite__dimension__def,axiom,
! [K2: set_nat,E: set_nat] :
( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ E )
= ( ? [N2: nat] : ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N2 @ K2 @ E ) ) ) ).
% s.finite_dimension_def
thf(fact_667_s_Oa__r__coset__subset__G,axiom,
! [H2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ord_less_eq_set_nat @ ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ H2 @ X ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.a_r_coset_subset_G
thf(fact_668_s_Ospace__subgroup__props_I2_J,axiom,
! [K2: set_nat,N: nat,E: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ E ) ) ) ).
% s.space_subgroup_props(2)
thf(fact_669_s_Ospace__subgroup__props_I3_J,axiom,
! [K2: set_nat,N: nat,E: set_nat,V1: nat,V22: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( ( member_nat @ V1 @ E )
=> ( ( member_nat @ V22 @ E )
=> ( member_nat @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ V1 @ V22 ) @ E ) ) ) ) ) ).
% s.space_subgroup_props(3)
thf(fact_670_s_Ospace__subgroup__props_I5_J,axiom,
! [K2: set_nat,N: nat,E: set_nat,K: nat,V3: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( ( member_nat @ K @ K2 )
=> ( ( member_nat @ V3 @ E )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ K @ V3 ) @ E ) ) ) ) ) ).
% s.space_subgroup_props(5)
thf(fact_671_s_Ospace__subgroup__props_I4_J,axiom,
! [K2: set_nat,N: nat,E: set_nat,V3: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( ( member_nat @ V3 @ E )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ V3 ) @ E ) ) ) ) ).
% s.space_subgroup_props(4)
thf(fact_672_s_Ounique__dimension,axiom,
! [K2: set_nat,E: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ E )
=> ? [X2: nat] :
( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ X2 @ K2 @ E )
& ! [Y6: nat] :
( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ Y6 @ K2 @ E )
=> ( Y6 = X2 ) ) ) ) ) ).
% s.unique_dimension
thf(fact_673_s_Oa__coset__add__assoc,axiom,
! [M2: set_nat,G: nat,H: nat] :
( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ G @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ H @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ G ) @ H )
= ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ G @ H ) ) ) ) ) ) ).
% s.a_coset_add_assoc
thf(fact_674_s_Oa__rcosI,axiom,
! [H: nat,H2: set_nat,X: nat] :
( ( member_nat @ H @ H2 )
=> ( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ H @ X ) @ ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ H2 @ X ) ) ) ) ) ).
% s.a_rcosI
thf(fact_675_s_Ospace__subgroup__props_I1_J,axiom,
! [K2: set_nat,N: nat,E: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( ord_less_eq_set_nat @ E @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.space_subgroup_props(1)
thf(fact_676_s_Oa__setmult__rcos__assoc,axiom,
! [H2: set_nat,K2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ H2 @ ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ K2 @ X ) )
= ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ H2 @ K2 ) @ X ) ) ) ) ) ).
% s.a_setmult_rcos_assoc
thf(fact_677_s_Oa__coset__add__inv1,axiom,
! [M2: set_nat,X: nat,Y: nat] :
( ( ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ Y ) ) )
= M2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ X )
= ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ Y ) ) ) ) ) ) ).
% s.a_coset_add_inv1
thf(fact_678_s_Oa__coset__add__inv2,axiom,
! [M2: set_nat,X: nat,Y: nat] :
( ( ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ X )
= ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ Y ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ Y ) ) )
= M2 ) ) ) ) ) ).
% s.a_coset_add_inv2
thf(fact_679_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_680_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_681_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_682_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_683_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_684_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_685_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_686_s_Oa__coset__add__zero,axiom,
! [M2: set_nat] :
( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
= M2 ) ) ).
% s.a_coset_add_zero
thf(fact_687_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_688_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_689_ring_Odimension_Ocong,axiom,
embedd5688180257602933782t_unit = embedd5688180257602933782t_unit ).
% ring.dimension.cong
thf(fact_690_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_691_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_692_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_693_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_694_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_695_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_696_subcringE_I2_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ H2 ) ) ).
% subcringE(2)
thf(fact_697_subcringE_I7_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H1: nat,H22: nat] :
( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ H1 @ H22 ) @ H2 ) ) ) ) ).
% subcringE(7)
thf(fact_698_subcringE_I4_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( H2 != bot_bot_set_nat ) ) ).
% subcringE(4)
thf(fact_699_subcringE_I6_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H1: nat,H22: nat] :
( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ H1 @ H22 ) @ H2 ) ) ) ) ).
% subcringE(6)
thf(fact_700_subcring_Osub__m__comm,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H1: nat,H22: nat] :
( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ H1 @ H22 )
= ( mult_n6028127365542633569t_unit @ R2 @ H22 @ H1 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_701_subfieldE_I2_J,axiom,
! [K2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( subcri1627753237249443161t_unit @ K2 @ R2 ) ) ).
% subfieldE(2)
thf(fact_702_subcringE_I3_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ R2 ) @ H2 ) ) ).
% subcringE(3)
thf(fact_703_subcringE_I5_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H: nat] :
( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( ( member_nat @ H @ H2 )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ R2 @ H ) @ H2 ) ) ) ).
% subcringE(5)
thf(fact_704_group__cancel_Osub2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_705_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A5: int,B6: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B6 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_706_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A5: int,B6: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B6 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_707_ring_Odimension__is__inj,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int,M: nat] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( ( embedd646006463343340164t_unit @ R2 @ M @ K2 @ E )
=> ( N = M ) ) ) ) ) ).
% ring.dimension_is_inj
thf(fact_708_ring_Odimension__is__inj,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat,M: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( ( embedd5688180257602933782t_unit @ R2 @ M @ K2 @ E )
=> ( N = M ) ) ) ) ) ).
% ring.dimension_is_inj
thf(fact_709_ring_Ofinite__dimension__def,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( embedd8246663962306818995t_unit @ R2 @ K2 @ E )
= ( ? [N2: nat] : ( embedd646006463343340164t_unit @ R2 @ N2 @ K2 @ E ) ) ) ) ).
% ring.finite_dimension_def
thf(fact_710_ring_Ofinite__dimension__def,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( embedd6096501799845681477t_unit @ R2 @ K2 @ E )
= ( ? [N2: nat] : ( embedd5688180257602933782t_unit @ R2 @ N2 @ K2 @ E ) ) ) ) ).
% ring.finite_dimension_def
thf(fact_711_ring_Ofinite__dimensionE_H,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( embedd8246663962306818995t_unit @ R2 @ K2 @ E )
=> ~ ! [N3: nat] :
~ ( embedd646006463343340164t_unit @ R2 @ N3 @ K2 @ E ) ) ) ).
% ring.finite_dimensionE'
thf(fact_712_ring_Ofinite__dimensionE_H,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( embedd6096501799845681477t_unit @ R2 @ K2 @ E )
=> ~ ! [N3: nat] :
~ ( embedd5688180257602933782t_unit @ R2 @ N3 @ K2 @ E ) ) ) ).
% ring.finite_dimensionE'
thf(fact_713_ring_Ofinite__dimensionI,axiom,
! [R2: partia4934656038542163276t_unit,N: nat,K2: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( embedd8246663962306818995t_unit @ R2 @ K2 @ E ) ) ) ).
% ring.finite_dimensionI
thf(fact_714_ring_Ofinite__dimensionI,axiom,
! [R2: partia4692342223508353374t_unit,N: nat,K2: set_nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( embedd6096501799845681477t_unit @ R2 @ K2 @ E ) ) ) ).
% ring.finite_dimensionI
thf(fact_715_subcringE_I1_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% subcringE(1)
thf(fact_716_subcringE_I1_J,axiom,
! [H2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( subcri1024317279029940167t_unit @ H2 @ R2 )
=> ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% subcringE(1)
thf(fact_717_ring_Ospace__subgroup__props_I2_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ E ) ) ) ) ).
% ring.space_subgroup_props(2)
thf(fact_718_ring_Ospace__subgroup__props_I2_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ E ) ) ) ) ).
% ring.space_subgroup_props(2)
thf(fact_719_ring_Ospace__subgroup__props_I3_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int,V1: set_int,V22: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( ( member_set_int @ V1 @ E )
=> ( ( member_set_int @ V22 @ E )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R2 @ V1 @ V22 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(3)
thf(fact_720_ring_Ospace__subgroup__props_I3_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat,V1: nat,V22: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( ( member_nat @ V1 @ E )
=> ( ( member_nat @ V22 @ E )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ V1 @ V22 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(3)
thf(fact_721_ring_Ospace__subgroup__props_I5_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int,K: set_int,V3: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( ( member_set_int @ K @ K2 )
=> ( ( member_set_int @ V3 @ E )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ K @ V3 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(5)
thf(fact_722_ring_Ospace__subgroup__props_I5_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat,K: nat,V3: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( ( member_nat @ K @ K2 )
=> ( ( member_nat @ V3 @ E )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ K @ V3 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(5)
thf(fact_723_ring_Ospace__subgroup__props_I4_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int,V3: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( ( member_set_int @ V3 @ E )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R2 @ V3 ) @ E ) ) ) ) ) ).
% ring.space_subgroup_props(4)
thf(fact_724_ring_Ospace__subgroup__props_I4_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat,V3: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( ( member_nat @ V3 @ E )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ R2 @ V3 ) @ E ) ) ) ) ) ).
% ring.space_subgroup_props(4)
thf(fact_725_ring_Ounique__dimension,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd8246663962306818995t_unit @ R2 @ K2 @ E )
=> ? [X2: nat] :
( ( embedd646006463343340164t_unit @ R2 @ X2 @ K2 @ E )
& ! [Y6: nat] :
( ( embedd646006463343340164t_unit @ R2 @ Y6 @ K2 @ E )
=> ( Y6 = X2 ) ) ) ) ) ) ).
% ring.unique_dimension
thf(fact_726_ring_Ounique__dimension,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd6096501799845681477t_unit @ R2 @ K2 @ E )
=> ? [X2: nat] :
( ( embedd5688180257602933782t_unit @ R2 @ X2 @ K2 @ E )
& ! [Y6: nat] :
( ( embedd5688180257602933782t_unit @ R2 @ Y6 @ K2 @ E )
=> ( Y6 = X2 ) ) ) ) ) ) ).
% ring.unique_dimension
thf(fact_727_ring_OsubcringI,axiom,
! [R2: partia4934656038542163276t_unit,H2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subrin7689096310803670856t_unit @ H2 @ R2 )
=> ( ! [H12: set_int,H23: set_int] :
( ( member_set_int @ H12 @ H2 )
=> ( ( member_set_int @ H23 @ H2 )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ H12 @ H23 )
= ( mult_s3864001451298473021t_unit @ R2 @ H23 @ H12 ) ) ) )
=> ( subcri1024317279029940167t_unit @ H2 @ R2 ) ) ) ) ).
% ring.subcringI
thf(fact_728_ring_OsubcringI,axiom,
! [R2: partia4692342223508353374t_unit,H2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subrin2893992908230074586t_unit @ H2 @ R2 )
=> ( ! [H12: nat,H23: nat] :
( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ H12 @ H23 )
= ( mult_n6028127365542633569t_unit @ R2 @ H23 @ H12 ) ) ) )
=> ( subcri1627753237249443161t_unit @ H2 @ R2 ) ) ) ) ).
% ring.subcringI
thf(fact_729_ring_Ospace__subgroup__props_I1_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( ord_less_eq_set_nat @ E @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% ring.space_subgroup_props(1)
thf(fact_730_ring_Ospace__subgroup__props_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% ring.space_subgroup_props(1)
thf(fact_731_ring_Ospace__subgroup__props_I6_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat,K: nat,A: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ K @ A ) @ E )
=> ( member_nat @ A @ E ) ) ) ) ) ) ) ).
% ring.space_subgroup_props(6)
thf(fact_732_ring_Ospace__subgroup__props_I6_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int,K: set_int,A: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( ( member_set_int @ K @ ( minus_8897228262479074673et_int @ K2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ K @ A ) @ E )
=> ( member_set_int @ A @ E ) ) ) ) ) ) ) ).
% ring.space_subgroup_props(6)
thf(fact_733_s_OsubdomainI,axiom,
! [H2: set_nat] :
( ( subcri1627753237249443161t_unit @ H2 @ ( mod_ring @ n ) )
=> ( ( ( one_na902338870878123981t_unit @ ( mod_ring @ n ) )
!= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
=> ( ! [H12: nat,H23: nat] :
( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ H12 @ H23 )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
=> ( ( H12
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
| ( H23
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ) )
=> ( subdom2148668005855505734t_unit @ H2 @ ( mod_ring @ n ) ) ) ) ) ).
% s.subdomainI
thf(fact_734_s_Oa__rcosetsI,axiom,
! [H2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_set_nat @ ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ H2 @ X ) @ ( a_RCOS6328597822339572043t_unit @ ( mod_ring @ n ) @ H2 ) ) ) ) ).
% s.a_rcosetsI
thf(fact_735_s_Odimension__zero,axiom,
! [K2: set_nat,E: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ zero_zero_nat @ K2 @ E )
=> ( E
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) ) ) ).
% s.dimension_zero
thf(fact_736_s_Odimension__direct__sum__space,axiom,
! [K2: set_nat,N: nat,E: set_nat,M: nat,F2: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ M @ K2 @ F2 )
=> ( ( ( inf_inf_set_nat @ E @ F2 )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) )
=> ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ ( plus_plus_nat @ N @ M ) @ K2 @ ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ E @ F2 ) ) ) ) ) ) ).
% s.dimension_direct_sum_space
thf(fact_737_s_Ozero__dim,axiom,
! [K2: set_nat] : ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ zero_zero_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) ).
% s.zero_dim
thf(fact_738_Compl__anti__mono,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B3 ) @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_739_Compl__anti__mono,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ord_le6893508408891458716et_nat @ ( uminus613421341184616069et_nat @ B3 ) @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_740_Compl__subset__Compl__iff,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B3 ) )
= ( ord_less_eq_set_nat @ B3 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_741_Compl__subset__Compl__iff,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( uminus613421341184616069et_nat @ A2 ) @ ( uminus613421341184616069et_nat @ B3 ) )
= ( ord_le6893508408891458716et_nat @ B3 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_742_Int__iff,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( inf_in1752217752563533465et_int @ A2 @ B3 ) )
= ( ( member_nat_set_int @ C @ A2 )
& ( member_nat_set_int @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_743_Int__iff,axiom,
! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B3 ) )
= ( ( member_set_nat @ C @ A2 )
& ( member_set_nat @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_744_Int__iff,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B3 ) )
= ( ( member_nat @ C @ A2 )
& ( member_nat @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_745_IntI,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ C @ A2 )
=> ( ( member_nat_set_int @ C @ B3 )
=> ( member_nat_set_int @ C @ ( inf_in1752217752563533465et_int @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_746_IntI,axiom,
! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ A2 )
=> ( ( member_set_nat @ C @ B3 )
=> ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_747_IntI,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_748_s_Osubring__inter,axiom,
! [I2: set_nat,J2: set_nat] :
( ( subrin2893992908230074586t_unit @ I2 @ ( mod_ring @ n ) )
=> ( ( subrin2893992908230074586t_unit @ J2 @ ( mod_ring @ n ) )
=> ( subrin2893992908230074586t_unit @ ( inf_inf_set_nat @ I2 @ J2 ) @ ( mod_ring @ n ) ) ) ) ).
% s.subring_inter
thf(fact_749_s_Osubalgebra__inter,axiom,
! [K2: set_nat,V: set_nat,V4: set_nat] :
( ( embedd2609395410403458802t_unit @ K2 @ V @ ( mod_ring @ n ) )
=> ( ( embedd2609395410403458802t_unit @ K2 @ V4 @ ( mod_ring @ n ) )
=> ( embedd2609395410403458802t_unit @ K2 @ ( inf_inf_set_nat @ V @ V4 ) @ ( mod_ring @ n ) ) ) ) ).
% s.subalgebra_inter
thf(fact_750_s_Osubcring__inter,axiom,
! [I2: set_nat,J2: set_nat] :
( ( subcri1627753237249443161t_unit @ I2 @ ( mod_ring @ n ) )
=> ( ( subcri1627753237249443161t_unit @ J2 @ ( mod_ring @ n ) )
=> ( subcri1627753237249443161t_unit @ ( inf_inf_set_nat @ I2 @ J2 ) @ ( mod_ring @ n ) ) ) ) ).
% s.subcring_inter
thf(fact_751_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_752_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_753_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_754_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_755_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_756_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_757_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_758_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_759_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_760_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_761_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_762_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_763_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_764_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_765_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_766_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_767_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_768_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_769_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_770_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_771_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_772_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_773_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_774_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_775_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_776_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_777_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_778_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_779_Int__subset__iff,axiom,
! [C3: set_nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ ( inf_inf_set_nat @ A2 @ B3 ) )
= ( ( ord_less_eq_set_nat @ C3 @ A2 )
& ( ord_less_eq_set_nat @ C3 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_780_Int__subset__iff,axiom,
! [C3: set_set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ ( inf_inf_set_set_nat @ A2 @ B3 ) )
= ( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
& ( ord_le6893508408891458716et_nat @ C3 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_781_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_782_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_783_Compl__disjoint2,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ A2 )
= bot_bot_set_nat ) ).
% Compl_disjoint2
thf(fact_784_Compl__disjoint,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= bot_bot_set_nat ) ).
% Compl_disjoint
thf(fact_785_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_786_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_787_Int__insert__left__if0,axiom,
! [A: nat > set_int,C3: set_nat_set_int,B3: set_nat_set_int] :
( ~ ( member_nat_set_int @ A @ C3 )
=> ( ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ B3 ) @ C3 )
= ( inf_in1752217752563533465et_int @ B3 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_788_Int__insert__left__if0,axiom,
! [A: set_nat,C3: set_set_nat,B3: set_set_nat] :
( ~ ( member_set_nat @ A @ C3 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B3 ) @ C3 )
= ( inf_inf_set_set_nat @ B3 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_789_Int__insert__left__if0,axiom,
! [A: nat,C3: set_nat,B3: set_nat] :
( ~ ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B3 ) @ C3 )
= ( inf_inf_set_nat @ B3 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_790_Int__insert__left__if1,axiom,
! [A: nat > set_int,C3: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ A @ C3 )
=> ( ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ B3 ) @ C3 )
= ( insert_nat_set_int @ A @ ( inf_in1752217752563533465et_int @ B3 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_791_Int__insert__left__if1,axiom,
! [A: set_nat,C3: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ A @ C3 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B3 ) @ C3 )
= ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ B3 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_792_Int__insert__left__if1,axiom,
! [A: nat,C3: set_nat,B3: set_nat] :
( ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B3 ) @ C3 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B3 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_793_insert__inter__insert,axiom,
! [A: nat,A2: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ ( insert_nat @ A @ B3 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_794_Int__insert__right__if0,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ~ ( member_nat_set_int @ A @ A2 )
=> ( ( inf_in1752217752563533465et_int @ A2 @ ( insert_nat_set_int @ A @ B3 ) )
= ( inf_in1752217752563533465et_int @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_795_Int__insert__right__if0,axiom,
! [A: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ~ ( member_set_nat @ A @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B3 ) )
= ( inf_inf_set_set_nat @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_796_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B3: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B3 ) )
= ( inf_inf_set_nat @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_797_Int__insert__right__if1,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ A @ A2 )
=> ( ( inf_in1752217752563533465et_int @ A2 @ ( insert_nat_set_int @ A @ B3 ) )
= ( insert_nat_set_int @ A @ ( inf_in1752217752563533465et_int @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_798_Int__insert__right__if1,axiom,
! [A: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B3 ) )
= ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_799_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B3 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_800_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_801_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_802_Diff__Compl,axiom,
! [A2: set_nat,B3: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ B3 ) )
= ( inf_inf_set_nat @ A2 @ B3 ) ) ).
% Diff_Compl
thf(fact_803_s_Odimension__sum__space,axiom,
! [K2: set_nat,N: nat,E: set_nat,M: nat,F2: set_nat,K: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ M @ K2 @ F2 )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ K @ K2 @ ( inf_inf_set_nat @ E @ F2 ) )
=> ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) @ K2 @ ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ E @ F2 ) ) ) ) ) ) ).
% s.dimension_sum_space
thf(fact_804_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_805_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_806_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_807_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_808_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_809_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_810_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_811_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_812_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_813_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_814_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_815_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_816_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_817_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_818_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_819_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_820_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_821_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_822_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_823_subset__Compl__singleton,axiom,
! [A2: set_nat_set_int,B: nat > set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ ( uminus4718767861138198480et_int @ ( insert_nat_set_int @ B @ bot_bo8417611410066262939et_int ) ) )
= ( ~ ( member_nat_set_int @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_824_subset__Compl__singleton,axiom,
! [A2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_825_subset__Compl__singleton,axiom,
! [A2: set_set_nat,B: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
= ( ~ ( member_set_nat @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_826_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_827_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_828_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_829_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_830_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_831_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_832_insert__disjoint_I1_J,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ A2 ) @ B3 )
= bot_bo8417611410066262939et_int )
= ( ~ ( member_nat_set_int @ A @ B3 )
& ( ( inf_in1752217752563533465et_int @ A2 @ B3 )
= bot_bo8417611410066262939et_int ) ) ) ).
% insert_disjoint(1)
thf(fact_833_insert__disjoint_I1_J,axiom,
! [A: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ A2 ) @ B3 )
= bot_bot_set_set_nat )
= ( ~ ( member_set_nat @ A @ B3 )
& ( ( inf_inf_set_set_nat @ A2 @ B3 )
= bot_bot_set_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_834_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B3 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B3 )
& ( ( inf_inf_set_nat @ A2 @ B3 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_835_insert__disjoint_I2_J,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( bot_bo8417611410066262939et_int
= ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ A2 ) @ B3 ) )
= ( ~ ( member_nat_set_int @ A @ B3 )
& ( bot_bo8417611410066262939et_int
= ( inf_in1752217752563533465et_int @ A2 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_836_insert__disjoint_I2_J,axiom,
! [A: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ A2 ) @ B3 ) )
= ( ~ ( member_set_nat @ A @ B3 )
& ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A2 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_837_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B3: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B3 ) )
= ( ~ ( member_nat @ A @ B3 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_838_disjoint__insert_I1_J,axiom,
! [B3: set_nat_set_int,A: nat > set_int,A2: set_nat_set_int] :
( ( ( inf_in1752217752563533465et_int @ B3 @ ( insert_nat_set_int @ A @ A2 ) )
= bot_bo8417611410066262939et_int )
= ( ~ ( member_nat_set_int @ A @ B3 )
& ( ( inf_in1752217752563533465et_int @ B3 @ A2 )
= bot_bo8417611410066262939et_int ) ) ) ).
% disjoint_insert(1)
thf(fact_839_disjoint__insert_I1_J,axiom,
! [B3: set_set_nat,A: set_nat,A2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ B3 @ ( insert_set_nat @ A @ A2 ) )
= bot_bot_set_set_nat )
= ( ~ ( member_set_nat @ A @ B3 )
& ( ( inf_inf_set_set_nat @ B3 @ A2 )
= bot_bot_set_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_840_disjoint__insert_I1_J,axiom,
! [B3: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B3 @ ( insert_nat @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B3 )
& ( ( inf_inf_set_nat @ B3 @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_841_disjoint__insert_I2_J,axiom,
! [A2: set_nat_set_int,B: nat > set_int,B3: set_nat_set_int] :
( ( bot_bo8417611410066262939et_int
= ( inf_in1752217752563533465et_int @ A2 @ ( insert_nat_set_int @ B @ B3 ) ) )
= ( ~ ( member_nat_set_int @ B @ A2 )
& ( bot_bo8417611410066262939et_int
= ( inf_in1752217752563533465et_int @ A2 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_842_disjoint__insert_I2_J,axiom,
! [A2: set_set_nat,B: set_nat,B3: set_set_nat] :
( ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ B @ B3 ) ) )
= ( ~ ( member_set_nat @ B @ A2 )
& ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A2 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_843_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B: nat,B3: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat @ B @ B3 ) ) )
= ( ~ ( member_nat @ B @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_844_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_845_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_846_Diff__disjoint,axiom,
! [A2: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B3 @ A2 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_847_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_848_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_849_Diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( inf_inf_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ) ).
% Diff_eq
thf(fact_850_disjoint__eq__subset__Compl,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ B3 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_851_disjoint__eq__subset__Compl,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A2 @ B3 )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ B3 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_852_Int__Collect__mono,axiom,
! [A2: set_nat_set_int,B3: set_nat_set_int,P: ( nat > set_int ) > $o,Q: ( nat > set_int ) > $o] :
( ( ord_le5995675665013768039et_int @ A2 @ B3 )
=> ( ! [X2: nat > set_int] :
( ( member_nat_set_int @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le5995675665013768039et_int @ ( inf_in1752217752563533465et_int @ A2 @ ( collect_nat_set_int @ P ) ) @ ( inf_in1752217752563533465et_int @ B3 @ ( collect_nat_set_int @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_853_Int__Collect__mono,axiom,
! [A2: set_nat,B3: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B3 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_854_Int__Collect__mono,axiom,
! [A2: set_set_nat,B3: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B3 @ ( collect_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_855_Int__greatest,axiom,
! [C3: set_nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( ( ord_less_eq_set_nat @ C3 @ B3 )
=> ( ord_less_eq_set_nat @ C3 @ ( inf_inf_set_nat @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_856_Int__greatest,axiom,
! [C3: set_set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C3 @ B3 )
=> ( ord_le6893508408891458716et_nat @ C3 @ ( inf_inf_set_set_nat @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_857_Int__absorb2,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( inf_inf_set_nat @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_858_Int__absorb2,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( inf_inf_set_set_nat @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_859_Int__absorb1,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_860_Int__absorb1,axiom,
! [B3: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_861_Int__lower2,axiom,
! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_862_Int__lower2,axiom,
! [A2: set_set_nat,B3: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_863_Int__lower1,axiom,
! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_864_Int__lower1,axiom,
! [A2: set_set_nat,B3: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_865_Int__mono,axiom,
! [A2: set_nat,C3: set_nat,B3: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ B3 @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ ( inf_inf_set_nat @ C3 @ D ) ) ) ) ).
% Int_mono
thf(fact_866_Int__mono,axiom,
! [A2: set_set_nat,C3: set_set_nat,B3: set_set_nat,D: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C3 )
=> ( ( ord_le6893508408891458716et_nat @ B3 @ D )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B3 ) @ ( inf_inf_set_set_nat @ C3 @ D ) ) ) ) ).
% Int_mono
thf(fact_867_disjoint__iff__not__equal,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B3 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_868_Int__empty__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_869_Int__empty__left,axiom,
! [B3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B3 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_870_disjoint__iff,axiom,
! [A2: set_nat_set_int,B3: set_nat_set_int] :
( ( ( inf_in1752217752563533465et_int @ A2 @ B3 )
= bot_bo8417611410066262939et_int )
= ( ! [X3: nat > set_int] :
( ( member_nat_set_int @ X3 @ A2 )
=> ~ ( member_nat_set_int @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_871_disjoint__iff,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A2 @ B3 )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ~ ( member_set_nat @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_872_disjoint__iff,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B3 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_873_Int__emptyI,axiom,
! [A2: set_nat_set_int,B3: set_nat_set_int] :
( ! [X2: nat > set_int] :
( ( member_nat_set_int @ X2 @ A2 )
=> ~ ( member_nat_set_int @ X2 @ B3 ) )
=> ( ( inf_in1752217752563533465et_int @ A2 @ B3 )
= bot_bo8417611410066262939et_int ) ) ).
% Int_emptyI
thf(fact_874_Int__emptyI,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ~ ( member_set_nat @ X2 @ B3 ) )
=> ( ( inf_inf_set_set_nat @ A2 @ B3 )
= bot_bot_set_set_nat ) ) ).
% Int_emptyI
thf(fact_875_Int__emptyI,axiom,
! [A2: set_nat,B3: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ X2 @ B3 ) )
=> ( ( inf_inf_set_nat @ A2 @ B3 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_876_Int__insert__right,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( ( member_nat_set_int @ A @ A2 )
=> ( ( inf_in1752217752563533465et_int @ A2 @ ( insert_nat_set_int @ A @ B3 ) )
= ( insert_nat_set_int @ A @ ( inf_in1752217752563533465et_int @ A2 @ B3 ) ) ) )
& ( ~ ( member_nat_set_int @ A @ A2 )
=> ( ( inf_in1752217752563533465et_int @ A2 @ ( insert_nat_set_int @ A @ B3 ) )
= ( inf_in1752217752563533465et_int @ A2 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_877_Int__insert__right,axiom,
! [A: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( ( member_set_nat @ A @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B3 ) )
= ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ A2 @ B3 ) ) ) )
& ( ~ ( member_set_nat @ A @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B3 ) )
= ( inf_inf_set_set_nat @ A2 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_878_Int__insert__right,axiom,
! [A: nat,A2: set_nat,B3: set_nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B3 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B3 ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B3 ) )
= ( inf_inf_set_nat @ A2 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_879_Int__insert__left,axiom,
! [A: nat > set_int,C3: set_nat_set_int,B3: set_nat_set_int] :
( ( ( member_nat_set_int @ A @ C3 )
=> ( ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ B3 ) @ C3 )
= ( insert_nat_set_int @ A @ ( inf_in1752217752563533465et_int @ B3 @ C3 ) ) ) )
& ( ~ ( member_nat_set_int @ A @ C3 )
=> ( ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ B3 ) @ C3 )
= ( inf_in1752217752563533465et_int @ B3 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_880_Int__insert__left,axiom,
! [A: set_nat,C3: set_set_nat,B3: set_set_nat] :
( ( ( member_set_nat @ A @ C3 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B3 ) @ C3 )
= ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ B3 @ C3 ) ) ) )
& ( ~ ( member_set_nat @ A @ C3 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B3 ) @ C3 )
= ( inf_inf_set_set_nat @ B3 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_881_Int__insert__left,axiom,
! [A: nat,C3: set_nat,B3: set_nat] :
( ( ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B3 ) @ C3 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B3 @ C3 ) ) ) )
& ( ~ ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B3 ) @ C3 )
= ( inf_inf_set_nat @ B3 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_882_Int__Diff,axiom,
! [A2: set_nat,B3: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C3 )
= ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B3 @ C3 ) ) ) ).
% Int_Diff
thf(fact_883_Diff__Int2,axiom,
! [A2: set_nat,C3: set_nat,B3: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C3 ) @ ( inf_inf_set_nat @ B3 @ C3 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C3 ) @ B3 ) ) ).
% Diff_Int2
thf(fact_884_Diff__Diff__Int,axiom,
! [A2: set_nat,B3: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( minus_minus_set_nat @ A2 @ B3 ) )
= ( inf_inf_set_nat @ A2 @ B3 ) ) ).
% Diff_Diff_Int
thf(fact_885_Diff__Int__distrib,axiom,
! [C3: set_nat,A2: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ C3 @ ( minus_minus_set_nat @ A2 @ B3 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C3 @ A2 ) @ ( inf_inf_set_nat @ C3 @ B3 ) ) ) ).
% Diff_Int_distrib
thf(fact_886_Diff__Int__distrib2,axiom,
! [A2: set_nat,B3: set_nat,C3: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ C3 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C3 ) @ ( inf_inf_set_nat @ B3 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_887_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_888_Int__left__commute,axiom,
! [A2: set_nat,B3: set_nat,C3: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C3 ) )
= ( inf_inf_set_nat @ B3 @ ( inf_inf_set_nat @ A2 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_889_Int__left__absorb,axiom,
! [A2: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B3 ) )
= ( inf_inf_set_nat @ A2 @ B3 ) ) ).
% Int_left_absorb
thf(fact_890_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( inf_inf_set_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_891_Int__absorb,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_892_Int__assoc,axiom,
! [A2: set_nat,B3: set_nat,C3: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C3 )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C3 ) ) ) ).
% Int_assoc
thf(fact_893_IntD2,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( inf_in1752217752563533465et_int @ A2 @ B3 ) )
=> ( member_nat_set_int @ C @ B3 ) ) ).
% IntD2
thf(fact_894_IntD2,axiom,
! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B3 ) )
=> ( member_set_nat @ C @ B3 ) ) ).
% IntD2
thf(fact_895_IntD2,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B3 ) )
=> ( member_nat @ C @ B3 ) ) ).
% IntD2
thf(fact_896_IntD1,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( inf_in1752217752563533465et_int @ A2 @ B3 ) )
=> ( member_nat_set_int @ C @ A2 ) ) ).
% IntD1
thf(fact_897_IntD1,axiom,
! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B3 ) )
=> ( member_set_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_898_IntD1,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B3 ) )
=> ( member_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_899_IntE,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( inf_in1752217752563533465et_int @ A2 @ B3 ) )
=> ~ ( ( member_nat_set_int @ C @ A2 )
=> ~ ( member_nat_set_int @ C @ B3 ) ) ) ).
% IntE
thf(fact_900_IntE,axiom,
! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B3 ) )
=> ~ ( ( member_set_nat @ C @ A2 )
=> ~ ( member_set_nat @ C @ B3 ) ) ) ).
% IntE
thf(fact_901_IntE,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B3 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ~ ( member_nat @ C @ B3 ) ) ) ).
% IntE
thf(fact_902_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_903_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_904_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_905_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_906_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_907_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_908_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_909_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A5: int,B6: int] :
( ( minus_minus_int @ A5 @ B6 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_910_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_911_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_912_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_913_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_914_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_915_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_916_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_917_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_918_subdomainE_I2_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ H2 ) ) ).
% subdomainE(2)
thf(fact_919_subdomainE_I7_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H1: nat,H22: nat] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ H1 @ H22 ) @ H2 ) ) ) ) ).
% subdomainE(7)
thf(fact_920_subdomainE_I4_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( H2 != bot_bot_set_nat ) ) ).
% subdomainE(4)
thf(fact_921_subdomainE_I6_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H1: nat,H22: nat] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ H1 @ H22 ) @ H2 ) ) ) ) ).
% subdomainE(6)
thf(fact_922_subdomainE_I8_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H1: nat,H22: nat] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ H1 @ H22 )
= ( mult_n6028127365542633569t_unit @ R2 @ H22 @ H1 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_923_subfield_Oaxioms_I1_J,axiom,
! [K2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( subdom2148668005855505734t_unit @ K2 @ R2 ) ) ).
% subfield.axioms(1)
thf(fact_924_subdomainE_I3_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ R2 ) @ H2 ) ) ).
% subdomainE(3)
thf(fact_925_subdomainE_I5_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H: nat] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( ( member_nat @ H @ H2 )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ R2 @ H ) @ H2 ) ) ) ).
% subdomainE(5)
thf(fact_926_subset__Compl__self__eq,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_927_subset__Compl__self__eq,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_928_Diff__triv,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B3 )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A2 @ B3 )
= A2 ) ) ).
% Diff_triv
thf(fact_929_Int__Diff__disjoint,axiom,
! [A2: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ ( minus_minus_set_nat @ A2 @ B3 ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_930_ring_Osubring__inter,axiom,
! [R2: partia4934656038542163276t_unit,I2: set_set_int,J2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subrin7689096310803670856t_unit @ I2 @ R2 )
=> ( ( subrin7689096310803670856t_unit @ J2 @ R2 )
=> ( subrin7689096310803670856t_unit @ ( inf_inf_set_set_int @ I2 @ J2 ) @ R2 ) ) ) ) ).
% ring.subring_inter
thf(fact_931_ring_Osubring__inter,axiom,
! [R2: partia4692342223508353374t_unit,I2: set_nat,J2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subrin2893992908230074586t_unit @ I2 @ R2 )
=> ( ( subrin2893992908230074586t_unit @ J2 @ R2 )
=> ( subrin2893992908230074586t_unit @ ( inf_inf_set_nat @ I2 @ J2 ) @ R2 ) ) ) ) ).
% ring.subring_inter
thf(fact_932_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_933_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_934_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_935_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_936_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_937_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_938_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_939_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_940_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_941_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_942_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_943_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_944_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_945_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_946_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_947_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_948_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B6: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B6 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_949_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_950_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_951_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_952_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_953_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_954_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_955_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_956_ring_Osubalgebra__inter,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,V: set_set_int,V4: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( embedd2743979684206749024t_unit @ K2 @ V @ R2 )
=> ( ( embedd2743979684206749024t_unit @ K2 @ V4 @ R2 )
=> ( embedd2743979684206749024t_unit @ K2 @ ( inf_inf_set_set_int @ V @ V4 ) @ R2 ) ) ) ) ).
% ring.subalgebra_inter
thf(fact_957_ring_Osubalgebra__inter,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,V: set_nat,V4: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( embedd2609395410403458802t_unit @ K2 @ V @ R2 )
=> ( ( embedd2609395410403458802t_unit @ K2 @ V4 @ R2 )
=> ( embedd2609395410403458802t_unit @ K2 @ ( inf_inf_set_nat @ V @ V4 ) @ R2 ) ) ) ) ).
% ring.subalgebra_inter
thf(fact_958_ring_Osubcring__inter,axiom,
! [R2: partia4934656038542163276t_unit,I2: set_set_int,J2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subcri1024317279029940167t_unit @ I2 @ R2 )
=> ( ( subcri1024317279029940167t_unit @ J2 @ R2 )
=> ( subcri1024317279029940167t_unit @ ( inf_inf_set_set_int @ I2 @ J2 ) @ R2 ) ) ) ) ).
% ring.subcring_inter
thf(fact_959_ring_Osubcring__inter,axiom,
! [R2: partia4692342223508353374t_unit,I2: set_nat,J2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subcri1627753237249443161t_unit @ I2 @ R2 )
=> ( ( subcri1627753237249443161t_unit @ J2 @ R2 )
=> ( subcri1627753237249443161t_unit @ ( inf_inf_set_nat @ I2 @ J2 ) @ R2 ) ) ) ) ).
% ring.subcring_inter
thf(fact_960_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_961_subdomainE_I1_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% subdomainE(1)
thf(fact_962_subdomainE_I1_J,axiom,
! [H2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( subdom1520866149873910708t_unit @ H2 @ R2 )
=> ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% subdomainE(1)
thf(fact_963_subdomain_Osubintegral,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H1: nat,H22: nat] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( ( ( mult_n6028127365542633569t_unit @ R2 @ H1 @ H22 )
= ( zero_n5149899317435570679t_unit @ R2 ) )
=> ( ( H1
= ( zero_n5149899317435570679t_unit @ R2 ) )
| ( H22
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_964_subdomain_Osub__one__not__zero,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( ( one_na902338870878123981t_unit @ R2 )
!= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_965_Compl__insert,axiom,
! [X: nat,A2: set_nat] :
( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A2 ) )
= ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% Compl_insert
thf(fact_966_ring_Ozero__dim,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( embedd646006463343340164t_unit @ R2 @ zero_zero_nat @ K2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) ) ).
% ring.zero_dim
thf(fact_967_ring_Ozero__dim,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( embedd5688180257602933782t_unit @ R2 @ zero_zero_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) ) ).
% ring.zero_dim
thf(fact_968_ring_Odimension__sum__space,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int,M: nat,F2: set_set_int,K: nat] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( ( embedd646006463343340164t_unit @ R2 @ M @ K2 @ F2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ K @ K2 @ ( inf_inf_set_set_int @ E @ F2 ) )
=> ( embedd646006463343340164t_unit @ R2 @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) @ K2 @ ( set_ad273131178244904872t_unit @ R2 @ E @ F2 ) ) ) ) ) ) ) ).
% ring.dimension_sum_space
thf(fact_969_ring_Odimension__sum__space,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat,M: nat,F2: set_nat,K: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( ( embedd5688180257602933782t_unit @ R2 @ M @ K2 @ F2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ K @ K2 @ ( inf_inf_set_nat @ E @ F2 ) )
=> ( embedd5688180257602933782t_unit @ R2 @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) @ K2 @ ( set_ad320919470248169786t_unit @ R2 @ E @ F2 ) ) ) ) ) ) ) ).
% ring.dimension_sum_space
thf(fact_970_ring_Odimension__zero,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ zero_zero_nat @ K2 @ E )
=> ( E
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) ) ) ) ).
% ring.dimension_zero
thf(fact_971_ring_Odimension__zero,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ zero_zero_nat @ K2 @ E )
=> ( E
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) ) ) ) ).
% ring.dimension_zero
thf(fact_972_ring_OsubdomainI,axiom,
! [R2: partia4934656038542163276t_unit,H2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subcri1024317279029940167t_unit @ H2 @ R2 )
=> ( ( ( one_se8065767436706823081t_unit @ R2 )
!= ( zero_s6269048424454532197t_unit @ R2 ) )
=> ( ! [H12: set_int,H23: set_int] :
( ( member_set_int @ H12 @ H2 )
=> ( ( member_set_int @ H23 @ H2 )
=> ( ( ( mult_s3864001451298473021t_unit @ R2 @ H12 @ H23 )
= ( zero_s6269048424454532197t_unit @ R2 ) )
=> ( ( H12
= ( zero_s6269048424454532197t_unit @ R2 ) )
| ( H23
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ) )
=> ( subdom1520866149873910708t_unit @ H2 @ R2 ) ) ) ) ) ).
% ring.subdomainI
thf(fact_973_ring_OsubdomainI,axiom,
! [R2: partia4692342223508353374t_unit,H2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( ( ( one_na902338870878123981t_unit @ R2 )
!= ( zero_n5149899317435570679t_unit @ R2 ) )
=> ( ! [H12: nat,H23: nat] :
( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( ( ( mult_n6028127365542633569t_unit @ R2 @ H12 @ H23 )
= ( zero_n5149899317435570679t_unit @ R2 ) )
=> ( ( H12
= ( zero_n5149899317435570679t_unit @ R2 ) )
| ( H23
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ) )
=> ( subdom2148668005855505734t_unit @ H2 @ R2 ) ) ) ) ) ).
% ring.subdomainI
thf(fact_974_ring_Odimension__direct__sum__space,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int,M: nat,F2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( ( embedd646006463343340164t_unit @ R2 @ M @ K2 @ F2 )
=> ( ( ( inf_inf_set_set_int @ E @ F2 )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) )
=> ( embedd646006463343340164t_unit @ R2 @ ( plus_plus_nat @ N @ M ) @ K2 @ ( set_ad273131178244904872t_unit @ R2 @ E @ F2 ) ) ) ) ) ) ) ).
% ring.dimension_direct_sum_space
thf(fact_975_ring_Odimension__direct__sum__space,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat,M: nat,F2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( ( embedd5688180257602933782t_unit @ R2 @ M @ K2 @ F2 )
=> ( ( ( inf_inf_set_nat @ E @ F2 )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) )
=> ( embedd5688180257602933782t_unit @ R2 @ ( plus_plus_nat @ N @ M ) @ K2 @ ( set_ad320919470248169786t_unit @ R2 @ E @ F2 ) ) ) ) ) ) ) ).
% ring.dimension_direct_sum_space
thf(fact_976_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_977_s_Odimension_Osimps,axiom,
! [A1: nat,A22: set_nat,A32: set_nat] :
( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ A1 @ A22 @ A32 )
= ( ? [K5: set_nat] :
( ( A1 = zero_zero_nat )
& ( A22 = K5 )
& ( A32
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) )
| ? [V5: nat,E2: set_nat,N2: nat,K5: set_nat] :
( ( A1
= ( suc @ N2 ) )
& ( A22 = K5 )
& ( A32
= ( embedd838748496991043025t_unit @ ( mod_ring @ n ) @ K5 @ V5 @ E2 ) )
& ( member_nat @ V5 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
& ~ ( member_nat @ V5 @ E2 )
& ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N2 @ K5 @ E2 ) ) ) ) ).
% s.dimension.simps
thf(fact_978_s_Odimension_Ocases,axiom,
! [A1: nat,A22: set_nat,A32: set_nat] :
( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) )
=> ~ ! [V2: nat,E3: set_nat,N3: nat] :
( ( A1
= ( suc @ N3 ) )
=> ( ( A32
= ( embedd838748496991043025t_unit @ ( mod_ring @ n ) @ A22 @ V2 @ E3 ) )
=> ( ( member_nat @ V2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ~ ( member_nat @ V2 @ E3 )
=> ~ ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N3 @ A22 @ E3 ) ) ) ) ) ) ) ).
% s.dimension.cases
thf(fact_979_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_980_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_981_ComplI,axiom,
! [C: nat,A2: set_nat] :
( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% ComplI
thf(fact_982_ComplI,axiom,
! [C: nat > set_int,A2: set_nat_set_int] :
( ~ ( member_nat_set_int @ C @ A2 )
=> ( member_nat_set_int @ C @ ( uminus4718767861138198480et_int @ A2 ) ) ) ).
% ComplI
thf(fact_983_ComplI,axiom,
! [C: set_nat,A2: set_set_nat] :
( ~ ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).
% ComplI
thf(fact_984_Compl__iff,axiom,
! [C: nat,A2: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( ~ ( member_nat @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_985_Compl__iff,axiom,
! [C: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( uminus4718767861138198480et_int @ A2 ) )
= ( ~ ( member_nat_set_int @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_986_Compl__iff,axiom,
! [C: set_nat,A2: set_set_nat] :
( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
= ( ~ ( member_set_nat @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_987_s_OSuc__dim,axiom,
! [V3: nat,E: set_nat,N: nat,K2: set_nat] :
( ( member_nat @ V3 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ~ ( member_nat @ V3 @ E )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ ( suc @ N ) @ K2 @ ( embedd838748496991043025t_unit @ ( mod_ring @ n ) @ K2 @ V3 @ E ) ) ) ) ) ).
% s.Suc_dim
thf(fact_988_s_Odimension__backwards,axiom,
! [K2: set_nat,N: nat,E: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ ( suc @ N ) @ K2 @ E )
=> ? [X2: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
& ? [E4: set_nat] :
( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E4 )
& ~ ( member_nat @ X2 @ E4 )
& ( E
= ( embedd838748496991043025t_unit @ ( mod_ring @ n ) @ K2 @ X2 @ E4 ) ) ) ) ) ) ).
% s.dimension_backwards
thf(fact_989_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_990_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_991_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_992_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_993_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_994_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_995_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_996_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_997_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_998_ComplD,axiom,
! [C: nat,A2: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
=> ~ ( member_nat @ C @ A2 ) ) ).
% ComplD
thf(fact_999_ComplD,axiom,
! [C: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( uminus4718767861138198480et_int @ A2 ) )
=> ~ ( member_nat_set_int @ C @ A2 ) ) ).
% ComplD
thf(fact_1000_ComplD,axiom,
! [C: set_nat,A2: set_set_nat] :
( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
=> ~ ( member_set_nat @ C @ A2 ) ) ).
% ComplD
thf(fact_1001_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1002_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1003_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1004_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1005_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1006_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y5: nat] : ( P @ zero_zero_nat @ ( suc @ Y5 ) )
=> ( ! [X2: nat,Y5: nat] :
( ( P @ X2 @ Y5 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y5 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1007_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1008_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1009_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1010_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1011_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1012_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1013_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1014_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1015_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1016_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1017_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1018_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1019_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1020_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1021_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1022_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1023_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R2 @ X2 @ X2 )
=> ( ! [X2: nat,Y5: nat,Z3: nat] :
( ( R2 @ X2 @ Y5 )
=> ( ( R2 @ Y5 @ Z3 )
=> ( R2 @ X2 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1024_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1025_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1026_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1027_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1028_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1029_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1030_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1031_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1032_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1033_int__cases,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_1034_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_1035_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1036_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1037_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1038_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1039_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1040_lift__Suc__antimono__le,axiom,
! [F: nat > set_set_nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_le6893508408891458716et_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1041_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1042_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1043_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1044_lift__Suc__mono__le,axiom,
! [F: nat > set_set_nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_le6893508408891458716et_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1045_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_1046_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_1047_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1048_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1049_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1050_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_1051_ring_OSuc__dim,axiom,
! [R2: partia4934656038542163276t_unit,V3: set_int,E: set_set_int,N: nat,K2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ V3 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ~ ( member_set_int @ V3 @ E )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( embedd646006463343340164t_unit @ R2 @ ( suc @ N ) @ K2 @ ( embedd4283282269743769663t_unit @ R2 @ K2 @ V3 @ E ) ) ) ) ) ) ).
% ring.Suc_dim
thf(fact_1052_ring_OSuc__dim,axiom,
! [R2: partia4692342223508353374t_unit,V3: nat,E: set_nat,N: nat,K2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ V3 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ~ ( member_nat @ V3 @ E )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( embedd5688180257602933782t_unit @ R2 @ ( suc @ N ) @ K2 @ ( embedd838748496991043025t_unit @ R2 @ K2 @ V3 @ E ) ) ) ) ) ) ).
% ring.Suc_dim
thf(fact_1053_ring_Odimension__backwards,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ ( suc @ N ) @ K2 @ E )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R2 ) )
& ? [E4: set_set_int] :
( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E4 )
& ~ ( member_set_int @ X2 @ E4 )
& ( E
= ( embedd4283282269743769663t_unit @ R2 @ K2 @ X2 @ E4 ) ) ) ) ) ) ) ).
% ring.dimension_backwards
thf(fact_1054_ring_Odimension__backwards,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ ( suc @ N ) @ K2 @ E )
=> ? [X2: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R2 ) )
& ? [E4: set_nat] :
( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E4 )
& ~ ( member_nat @ X2 @ E4 )
& ( E
= ( embedd838748496991043025t_unit @ R2 @ K2 @ X2 @ E4 ) ) ) ) ) ) ) ).
% ring.dimension_backwards
thf(fact_1055_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W: int,Z4: int] :
? [N2: nat] :
( Z4
= ( plus_plus_int @ W @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1056_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1057_int__cases2,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_1058_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N3: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_1059_ring_Odimension_Ocases,axiom,
! [R2: partia4934656038542163276t_unit,A1: nat,A22: set_set_int,A32: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) )
=> ~ ! [V2: set_int,E3: set_set_int,N3: nat] :
( ( A1
= ( suc @ N3 ) )
=> ( ( A32
= ( embedd4283282269743769663t_unit @ R2 @ A22 @ V2 @ E3 ) )
=> ( ( member_set_int @ V2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ~ ( member_set_int @ V2 @ E3 )
=> ~ ( embedd646006463343340164t_unit @ R2 @ N3 @ A22 @ E3 ) ) ) ) ) ) ) ) ).
% ring.dimension.cases
thf(fact_1060_ring_Odimension_Ocases,axiom,
! [R2: partia4692342223508353374t_unit,A1: nat,A22: set_nat,A32: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) )
=> ~ ! [V2: nat,E3: set_nat,N3: nat] :
( ( A1
= ( suc @ N3 ) )
=> ( ( A32
= ( embedd838748496991043025t_unit @ R2 @ A22 @ V2 @ E3 ) )
=> ( ( member_nat @ V2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ~ ( member_nat @ V2 @ E3 )
=> ~ ( embedd5688180257602933782t_unit @ R2 @ N3 @ A22 @ E3 ) ) ) ) ) ) ) ) ).
% ring.dimension.cases
thf(fact_1061_ring_Odimension_Osimps,axiom,
! [R2: partia4934656038542163276t_unit,A1: nat,A22: set_set_int,A32: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ A1 @ A22 @ A32 )
= ( ? [K5: set_set_int] :
( ( A1 = zero_zero_nat )
& ( A22 = K5 )
& ( A32
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) )
| ? [V5: set_int,E2: set_set_int,N2: nat,K5: set_set_int] :
( ( A1
= ( suc @ N2 ) )
& ( A22 = K5 )
& ( A32
= ( embedd4283282269743769663t_unit @ R2 @ K5 @ V5 @ E2 ) )
& ( member_set_int @ V5 @ ( partia966996272515721803t_unit @ R2 ) )
& ~ ( member_set_int @ V5 @ E2 )
& ( embedd646006463343340164t_unit @ R2 @ N2 @ K5 @ E2 ) ) ) ) ) ).
% ring.dimension.simps
thf(fact_1062_ring_Odimension_Osimps,axiom,
! [R2: partia4692342223508353374t_unit,A1: nat,A22: set_nat,A32: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ A1 @ A22 @ A32 )
= ( ? [K5: set_nat] :
( ( A1 = zero_zero_nat )
& ( A22 = K5 )
& ( A32
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) )
| ? [V5: nat,E2: set_nat,N2: nat,K5: set_nat] :
( ( A1
= ( suc @ N2 ) )
& ( A22 = K5 )
& ( A32
= ( embedd838748496991043025t_unit @ R2 @ K5 @ V5 @ E2 ) )
& ( member_nat @ V5 @ ( partia3499330772048238685t_unit @ R2 ) )
& ~ ( member_nat @ V5 @ E2 )
& ( embedd5688180257602933782t_unit @ R2 @ N2 @ K5 @ E2 ) ) ) ) ) ).
% ring.dimension.simps
thf(fact_1063_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_1064_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1065_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1066_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1067_inf__compl__bot__right,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_right
thf(fact_1068_compl__le__compl__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
= ( ord_less_eq_set_nat @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_1069_compl__le__compl__iff,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( uminus613421341184616069et_nat @ X ) @ ( uminus613421341184616069et_nat @ Y ) )
= ( ord_le6893508408891458716et_nat @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_1070_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_1071_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_1072_inf__compl__bot__left1,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( inf_inf_set_nat @ X @ Y ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left1
thf(fact_1073_inf__compl__bot__left2,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
= bot_bot_set_nat ) ).
% inf_compl_bot_left2
thf(fact_1074_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1075_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1076_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1077_compl__le__swap2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1078_compl__le__swap2,axiom,
! [Y: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( uminus613421341184616069et_nat @ Y ) @ X )
=> ( ord_le6893508408891458716et_nat @ ( uminus613421341184616069et_nat @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1079_compl__le__swap1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
=> ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1080_compl__le__swap1,axiom,
! [Y: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ ( uminus613421341184616069et_nat @ X ) )
=> ( ord_le6893508408891458716et_nat @ X @ ( uminus613421341184616069et_nat @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1081_compl__mono,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).
% compl_mono
thf(fact_1082_compl__mono,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( uminus613421341184616069et_nat @ Y ) @ ( uminus613421341184616069et_nat @ X ) ) ) ).
% compl_mono
thf(fact_1083_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1084_diff__shunt__var,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ( minus_2163939370556025621et_nat @ X @ Y )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1085_inf__cancel__left1,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ A ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B ) )
= bot_bot_set_nat ) ).
% inf_cancel_left1
thf(fact_1086_inf__cancel__left2,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A ) @ ( inf_inf_set_nat @ X @ B ) )
= bot_bot_set_nat ) ).
% inf_cancel_left2
thf(fact_1087_inf__shunt,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% inf_shunt
thf(fact_1088_inf__shunt,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ( inf_inf_set_set_nat @ X @ Y )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ X @ ( uminus613421341184616069et_nat @ Y ) ) ) ).
% inf_shunt
thf(fact_1089_s_Oadd_Oint__pow__neg__int,axiom,
! [X: nat,N: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ X )
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) ) ) ) ).
% s.add.int_pow_neg_int
thf(fact_1090_s_Orcosets__subset__PowG,axiom,
! [H2: set_nat] :
( ( additi4297497278381856430t_unit @ H2 @ ( mod_ring @ n ) )
=> ( ord_le6893508408891458716et_nat @ ( a_RCOS6328597822339572043t_unit @ ( mod_ring @ n ) @ H2 ) @ ( pow_nat @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.rcosets_subset_PowG
thf(fact_1091_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_1092_s_Oadd_Ogroup__commutes__pow,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ X ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) @ Y )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) ) ) ) ) ) ).
% s.add.group_commutes_pow
thf(fact_1093_s_Oadd_Onat__pow__comm,axiom,
! [X: nat,N: nat,M: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ M @ X ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ M @ X ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) ) ) ) ).
% s.add.nat_pow_comm
thf(fact_1094_s_Oadd_Onat__pow__distrib,axiom,
! [X: nat,Y: nat,N: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ Y ) ) ) ) ) ).
% s.add.nat_pow_distrib
thf(fact_1095_s_Oadd_Opow__mult__distrib,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ X ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ Y ) ) ) ) ) ) ).
% s.add.pow_mult_distrib
thf(fact_1096_s_Oadd__pow__ldistr,axiom,
! [A: nat,B: nat,K: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ K @ A ) @ B )
= ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ K @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ B ) ) ) ) ) ).
% s.add_pow_ldistr
thf(fact_1097_s_Oadd__pow__rdistr,axiom,
! [A: nat,B: nat,K: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ K @ B ) )
= ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ K @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ B ) ) ) ) ) ).
% s.add_pow_rdistr
thf(fact_1098_s_Oadd_Onat__pow__inv,axiom,
! [X: nat,I: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ I @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) )
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ I @ X ) ) ) ) ).
% s.add.nat_pow_inv
thf(fact_1099_s_Oadd_Onat__pow__Suc2,axiom,
! [X: nat,N: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ ( suc @ N ) @ X )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) ) ) ) ).
% s.add.nat_pow_Suc2
thf(fact_1100_s_Oadd_Opow__eq__div2,axiom,
! [X: nat,M: nat,N: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ M @ X )
= ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) )
=> ( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ ( minus_minus_nat @ M @ N ) @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.add.pow_eq_div2
thf(fact_1101_s_Oadd_Onat__pow__mult,axiom,
! [X: nat,N: nat,M: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ M @ X ) )
= ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ ( plus_plus_nat @ N @ M ) @ X ) ) ) ).
% s.add.nat_pow_mult
thf(fact_1102_le__inf__iff,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( ( ord_less_eq_set_nat @ X @ Y )
& ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_1103_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_1104_le__inf__iff,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) )
= ( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_1105_le__inf__iff,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) )
= ( ( ord_le6893508408891458716et_nat @ X @ Y )
& ( ord_le6893508408891458716et_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_1106_inf_Obounded__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
= ( ( ord_less_eq_set_nat @ A @ B )
& ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1107_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1108_inf_Obounded__iff,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1109_inf_Obounded__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( inf_inf_set_set_nat @ B @ C ) )
= ( ( ord_le6893508408891458716et_nat @ A @ B )
& ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1110_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_1111_Pow__iff,axiom,
! [A2: set_nat,B3: set_nat] :
( ( member_set_nat @ A2 @ ( pow_nat @ B3 ) )
= ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% Pow_iff
thf(fact_1112_Pow__iff,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( member_set_set_nat @ A2 @ ( pow_set_nat @ B3 ) )
= ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ).
% Pow_iff
thf(fact_1113_PowI,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( member_set_nat @ A2 @ ( pow_nat @ B3 ) ) ) ).
% PowI
thf(fact_1114_PowI,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( member_set_set_nat @ A2 @ ( pow_set_nat @ B3 ) ) ) ).
% PowI
thf(fact_1115_Pow__singleton__iff,axiom,
! [X4: set_nat,Y7: set_nat] :
( ( ( pow_nat @ X4 )
= ( insert_set_nat @ Y7 @ bot_bot_set_set_nat ) )
= ( ( X4 = bot_bot_set_nat )
& ( Y7 = bot_bot_set_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_1116_Pow__empty,axiom,
( ( pow_nat @ bot_bot_set_nat )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_empty
thf(fact_1117_Pow__Int__eq,axiom,
! [A2: set_nat,B3: set_nat] :
( ( pow_nat @ ( inf_inf_set_nat @ A2 @ B3 ) )
= ( inf_inf_set_set_nat @ ( pow_nat @ A2 ) @ ( pow_nat @ B3 ) ) ) ).
% Pow_Int_eq
thf(fact_1118_s_Oadd_Onat__pow__closed,axiom,
! [X: nat,N: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.add.nat_pow_closed
thf(fact_1119_s_Oadd_Onat__pow__one,axiom,
! [N: nat] :
( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ).
% s.add.nat_pow_one
thf(fact_1120_s_Oadd_Onat__pow__0,axiom,
! [X: nat] :
( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ zero_zero_nat @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ).
% s.add.nat_pow_0
thf(fact_1121_s_Oadd_Onat__pow__Suc,axiom,
! [N: nat,X: nat] :
( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ ( suc @ N ) @ X )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) @ X ) ) ).
% s.add.nat_pow_Suc
thf(fact_1122_Pow__mono,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_le6893508408891458716et_nat @ ( pow_nat @ A2 ) @ ( pow_nat @ B3 ) ) ) ).
% Pow_mono
thf(fact_1123_Pow__mono,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ord_le9131159989063066194et_nat @ ( pow_set_nat @ A2 ) @ ( pow_set_nat @ B3 ) ) ) ).
% Pow_mono
thf(fact_1124_PowD,axiom,
! [A2: set_nat,B3: set_nat] :
( ( member_set_nat @ A2 @ ( pow_nat @ B3 ) )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% PowD
thf(fact_1125_PowD,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( member_set_set_nat @ A2 @ ( pow_set_nat @ B3 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ).
% PowD
thf(fact_1126_Pow__bottom,axiom,
! [B3: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B3 ) ) ).
% Pow_bottom
thf(fact_1127_Pow__not__empty,axiom,
! [A2: set_nat] :
( ( pow_nat @ A2 )
!= bot_bot_set_set_nat ) ).
% Pow_not_empty
thf(fact_1128_Pow__top,axiom,
! [A2: set_nat] : ( member_set_nat @ A2 @ ( pow_nat @ A2 ) ) ).
% Pow_top
thf(fact_1129_inf_OcoboundedI2,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1130_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1131_inf_OcoboundedI2,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1132_inf_OcoboundedI2,axiom,
! [B: set_set_nat,C: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1133_inf_OcoboundedI1,axiom,
! [A: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1134_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1135_inf_OcoboundedI1,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1136_inf_OcoboundedI1,axiom,
! [A: set_set_nat,C: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1137_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B6: set_nat,A5: set_nat] :
( ( inf_inf_set_nat @ A5 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_1138_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_1139_inf_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [B6: int,A5: int] :
( ( inf_inf_int @ A5 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_1140_inf_Oabsorb__iff2,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B6: set_set_nat,A5: set_set_nat] :
( ( inf_inf_set_set_nat @ A5 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_1141_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( ( inf_inf_set_nat @ A5 @ B6 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_1142_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B6: nat] :
( ( inf_inf_nat @ A5 @ B6 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_1143_inf_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B6: int] :
( ( inf_inf_int @ A5 @ B6 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_1144_inf_Oabsorb__iff1,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B6: set_set_nat] :
( ( inf_inf_set_set_nat @ A5 @ B6 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_1145_inf_Ocobounded2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1146_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1147_inf_Ocobounded2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1148_inf_Ocobounded2,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1149_inf_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_1150_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_1151_inf_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_1152_inf_Ocobounded1,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_1153_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( A5
= ( inf_inf_set_nat @ A5 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_1154_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B6: nat] :
( A5
= ( inf_inf_nat @ A5 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_1155_inf_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B6: int] :
( A5
= ( inf_inf_int @ A5 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_1156_inf_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B6: set_set_nat] :
( A5
= ( inf_inf_set_set_nat @ A5 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_1157_inf__greatest,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Z )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1158_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1159_inf__greatest,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Z )
=> ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1160_inf__greatest,axiom,
! [X: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( ord_le6893508408891458716et_nat @ X @ Z )
=> ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_1161_inf_OboundedI,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1162_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1163_inf_OboundedI,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ A @ C )
=> ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1164_inf_OboundedI,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ C )
=> ( ord_le6893508408891458716et_nat @ A @ ( inf_inf_set_set_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1165_inf_OboundedE,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1166_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1167_inf_OboundedE,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) )
=> ~ ( ( ord_less_eq_int @ A @ B )
=> ~ ( ord_less_eq_int @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1168_inf_OboundedE,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( inf_inf_set_set_nat @ B @ C ) )
=> ~ ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ~ ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1169_inf__absorb2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( inf_inf_set_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1170_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1171_inf__absorb2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( inf_inf_int @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1172_inf__absorb2,axiom,
! [Y: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X )
=> ( ( inf_inf_set_set_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_1173_inf__absorb1,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( inf_inf_set_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1174_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1175_inf__absorb1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( inf_inf_int @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1176_inf__absorb1,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y )
=> ( ( inf_inf_set_set_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1177_inf_Oabsorb2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( inf_inf_int @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1178_inf_Oabsorb2,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( inf_inf_set_set_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1179_s_Otelescopic__base__aux,axiom,
! [K2: set_nat,F2: set_nat,N: nat,E: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( subfie4892355163478727762t_unit @ F2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ F2 )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ one_one_nat @ F2 @ E )
=> ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E ) ) ) ) ) ).
% s.telescopic_base_aux
thf(fact_1180_s_Ogroup__l__invI,axiom,
( ! [X2: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
& ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Xa2 @ X2 )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) )
=> ( group_2402336746480010918t_unit @ ( mod_ring @ n ) ) ) ).
% s.group_l_invI
thf(fact_1181_n__ge__1,axiom,
ord_less_nat @ one_one_nat @ n ).
% n_ge_1
thf(fact_1182_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1183_s_Oadd_Onat__pow__eone,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ one_one_nat @ X )
= X ) ) ).
% s.add.nat_pow_eone
thf(fact_1184_s_Odimension__one,axiom,
! [K2: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ one_one_nat @ K2 @ K2 ) ) ).
% s.dimension_one
thf(fact_1185_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1186_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1187_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1188_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1189_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1190_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1191_n__ge__0,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% n_ge_0
thf(fact_1192_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_1193_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1194_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1195_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1196_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1197_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1198_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1199_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1200_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1201_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1202_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_1203_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1204_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1205_s_Oadd_Oint__pow__1,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ one_one_int @ X )
= X ) ) ).
% s.add.int_pow_1
thf(fact_1206_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1207_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1208_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_1209_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1210_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_1211_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_1212_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1213_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1214_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1215_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1216_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1217_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1218_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1219_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1220_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1221_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1222_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1223_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1224_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1225_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1226_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1227_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1228_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1229_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1230_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1231_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1232_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1233_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1234_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1235_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1236_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1237_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1238_mod__ring__carr,axiom,
! [X: nat,N: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ N ) ) )
= ( ord_less_nat @ X @ N ) ) ).
% mod_ring_carr
thf(fact_1239_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1240_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_1241_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_1242_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1243_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1244_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1245_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1246_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1247_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_1248_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1249_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1250_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_1251_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1252_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1253_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1254_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1255_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1256_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1257_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1258_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1259_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I3 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1260_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1261_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1262_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1263_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1264_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
=> ( P @ I5 ) ) )
= ( ( P @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( P @ I5 ) ) ) ) ).
% All_less_Suc
thf(fact_1265_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1266_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1267_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I5: nat] :
( ( ord_less_nat @ I5 @ ( suc @ N ) )
& ( P @ I5 ) ) )
= ( ( P @ N )
| ? [I5: nat] :
( ( ord_less_nat @ I5 @ N )
& ( P @ I5 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1268_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1269_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1270_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
member_nat @ x @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ).
thf(conj_1,conjecture,
( ( hilber5958887739343024896nt_nat @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zfact_iso_inv @ n ) @ x )
= ( ring_zfact_iso @ n @ x ) ) ).
%------------------------------------------------------------------------------