TPTP Problem File: SLH0900^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Interpolation_Polynomials_HOL_Algebra/0000_Bounded_Degree_Polynomials/prob_00032_001492__16960370_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1388 ( 377 unt; 112 typ; 0 def)
% Number of atoms : 4274 (1334 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 15311 ( 253 ~; 46 |; 221 &;12344 @)
% ( 0 <=>;2447 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 273 ( 273 >; 0 *; 0 +; 0 <<)
% Number of symbols : 101 ( 100 usr; 8 con; 0-4 aty)
% Number of variables : 3437 ( 47 ^;3266 !; 124 ?;3437 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:30:14.589
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J_J,type,
partia2175431115845679010xt_a_b: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Product____Type__Ounit_J_J,type,
partia8223610829204095565t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (100)
thf(sy_c_AbelCoset_OA__RCOSETS_001tf__a_001tf__b,type,
a_RCOSETS_a_b: partia2175431115845679010xt_a_b > set_a > set_set_a ).
thf(sy_c_AbelCoset_Oa__l__coset_001tf__a_001tf__b,type,
a_l_coset_a_b: partia2175431115845679010xt_a_b > a > set_a > set_a ).
thf(sy_c_AbelCoset_Oa__r__coset_001tf__a_001tf__b,type,
a_r_coset_a_b: partia2175431115845679010xt_a_b > set_a > a > set_a ).
thf(sy_c_AbelCoset_Oadditive__subgroup_001tf__a_001tf__b,type,
additi2834746164131130830up_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_AbelCoset_Oset__add_001tf__a_001tf__b,type,
set_add_a_b: partia2175431115845679010xt_a_b > set_a > set_a > set_a ).
thf(sy_c_Bounded__Degree__Polynomials_Obounded__degree__polynomials_001tf__a_001tf__b,type,
bounde2262800523058855161ls_a_b: partia2175431115845679010xt_a_b > nat > set_list_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Product____Type__Ounit_J,type,
partia6735698275553448452t_unit: partia8223610829204095565t_unit > set_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
partia707051561876973205xt_a_b: partia2175431115845679010xt_a_b > set_a ).
thf(sy_c_Coset_Oorder_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
order_a_ring_ext_a_b: partia2175431115845679010xt_a_b > nat ).
thf(sy_c_Divisibility_Oassociated_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
associ5860276527279195403xt_a_b: partia2175431115845679010xt_a_b > a > a > $o ).
thf(sy_c_Divisibility_Oirreducible_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
irredu6211895646901577903xt_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Divisibility_Omonoid__cancel_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
monoid5798828371819920185xt_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Divisibility_Oproperfactor_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
proper19828929941537682xt_a_b: partia2175431115845679010xt_a_b > a > a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Odimension_001tf__a_001tf__b,type,
embedd2795209813406577254on_a_b: partia2175431115845679010xt_a_b > nat > set_a > set_a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Ofinite__dimension_001tf__a_001tf__b,type,
embedd8708762675212832759on_a_b: partia2175431115845679010xt_a_b > set_a > set_a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001tf__a_001tf__b,type,
embedd971793762689825387on_a_b: partia2175431115845679010xt_a_b > set_a > a > set_a > set_a ).
thf(sy_c_Embedded__Algebras_Osubalgebra_001tf__a_001tf__b,type,
embedd9027525575939734154ra_a_b: set_a > set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_Itf__a_J,type,
finite_card_list_a: set_list_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J,type,
finite_card_set_a: set_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Group_OUnits_001tf__a_001t__Product____Type__Ounit,type,
units_a_Product_unit: partia8223610829204095565t_unit > set_a ).
thf(sy_c_Group_OUnits_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
units_a_ring_ext_a_b: partia2175431115845679010xt_a_b > set_a ).
thf(sy_c_Group_Ogroup_001tf__a_001t__Product____Type__Ounit,type,
group_a_Product_unit: partia8223610829204095565t_unit > $o ).
thf(sy_c_Group_Ogroup_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
group_a_ring_ext_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Group_Om__inv_001tf__a_001t__Product____Type__Ounit,type,
m_inv_a_Product_unit: partia8223610829204095565t_unit > a > a ).
thf(sy_c_Group_Om__inv_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
m_inv_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > a ).
thf(sy_c_Group_Omonoid_001tf__a_001t__Product____Type__Ounit,type,
monoid2746444814143937472t_unit: partia8223610829204095565t_unit > $o ).
thf(sy_c_Group_Omonoid_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
monoid8385113658579753027xt_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Product____Type__Ounit,type,
mult_a_Product_unit: partia8223610829204095565t_unit > a > a > a ).
thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
mult_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Group_Omonoid_Oone_001tf__a_001t__Product____Type__Ounit,type,
one_a_Product_unit: partia8223610829204095565t_unit > a ).
thf(sy_c_Group_Omonoid_Oone_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
one_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Group_Ounits__of_001tf__a_001t__Product____Type__Ounit,type,
units_7501539392726747778t_unit: partia8223610829204095565t_unit > partia8223610829204095565t_unit ).
thf(sy_c_Group_Ounits__of_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
units_8174867845824275201xt_a_b: partia2175431115845679010xt_a_b > partia8223610829204095565t_unit ).
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_It__List__Olist_Itf__a_J_J,type,
minus_646659088055828811list_a: set_list_a > set_list_a > set_list_a ).
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_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Ideal_Ocgenideal_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
cgenid547466209912283029xt_a_b: partia2175431115845679010xt_a_b > a > set_a ).
thf(sy_c_Ideal_Ogenideal_001tf__a_001tf__b,type,
genideal_a_b: partia2175431115845679010xt_a_b > set_a > set_a ).
thf(sy_c_Ideal_Oprincipalideal_001tf__a_001tf__b,type,
principalideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
inf_inf_set_list_a: set_list_a > set_list_a > set_list_a ).
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_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bot_bot_set_list_a: set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_QuotRing_Oring__iso_001tf__a_001tf__b_001tf__a_001tf__b,type,
ring_iso_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > set_a_a ).
thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
a_inv_a_b: partia2175431115845679010xt_a_b > a > a ).
thf(sy_c_Ring_Oa__minus_001tf__a_001tf__b,type,
a_minus_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oabelian__group_001tf__a_001tf__b,type,
abelian_group_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oabelian__monoid_001tf__a_001tf__b,type,
abelian_monoid_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Int__Oint,type,
add_pow_a_b_int: partia2175431115845679010xt_a_b > int > a > a ).
thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Nat__Onat,type,
add_pow_a_b_nat: partia2175431115845679010xt_a_b > nat > a > a ).
thf(sy_c_Ring_Oring_001tf__a_001tf__b,type,
ring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
add_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oring_Omore_001tf__a_001tf__b,type,
more_a_b: partia2175431115845679010xt_a_b > b ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring_Oring__hom_001tf__a_001tf__b_001tf__a_001tf__b,type,
ring_hom_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > set_a_a ).
thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
semiring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Osemiring__axioms_001tf__a_001tf__b,type,
semiring_axioms_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_001tf__a_001tf__b,type,
subring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1275)
thf(fact_0_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_1_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_2_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_3_is__abelian__group,axiom,
abelian_group_a_b @ r ).
% is_abelian_group
thf(fact_4_associated__sym,axiom,
! [A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( associ5860276527279195403xt_a_b @ r @ B @ A ) ) ).
% associated_sym
thf(fact_5_monoid__axioms,axiom,
monoid8385113658579753027xt_a_b @ r ).
% monoid_axioms
thf(fact_6_assms,axiom,
finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ).
% assms
thf(fact_7_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_8_units__group,axiom,
group_a_Product_unit @ ( units_8174867845824275201xt_a_b @ r ) ).
% units_group
thf(fact_9_associated__trans,axiom,
! [A: a,B: a,C: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ( associ5860276527279195403xt_a_b @ r @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ C ) ) ) ) ) ).
% associated_trans
thf(fact_10_assoc__subst,axiom,
! [A: a,B: a,F: a > a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ! [A2: a,B2: a] :
( ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ A2 @ B2 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ ( F @ B2 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% assoc_subst
thf(fact_11_associated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ A ) ) ).
% associated_refl
thf(fact_12_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_13_monoid_Ounits__group,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( group_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G ) ) ) ).
% monoid.units_group
thf(fact_14_ring_Oonepideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( principalideal_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% ring.onepideal
thf(fact_15_monoid_Oassoc__subst,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,F: a > a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ! [A2: a,B2: a] :
( ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
& ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
& ( associ5860276527279195403xt_a_b @ G @ A2 @ B2 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ G ) )
& ( member_a @ ( F @ B2 ) @ ( partia707051561876973205xt_a_b @ G ) )
& ( associ5860276527279195403xt_a_b @ G @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ) ).
% monoid.assoc_subst
thf(fact_16_monoid_Oassociated__refl,axiom,
! [G: partia2175431115845679010xt_a_b,A: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A @ A ) ) ) ).
% monoid.associated_refl
thf(fact_17_monoid_Oassociated__trans,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ( associ5860276527279195403xt_a_b @ G @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A @ C ) ) ) ) ) ) ).
% monoid.associated_trans
thf(fact_18_properfactor__cong__r,axiom,
! [X: a,Y: a,Y2: a] :
( ( proper19828929941537682xt_a_b @ r @ X @ Y )
=> ( ( associ5860276527279195403xt_a_b @ r @ Y @ Y2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ X @ Y2 ) ) ) ) ) ) ).
% properfactor_cong_r
thf(fact_19_properfactor__cong__l,axiom,
! [X2: a,X: a,Y: a] :
( ( associ5860276527279195403xt_a_b @ r @ X2 @ X )
=> ( ( proper19828929941537682xt_a_b @ r @ X @ Y )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ X2 @ Y ) ) ) ) ) ) ).
% properfactor_cong_l
thf(fact_20_mult__cong__r,axiom,
! [B: a,B3: a,A: a] :
( ( associ5860276527279195403xt_a_b @ r @ B @ B3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B3 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_21_semiring_Oaxioms_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( monoid8385113658579753027xt_a_b @ R ) ) ).
% semiring.axioms(2)
thf(fact_22_m__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_23_properfactor__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( proper19828929941537682xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% properfactor_prod_r
thf(fact_24_m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_25_ring__hom__mult,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_26_units__of__mult,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( mult_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G ) )
= ( mult_a_ring_ext_a_b @ G ) ) ).
% units_of_mult
thf(fact_27_ring__hom__closed,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_28_monoid_Oproperfactor__prod__r,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( proper19828929941537682xt_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( proper19828929941537682xt_a_b @ G @ A @ ( mult_a_ring_ext_a_b @ G @ B @ C ) ) ) ) ) ) ) ).
% monoid.properfactor_prod_r
thf(fact_29_Group_Ogroup_Oright__cancel,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a,Z: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ Y @ X )
= ( mult_a_Product_unit @ G @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% Group.group.right_cancel
thf(fact_30_Group_Ogroup_Oright__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
= ( mult_a_ring_ext_a_b @ G @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% Group.group.right_cancel
thf(fact_31_ring_Oring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_32_ring_Oring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_33_monoid_Om__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% monoid.m_closed
thf(fact_34_monoid_Om__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ G @ X @ ( mult_a_ring_ext_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).
% monoid.m_assoc
thf(fact_35_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_36_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_37_ring__iso__memE_I2_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_38_ring_Ocgenideal__self,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ R @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_39_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_40_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_41_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_42_Collect__mem__eq,axiom,
! [A3: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_43_monoid_Omult__cong__r,axiom,
! [G: partia2175431115845679010xt_a_b,B: a,B3: a,A: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ B @ B3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( mult_a_ring_ext_a_b @ G @ A @ B3 ) ) ) ) ) ) ) ).
% monoid.mult_cong_r
thf(fact_44_monoid_Oproperfactor__cong__r,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Y2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( proper19828929941537682xt_a_b @ G @ X @ Y )
=> ( ( associ5860276527279195403xt_a_b @ G @ Y @ Y2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( proper19828929941537682xt_a_b @ G @ X @ Y2 ) ) ) ) ) ) ) ).
% monoid.properfactor_cong_r
thf(fact_45_monoid_Oproperfactor__cong__l,axiom,
! [G: partia2175431115845679010xt_a_b,X2: a,X: a,Y: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ X2 @ X )
=> ( ( proper19828929941537682xt_a_b @ G @ X @ Y )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( proper19828929941537682xt_a_b @ G @ X2 @ Y ) ) ) ) ) ) ) ).
% monoid.properfactor_cong_l
thf(fact_46_group_Ois__group,axiom,
! [G: partia8223610829204095565t_unit] :
( ( group_a_Product_unit @ G )
=> ( group_a_Product_unit @ G ) ) ).
% group.is_group
thf(fact_47_group_Ois__group,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( group_a_ring_ext_a_b @ G )
=> ( group_a_ring_ext_a_b @ G ) ) ).
% group.is_group
thf(fact_48_principalideal_Ois__principalideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ( principalideal_a_b @ I2 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_49_group_Ois__monoid,axiom,
! [G: partia8223610829204095565t_unit] :
( ( group_a_Product_unit @ G )
=> ( monoid2746444814143937472t_unit @ G ) ) ).
% group.is_monoid
thf(fact_50_group_Ois__monoid,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( group_a_ring_ext_a_b @ G )
=> ( monoid8385113658579753027xt_a_b @ G ) ) ).
% group.is_monoid
thf(fact_51_monoid_Oassociated__sym,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( associ5860276527279195403xt_a_b @ G @ B @ A ) ) ) ).
% monoid.associated_sym
thf(fact_52_ring_Ois__abelian__group,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( abelian_group_a_b @ R ) ) ).
% ring.is_abelian_group
thf(fact_53_ring__iso__memE_I1_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_54_abelian__group_Oaxioms_I1_J,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ G )
=> ( abelian_monoid_a_b @ G ) ) ).
% abelian_group.axioms(1)
thf(fact_55_ring_Ois__monoid,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( monoid8385113658579753027xt_a_b @ R ) ) ).
% ring.is_monoid
thf(fact_56_semiring_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( abelian_monoid_a_b @ R ) ) ).
% semiring.axioms(1)
thf(fact_57_monoid__cancelI,axiom,
( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A2 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A2 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B2 @ C2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_58_add__pow__ldistr__int,axiom,
! [A: a,B: a,K: int] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ K @ A ) @ B )
= ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_ldistr_int
thf(fact_59_add__pow__rdistr__int,axiom,
! [A: a,B: a,K: int] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ K @ B ) )
= ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_rdistr_int
thf(fact_60_associatedI2_H,axiom,
! [A: a,B: a,U: a] :
( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2'
thf(fact_61_associatedI2,axiom,
! [U: a,A: a,B: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2
thf(fact_62_l__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_63_r__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_64_semiring__def,axiom,
( semiring_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ R2 )
& ( monoid8385113658579753027xt_a_b @ R2 )
& ( semiring_axioms_a_b @ R2 ) ) ) ) ).
% semiring_def
thf(fact_65_semiring_Ointro,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ R )
=> ( ( monoid8385113658579753027xt_a_b @ R )
=> ( ( semiring_axioms_a_b @ R )
=> ( semiring_a_b @ R ) ) ) ) ).
% semiring.intro
thf(fact_66_add__pow__ldistr,axiom,
! [A: a,B: a,K: nat] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_nat @ r @ K @ A ) @ B )
= ( add_pow_a_b_nat @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_ldistr
thf(fact_67_add__pow__rdistr,axiom,
! [A: a,B: a,K: nat] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_nat @ r @ K @ B ) )
= ( add_pow_a_b_nat @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_rdistr
thf(fact_68_a__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_69_a__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) ) ) ) ).
% a_comm
thf(fact_70_a__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
= ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_71_add_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( add_a_b @ r @ A @ C )
= ( add_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.r_cancel
thf(fact_72_add_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( add_a_b @ r @ C @ A )
= ( add_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.l_cancel
thf(fact_73_Units__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_74_Units__assoc,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ).
% Units_assoc
thf(fact_75_add_Opow__mult__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ N @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ N @ Y ) ) ) ) ) ) ).
% add.pow_mult_distrib
thf(fact_76_add_Onat__pow__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ N @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ N @ Y ) ) ) ) ) ).
% add.nat_pow_distrib
thf(fact_77_add_Onat__pow__comm,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ M @ X ) )
= ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ M @ X ) @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ).
% add.nat_pow_comm
thf(fact_78_add_Ogroup__commutes__pow,axiom,
! [X: a,Y: a,N: nat] :
( ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ Y )
= ( add_a_b @ r @ Y @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ) ) ).
% add.group_commutes_pow
thf(fact_79_add_Oint__pow__mult__distrib,axiom,
! [X: a,Y: a,I: int] :
( ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ) ).
% add.int_pow_mult_distrib
thf(fact_80_add_Oint__pow__distrib,axiom,
! [X: a,Y: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ).
% add.int_pow_distrib
thf(fact_81_prod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_82_prod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_83_properfactor__unitE,axiom,
! [U: a,A: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( proper19828929941537682xt_a_b @ r @ A @ U )
=> ~ ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% properfactor_unitE
thf(fact_84_units__of__units,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( units_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G ) )
= ( units_a_ring_ext_a_b @ G ) ) ).
% units_of_units
thf(fact_85_a__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_86_local_Oadd_Oright__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X )
= ( add_a_b @ r @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_87_add_Onat__pow__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.nat_pow_closed
thf(fact_88_add_Oint__pow__closed,axiom,
! [X: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_pow_a_b_int @ r @ I @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.int_pow_closed
thf(fact_89_Units__m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_90_Units__l__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% Units_l_cancel
thf(fact_91_finite__ring__finite__units,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( finite_finite_a @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% finite_ring_finite_units
thf(fact_92_monoid__cancel_Ois__monoid__cancel,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( monoid5798828371819920185xt_a_b @ G ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_93_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_94_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_95_units__of__carrier,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( partia6735698275553448452t_unit @ ( units_8174867845824275201xt_a_b @ G ) )
= ( units_a_ring_ext_a_b @ G ) ) ).
% units_of_carrier
thf(fact_96_monoid__cancel_OassociatedD2,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( units_a_ring_ext_a_b @ G ) )
& ( A
= ( mult_a_ring_ext_a_b @ G @ B @ X4 ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_97_monoid__cancel_OassociatedE2,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ! [U2: a] :
( ( A
= ( mult_a_ring_ext_a_b @ G @ B @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_ring_ext_a_b @ G ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ~ ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_98_monoid__cancel_Oassociated__iff,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ G ) )
& ( A
= ( mult_a_ring_ext_a_b @ G @ B @ X3 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_99_monoid__cancel_Oaxioms_I1_J,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( monoid8385113658579753027xt_a_b @ G ) ) ).
% monoid_cancel.axioms(1)
thf(fact_100_group_OUnits__eq,axiom,
! [G: partia8223610829204095565t_unit] :
( ( group_a_Product_unit @ G )
=> ( ( units_a_Product_unit @ G )
= ( partia6735698275553448452t_unit @ G ) ) ) ).
% group.Units_eq
thf(fact_101_group_OUnits__eq,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( units_a_ring_ext_a_b @ G )
= ( partia707051561876973205xt_a_b @ G ) ) ) ).
% group.Units_eq
thf(fact_102_monoid_OUnits__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% monoid.Units_closed
thf(fact_103_monoid_OUnits__m__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.Units_m_closed
thf(fact_104_ring_Oring__simprules_I22_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_105_ring_Oring__simprules_I10_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y )
= ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_106_ring_Oring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_107_ring_Oring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_108_monoid_OUnits__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A @ B ) ) ) ) ).
% monoid.Units_assoc
thf(fact_109_abelian__groupE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y )
= ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_110_abelian__groupE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_111_abelian__groupE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_112_ring__hom__add,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_113_abelian__monoidE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y )
= ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_114_abelian__monoidE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_115_abelian__monoidE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_116_abelian__monoid_Oa__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X @ Y )
= ( add_a_b @ G @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_117_abelian__monoid_Oa__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( add_a_b @ G @ X @ Y ) @ Z )
= ( add_a_b @ G @ X @ ( add_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_118_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X @ ( add_a_b @ G @ Y @ Z ) )
= ( add_a_b @ G @ Y @ ( add_a_b @ G @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_119_abelian__monoid_Oa__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( add_a_b @ G @ X @ Y ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_120_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_121_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y )
= ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_122_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_123_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_124_ring__iso__memE_I3_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_125_monoid__cancel_Or__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,C: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ A @ C )
= ( mult_a_ring_ext_a_b @ G @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_126_monoid__cancel_Ol__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,C: a,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ C @ A )
= ( mult_a_ring_ext_a_b @ G @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_127_ring_Ofinite__ring__finite__units,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ R ) )
=> ( finite_finite_a @ ( units_a_ring_ext_a_b @ R ) ) ) ) ).
% ring.finite_ring_finite_units
thf(fact_128_monoid_OUnits__l__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
= ( mult_a_ring_ext_a_b @ G @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ).
% monoid.Units_l_cancel
thf(fact_129_monoid_Oprod__unit__l,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ) ).
% monoid.prod_unit_l
thf(fact_130_monoid_Oprod__unit__r,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ) ).
% monoid.prod_unit_r
thf(fact_131_monoid_Oproperfactor__unitE,axiom,
! [G: partia2175431115845679010xt_a_b,U: a,A: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( proper19828929941537682xt_a_b @ G @ A @ U )
=> ~ ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% monoid.properfactor_unitE
thf(fact_132_ring_Oring__simprules_I23_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_133_ring_Oring__simprules_I13_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_134_semiring_Ol__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_135_semiring_Or__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_136_semiring_Oaxioms_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( semiring_axioms_a_b @ R ) ) ).
% semiring.axioms(3)
thf(fact_137_monoid_OassociatedI2,axiom,
! [G: partia2175431115845679010xt_a_b,U: a,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ G @ B @ U ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A @ B ) ) ) ) ) ).
% monoid.associatedI2
thf(fact_138_monoid_OassociatedI2_H,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,U: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( A
= ( mult_a_ring_ext_a_b @ G @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A @ B ) ) ) ) ) ).
% monoid.associatedI2'
thf(fact_139_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,B3: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( mult_a_ring_ext_a_b @ G @ A @ B3 ) )
=> ( associ5860276527279195403xt_a_b @ G @ B @ B3 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_140_monoid_Omonoid__cancelI,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ G @ C2 @ A2 )
= ( mult_a_ring_ext_a_b @ G @ C2 @ B2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ G @ A2 @ C2 )
= ( mult_a_ring_ext_a_b @ G @ B2 @ C2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ G ) ) ) ) ).
% monoid.monoid_cancelI
thf(fact_141_monoid__cancel_Oproperfactor__mult__lI,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( proper19828929941537682xt_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( proper19828929941537682xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ C @ A ) @ ( mult_a_ring_ext_a_b @ G @ C @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_lI
thf(fact_142_monoid__cancel_Oproperfactor__mult__l,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( proper19828929941537682xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ C @ A ) @ ( mult_a_ring_ext_a_b @ G @ C @ B ) )
= ( proper19828929941537682xt_a_b @ G @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_l
thf(fact_143_ringI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ R )
=> ( ( monoid8385113658579753027xt_a_b @ R )
=> ( ! [X4: a,Y3: a,Z2: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X4 @ Y3 ) @ Z2 )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X4 @ Z2 ) @ ( mult_a_ring_ext_a_b @ R @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: a,Y3: a,Z2: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z2 @ ( add_a_b @ R @ X4 @ Y3 ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z2 @ X4 ) @ ( mult_a_ring_ext_a_b @ R @ Z2 @ Y3 ) ) ) ) ) )
=> ( ring_a_b @ R ) ) ) ) ) ).
% ringI
thf(fact_144_ring_Oadd__pow__ldistr__int,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,K: int] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_pow_a_b_int @ R @ K @ A ) @ B )
= ( add_pow_a_b_int @ R @ K @ ( mult_a_ring_ext_a_b @ R @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_ldistr_int
thf(fact_145_ring_Oadd__pow__rdistr__int,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,K: int] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ A @ ( add_pow_a_b_int @ R @ K @ B ) )
= ( add_pow_a_b_int @ R @ K @ ( mult_a_ring_ext_a_b @ R @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_rdistr_int
thf(fact_146_semiring_Oadd__pow__ldistr,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,K: nat] :
( ( semiring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_pow_a_b_nat @ R @ K @ A ) @ B )
= ( add_pow_a_b_nat @ R @ K @ ( mult_a_ring_ext_a_b @ R @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_ldistr
thf(fact_147_semiring_Oadd__pow__rdistr,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,K: nat] :
( ( semiring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ A @ ( add_pow_a_b_nat @ R @ K @ B ) )
= ( add_pow_a_b_nat @ R @ K @ ( mult_a_ring_ext_a_b @ R @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_rdistr
thf(fact_148_irreducible__prod__rI,axiom,
! [A: a,B: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_rI
thf(fact_149_line__extension__mem__iff,axiom,
! [U: a,K2: set_a,A: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ K2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ A ) @ Y4 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_150_add_Onat__pow__mult,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ M @ X ) )
= ( add_pow_a_b_nat @ r @ ( plus_plus_nat @ N @ M ) @ X ) ) ) ).
% add.nat_pow_mult
thf(fact_151_add_Oint__pow__mult,axiom,
! [X: a,I: int,J: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ ( plus_plus_int @ I @ J ) @ X )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ J @ X ) ) ) ) ).
% add.int_pow_mult
thf(fact_152_add_Onat__pow__Suc2,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ ( suc @ N ) @ X )
= ( add_a_b @ r @ X @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ).
% add.nat_pow_Suc2
thf(fact_153_Units__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X4 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_154_Units__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_155_add_Oint__pow__pow,axiom,
! [X: a,M: int,N: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ M @ ( add_pow_a_b_int @ r @ N @ X ) )
= ( add_pow_a_b_int @ r @ ( times_times_int @ N @ M ) @ X ) ) ) ).
% add.int_pow_pow
thf(fact_156_add_Onat__pow__pow,axiom,
! [X: a,M: nat,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ M @ ( add_pow_a_b_nat @ r @ N @ X ) )
= ( add_pow_a_b_nat @ r @ ( times_times_nat @ N @ M ) @ X ) ) ) ).
% add.nat_pow_pow
thf(fact_157_local_Ominus__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_158_add_Oinv__comm,axiom,
! [X: a,Y: a] :
( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_159_add_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X4 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_160_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X4 )
= X4 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_161_add_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X4 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_162_inv__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% inv_unique
thf(fact_163_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X4 )
= X4 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_164_group__l__invI,axiom,
( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ Xa @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) )
=> ( group_a_ring_ext_a_b @ r ) ) ).
% group_l_invI
thf(fact_165_Units__inv__comm,axiom,
! [X: a,Y: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_inv_comm
thf(fact_166_monoid_Oone__closed,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ).
% monoid.one_closed
thf(fact_167_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_168_add_Onat__pow__one,axiom,
! [N: nat] :
( ( add_pow_a_b_nat @ r @ N @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% add.nat_pow_one
thf(fact_169_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_170_add_Oint__pow__one,axiom,
! [Z: int] :
( ( add_pow_a_b_int @ r @ Z @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% add.int_pow_one
thf(fact_171_Units__one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_one_closed
thf(fact_172_monoid_Or__one,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X @ ( one_a_ring_ext_a_b @ G ) )
= X ) ) ) ).
% monoid.r_one
thf(fact_173_monoid_Ol__one,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X )
= X ) ) ) ).
% monoid.l_one
thf(fact_174_add_Ol__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ A )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_175_add_Ol__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ X @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_176_add_Or__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_177_add_Or__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ A @ X ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_178_l__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
= X ) ) ).
% l_zero
thf(fact_179_r__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
= X ) ) ).
% r_zero
thf(fact_180_l__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_181_r__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_182_add_Onat__pow__Suc,axiom,
! [N: nat,X: a] :
( ( add_pow_a_b_nat @ r @ ( suc @ N ) @ X )
= ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ X ) ) ).
% add.nat_pow_Suc
thf(fact_183_l__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
= X ) ) ).
% l_one
thf(fact_184_r__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
= X ) ) ).
% r_one
thf(fact_185_units__of__one,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( one_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G ) )
= ( one_a_ring_ext_a_b @ G ) ) ).
% units_of_one
thf(fact_186_ring__hom__one,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_hom_one
thf(fact_187_ring__iso__memE_I4_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_188_ring_Oring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.ring_simprules(6)
thf(fact_189_ring_Oring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.ring_simprules(2)
thf(fact_190_monoid_OUnits__one__closed,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( units_a_ring_ext_a_b @ G ) ) ) ).
% monoid.Units_one_closed
thf(fact_191_abelian__groupE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% abelian_groupE(2)
thf(fact_192_abelian__monoid_Ozero__closed,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ G )
=> ( member_a @ ( zero_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_193_abelian__monoidE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_194_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_195_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_196_ring__hom__zero,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( ring_a_b @ R )
=> ( ( ring_a_b @ S )
=> ( ( H @ ( zero_a_b @ R ) )
= ( zero_a_b @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_197_groupI,axiom,
! [G: partia8223610829204095565t_unit] :
( ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ ( mult_a_Product_unit @ G @ X4 @ Y3 ) @ ( partia6735698275553448452t_unit @ G ) ) ) )
=> ( ( member_a @ ( one_a_Product_unit @ G ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ! [Z2: a] :
( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X4 @ Y3 ) @ Z2 )
= ( mult_a_Product_unit @ G @ X4 @ ( mult_a_Product_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( one_a_Product_unit @ G ) @ X4 )
= X4 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia6735698275553448452t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ Xa @ X4 )
= ( one_a_Product_unit @ G ) ) ) )
=> ( group_a_Product_unit @ G ) ) ) ) ) ) ).
% groupI
thf(fact_198_groupI,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X4 @ Y3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ! [Z2: a] :
( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X4 @ Y3 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ G @ X4 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X4 )
= X4 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ Xa @ X4 )
= ( one_a_ring_ext_a_b @ G ) ) ) )
=> ( group_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% groupI
thf(fact_199_group_Or__cancel__one_H,axiom,
! [G: partia8223610829204095565t_unit,X: a,A: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( X
= ( mult_a_Product_unit @ G @ A @ X ) )
= ( A
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.r_cancel_one'
thf(fact_200_group_Or__cancel__one_H,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,A: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( X
= ( mult_a_ring_ext_a_b @ G @ A @ X ) )
= ( A
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% group.r_cancel_one'
thf(fact_201_group_Ol__cancel__one_H,axiom,
! [G: partia8223610829204095565t_unit,X: a,A: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( X
= ( mult_a_Product_unit @ G @ X @ A ) )
= ( A
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.l_cancel_one'
thf(fact_202_group_Ol__cancel__one_H,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,A: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( X
= ( mult_a_ring_ext_a_b @ G @ X @ A ) )
= ( A
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% group.l_cancel_one'
thf(fact_203_group_Or__cancel__one,axiom,
! [G: partia8223610829204095565t_unit,X: a,A: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ A @ X )
= X )
= ( A
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.r_cancel_one
thf(fact_204_group_Or__cancel__one,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,A: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ A @ X )
= X )
= ( A
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% group.r_cancel_one
thf(fact_205_group_Ol__cancel__one,axiom,
! [G: partia8223610829204095565t_unit,X: a,A: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ X @ A )
= X )
= ( A
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.l_cancel_one
thf(fact_206_group_Ol__cancel__one,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,A: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X @ A )
= X )
= ( A
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% group.l_cancel_one
thf(fact_207_group_Or__inv__ex,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ X @ X4 )
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% group.r_inv_ex
thf(fact_208_group_Or__inv__ex,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ X @ X4 )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% group.r_inv_ex
thf(fact_209_group_Ol__inv__ex,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ X4 @ X )
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% group.l_inv_ex
thf(fact_210_group_Ol__inv__ex,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ X4 @ X )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% group.l_inv_ex
thf(fact_211_group_Oinv__comm,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a] :
( ( group_a_Product_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ X @ Y )
= ( one_a_Product_unit @ G ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ Y @ X )
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.inv_comm
thf(fact_212_group_Oinv__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% group.inv_comm
thf(fact_213_Group_Omonoid__def,axiom,
( monoid8385113658579753027xt_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b] :
( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G2 @ X3 @ Y4 ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) )
& ! [X3: a,Y4: a,Z3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( mult_a_ring_ext_a_b @ G2 @ ( mult_a_ring_ext_a_b @ G2 @ X3 @ Y4 ) @ Z3 )
= ( mult_a_ring_ext_a_b @ G2 @ X3 @ ( mult_a_ring_ext_a_b @ G2 @ Y4 @ Z3 ) ) ) ) ) )
& ( member_a @ ( one_a_ring_ext_a_b @ G2 ) @ ( partia707051561876973205xt_a_b @ G2 ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( mult_a_ring_ext_a_b @ G2 @ ( one_a_ring_ext_a_b @ G2 ) @ X3 )
= X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( mult_a_ring_ext_a_b @ G2 @ X3 @ ( one_a_ring_ext_a_b @ G2 ) )
= X3 ) ) ) ) ) ).
% Group.monoid_def
thf(fact_214_monoidI,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X4 @ Y3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X4: a,Y3: a,Z2: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X4 @ Y3 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ G @ X4 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X4 )
= X4 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X4 @ ( one_a_ring_ext_a_b @ G ) )
= X4 ) )
=> ( monoid8385113658579753027xt_a_b @ G ) ) ) ) ) ) ).
% monoidI
thf(fact_215_monoid_Oone__unique,axiom,
! [G: partia2175431115845679010xt_a_b,U: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ U @ X4 )
= X4 ) )
=> ( U
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.one_unique
thf(fact_216_monoid_Oinv__unique,axiom,
! [G: partia2175431115845679010xt_a_b,Y: a,X: a,Y2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y2 )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% monoid.inv_unique
thf(fact_217_Group_Omonoid_Ointro,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X4 @ Y3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) )
=> ( ! [X4: a,Y3: a,Z2: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X4 @ Y3 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ G @ X4 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X4 )
= X4 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X4 @ ( one_a_ring_ext_a_b @ G ) )
= X4 ) )
=> ( monoid8385113658579753027xt_a_b @ G ) ) ) ) ) ) ).
% Group.monoid.intro
thf(fact_218_ring_Oring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X )
= X ) ) ) ).
% ring.ring_simprules(12)
thf(fact_219_ring_Oring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% ring.ring_simprules(8)
thf(fact_220_ring_Oring__simprules_I15_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
= X ) ) ) ).
% ring.ring_simprules(15)
thf(fact_221_monoid_OUnits__inv__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% monoid.Units_inv_comm
thf(fact_222_ring_Oring__simprules_I24_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_223_ring_Oring__simprules_I25_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_224_abelian__groupI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X4 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Z2: a] :
( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X4 @ Y3 ) @ Z2 )
= ( add_a_b @ R @ X4 @ ( add_a_b @ R @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X4 @ Y3 )
= ( add_a_b @ R @ Y3 @ X4 ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X4 )
= X4 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( add_a_b @ R @ Xa @ X4 )
= ( zero_a_b @ R ) ) ) )
=> ( abelian_group_a_b @ R ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_225_abelian__groupE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% abelian_groupE(5)
thf(fact_226_abelian__groupE_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( add_a_b @ R @ X4 @ X )
= ( zero_a_b @ R ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_227_abelian__monoidI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X4 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ! [X4: a,Y3: a,Z2: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X4 @ Y3 ) @ Z2 )
= ( add_a_b @ R @ X4 @ ( add_a_b @ R @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X4 )
= X4 ) )
=> ( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X4 @ Y3 )
= ( add_a_b @ R @ Y3 @ X4 ) ) ) )
=> ( abelian_monoid_a_b @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_228_abelian__monoid_Ominus__unique,axiom,
! [G: partia2175431115845679010xt_a_b,Y: a,X: a,Y2: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( ( add_a_b @ G @ Y @ X )
= ( zero_a_b @ G ) )
=> ( ( ( add_a_b @ G @ X @ Y2 )
= ( zero_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_229_abelian__monoid_Or__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X @ ( zero_a_b @ G ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_230_abelian__monoid_Ol__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( zero_a_b @ G ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_231_abelian__monoidE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_232_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_233_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_234_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_235_semiring_Or__null,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.r_null
thf(fact_236_semiring_Ol__null,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.l_null
thf(fact_237_irreducibleD,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( irredu6211895646901577903xt_a_b @ G @ A )
=> ( ( proper19828929941537682xt_a_b @ G @ B @ A )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ).
% irreducibleD
thf(fact_238_irreducibleE,axiom,
! [G: partia2175431115845679010xt_a_b,A: a] :
( ( irredu6211895646901577903xt_a_b @ G @ A )
=> ~ ( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
=> ~ ! [B4: a] :
( ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ G ) )
& ( proper19828929941537682xt_a_b @ G @ B4 @ A ) )
=> ( member_a @ B4 @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ).
% irreducibleE
thf(fact_239_irreducibleI,axiom,
! [A: a,G: partia2175431115845679010xt_a_b] :
( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( proper19828929941537682xt_a_b @ G @ B2 @ A )
=> ( member_a @ B2 @ ( units_a_ring_ext_a_b @ G ) ) ) )
=> ( irredu6211895646901577903xt_a_b @ G @ A ) ) ) ).
% irreducibleI
thf(fact_240_irreducible__def,axiom,
( irredu6211895646901577903xt_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b,A4: a] :
( ~ ( member_a @ A4 @ ( units_a_ring_ext_a_b @ G2 ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( proper19828929941537682xt_a_b @ G2 @ X3 @ A4 )
=> ( member_a @ X3 @ ( units_a_ring_ext_a_b @ G2 ) ) ) ) ) ) ) ).
% irreducible_def
thf(fact_241_monoid_Ogroup__l__invI,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia6735698275553448452t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ Xa @ X4 )
= ( one_a_Product_unit @ G ) ) ) )
=> ( group_a_Product_unit @ G ) ) ) ).
% monoid.group_l_invI
thf(fact_242_monoid_Ogroup__l__invI,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ Xa @ X4 )
= ( one_a_ring_ext_a_b @ G ) ) ) )
=> ( group_a_ring_ext_a_b @ G ) ) ) ).
% monoid.group_l_invI
thf(fact_243_monoid_OUnits__l__inv__ex,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ X4 @ X )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.Units_l_inv_ex
thf(fact_244_monoid_OUnits__r__inv__ex,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ X @ X4 )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.Units_r_inv_ex
thf(fact_245_monoid__cancel_Oirreducible__cong,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,A5: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( irredu6211895646901577903xt_a_b @ G @ A )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ A5 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( irredu6211895646901577903xt_a_b @ G @ A5 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_246_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H: a > a,S: partia2175431115845679010xt_a_b] :
( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X4 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X4 @ Y3 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X4 @ Y3 ) )
= ( add_a_b @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_247_semiring__axioms_Ointro,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X4: a,Y3: a,Z2: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X4 @ Y3 ) @ Z2 )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X4 @ Z2 ) @ ( mult_a_ring_ext_a_b @ R @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: a,Y3: a,Z2: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z2 @ ( add_a_b @ R @ X4 @ Y3 ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z2 @ X4 ) @ ( mult_a_ring_ext_a_b @ R @ Z2 @ Y3 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X4 )
= ( zero_a_b @ R ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X4 @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) )
=> ( semiring_axioms_a_b @ R ) ) ) ) ) ).
% semiring_axioms.intro
thf(fact_248_semiring__axioms__def,axiom,
( semiring_axioms_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b] :
( ! [X3: a,Y4: a,Z3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( add_a_b @ R2 @ X3 @ Y4 ) @ Z3 )
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ X3 @ Z3 ) @ ( mult_a_ring_ext_a_b @ R2 @ Y4 @ Z3 ) ) ) ) ) )
& ! [X3: a,Y4: a,Z3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ Z3 @ ( add_a_b @ R2 @ X3 @ Y4 ) )
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ Z3 @ X3 ) @ ( mult_a_ring_ext_a_b @ R2 @ Z3 @ Y4 ) ) ) ) ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( zero_a_b @ R2 ) @ X3 )
= ( zero_a_b @ R2 ) ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ X3 @ ( zero_a_b @ R2 ) )
= ( zero_a_b @ R2 ) ) ) ) ) ) ).
% semiring_axioms_def
thf(fact_249_monoid_Oirreducible__prod__rI,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( irredu6211895646901577903xt_a_b @ G @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( irredu6211895646901577903xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A @ B ) ) ) ) ) ) ) ).
% monoid.irreducible_prod_rI
thf(fact_250_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_251_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_252_ring_Oline__extension__mem__iff,axiom,
! [R: partia2175431115845679010xt_a_b,U: a,K2: set_a,A: a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ K2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ E )
& ( U
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ A ) @ Y4 ) ) ) ) ) ) ) ).
% ring.line_extension_mem_iff
thf(fact_253_add_Opower__order__eq__one,axiom,
! [A: a] :
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ ( finite_card_a @ ( partia707051561876973205xt_a_b @ r ) ) @ A )
= ( zero_a_b @ r ) ) ) ) ).
% add.power_order_eq_one
thf(fact_254_add_Opow__eq__div2,axiom,
! [X: a,M: nat,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_pow_a_b_nat @ r @ M @ X )
= ( add_pow_a_b_nat @ r @ N @ X ) )
=> ( ( add_pow_a_b_nat @ r @ ( minus_minus_nat @ M @ N ) @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.pow_eq_div2
thf(fact_255_line__extension__in__carrier,axiom,
! [K2: set_a,A: a,E: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_256_ring_Oequality,axiom,
! [R3: partia2175431115845679010xt_a_b,R4: partia2175431115845679010xt_a_b] :
( ( ( partia707051561876973205xt_a_b @ R3 )
= ( partia707051561876973205xt_a_b @ R4 ) )
=> ( ( ( mult_a_ring_ext_a_b @ R3 )
= ( mult_a_ring_ext_a_b @ R4 ) )
=> ( ( ( one_a_ring_ext_a_b @ R3 )
= ( one_a_ring_ext_a_b @ R4 ) )
=> ( ( ( zero_a_b @ R3 )
= ( zero_a_b @ R4 ) )
=> ( ( ( add_a_b @ R3 )
= ( add_a_b @ R4 ) )
=> ( ( ( more_a_b @ R3 )
= ( more_a_b @ R4 ) )
=> ( R3 = R4 ) ) ) ) ) ) ) ).
% ring.equality
thf(fact_257_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_258_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_259_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_260_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_261_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_262_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_263_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_264_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_265_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_266_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_267_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_268_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_269_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_270_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_271_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_272_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_273_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_274_infinite__arbitrarily__large,axiom,
! [A3: set_set_a,N: nat] :
( ~ ( finite_finite_set_a @ A3 )
=> ? [B5: set_set_a] :
( ( finite_finite_set_a @ B5 )
& ( ( finite_card_set_a @ B5 )
= N )
& ( ord_le3724670747650509150_set_a @ B5 @ A3 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_275_infinite__arbitrarily__large,axiom,
! [A3: set_list_a,N: nat] :
( ~ ( finite_finite_list_a @ A3 )
=> ? [B5: set_list_a] :
( ( finite_finite_list_a @ B5 )
& ( ( finite_card_list_a @ B5 )
= N )
& ( ord_le8861187494160871172list_a @ B5 @ A3 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_276_infinite__arbitrarily__large,axiom,
! [A3: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A3 )
=> ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ( finite_card_nat @ B5 )
= N )
& ( ord_less_eq_set_nat @ B5 @ A3 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_277_infinite__arbitrarily__large,axiom,
! [A3: set_a,N: nat] :
( ~ ( finite_finite_a @ A3 )
=> ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ( finite_card_a @ B5 )
= N )
& ( ord_less_eq_set_a @ B5 @ A3 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_278_card__subset__eq,axiom,
! [B6: set_set_a,A3: set_set_a] :
( ( finite_finite_set_a @ B6 )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ B6 )
=> ( ( ( finite_card_set_a @ A3 )
= ( finite_card_set_a @ B6 ) )
=> ( A3 = B6 ) ) ) ) ).
% card_subset_eq
thf(fact_279_card__subset__eq,axiom,
! [B6: set_list_a,A3: set_list_a] :
( ( finite_finite_list_a @ B6 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ B6 )
=> ( ( ( finite_card_list_a @ A3 )
= ( finite_card_list_a @ B6 ) )
=> ( A3 = B6 ) ) ) ) ).
% card_subset_eq
thf(fact_280_card__subset__eq,axiom,
! [B6: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B6 )
=> ( ( ord_less_eq_set_nat @ A3 @ B6 )
=> ( ( ( finite_card_nat @ A3 )
= ( finite_card_nat @ B6 ) )
=> ( A3 = B6 ) ) ) ) ).
% card_subset_eq
thf(fact_281_card__subset__eq,axiom,
! [B6: set_a,A3: set_a] :
( ( finite_finite_a @ B6 )
=> ( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( ( finite_card_a @ A3 )
= ( finite_card_a @ B6 ) )
=> ( A3 = B6 ) ) ) ) ).
% card_subset_eq
thf(fact_282_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_283_finite__has__minimal2,axiom,
! [A3: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ A @ A3 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A3 )
& ( ord_less_eq_set_a @ X4 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_284_finite__has__minimal2,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A3 )
& ( ord_less_eq_nat @ X4 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_285_finite__has__maximal2,axiom,
! [A3: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ A @ A3 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A3 )
& ( ord_less_eq_set_a @ A @ X4 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_286_finite__has__maximal2,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A3 )
& ( ord_less_eq_nat @ A @ X4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_287_rev__finite__subset,axiom,
! [B6: set_list_a,A3: set_list_a] :
( ( finite_finite_list_a @ B6 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ B6 )
=> ( finite_finite_list_a @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_288_rev__finite__subset,axiom,
! [B6: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B6 )
=> ( ( ord_less_eq_set_nat @ A3 @ B6 )
=> ( finite_finite_nat @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_289_rev__finite__subset,axiom,
! [B6: set_a,A3: set_a] :
( ( finite_finite_a @ B6 )
=> ( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( finite_finite_a @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_290_infinite__super,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_super
thf(fact_291_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_292_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_293_finite__subset,axiom,
! [A3: set_list_a,B6: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B6 )
=> ( ( finite_finite_list_a @ B6 )
=> ( finite_finite_list_a @ A3 ) ) ) ).
% finite_subset
thf(fact_294_finite__subset,axiom,
! [A3: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B6 )
=> ( ( finite_finite_nat @ B6 )
=> ( finite_finite_nat @ A3 ) ) ) ).
% finite_subset
thf(fact_295_finite__subset,axiom,
! [A3: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( finite_finite_a @ B6 )
=> ( finite_finite_a @ A3 ) ) ) ).
% finite_subset
thf(fact_296_ring_Oline__extension__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,A: a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_297_group_OUnits,axiom,
! [G: partia8223610829204095565t_unit] :
( ( group_a_Product_unit @ G )
=> ( ord_less_eq_set_a @ ( partia6735698275553448452t_unit @ G ) @ ( units_a_Product_unit @ G ) ) ) ).
% group.Units
thf(fact_298_group_OUnits,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( group_a_ring_ext_a_b @ G )
=> ( ord_less_eq_set_a @ ( partia707051561876973205xt_a_b @ G ) @ ( units_a_ring_ext_a_b @ G ) ) ) ).
% group.Units
thf(fact_299_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_300_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_301_ring_Oline__extension_Ocong,axiom,
embedd971793762689825387on_a_b = embedd971793762689825387on_a_b ).
% ring.line_extension.cong
thf(fact_302_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_303_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_304_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_305_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_306_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_307_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_308_a__card__cosets__equal,axiom,
! [C: set_a,H2: set_a] :
( ( member_set_a @ C @ ( a_RCOSETS_a_b @ r @ H2 ) )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_card_a @ C )
= ( finite_card_a @ H2 ) ) ) ) ) ).
% a_card_cosets_equal
thf(fact_309_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_310_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_311_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_312_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_313_a__lcos__mult__one,axiom,
! [M2: set_a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M2 )
= M2 ) ) ).
% a_lcos_mult_one
thf(fact_314_a__lcos__m__assoc,axiom,
! [M2: set_a,G3: a,H: a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G3 @ ( a_l_coset_a_b @ r @ H @ M2 ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G3 @ H ) @ M2 ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_315_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_316_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_317_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_318_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_319_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_320_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_321_finite__Diff,axiom,
! [A3: set_list_a,B6: set_list_a] :
( ( finite_finite_list_a @ A3 )
=> ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A3 @ B6 ) ) ) ).
% finite_Diff
thf(fact_322_finite__Diff,axiom,
! [A3: set_nat,B6: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B6 ) ) ) ).
% finite_Diff
thf(fact_323_finite__Diff,axiom,
! [A3: set_a,B6: set_a] :
( ( finite_finite_a @ A3 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A3 @ B6 ) ) ) ).
% finite_Diff
thf(fact_324_finite__Diff2,axiom,
! [B6: set_list_a,A3: set_list_a] :
( ( finite_finite_list_a @ B6 )
=> ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A3 @ B6 ) )
= ( finite_finite_list_a @ A3 ) ) ) ).
% finite_Diff2
thf(fact_325_finite__Diff2,axiom,
! [B6: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B6 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B6 ) )
= ( finite_finite_nat @ A3 ) ) ) ).
% finite_Diff2
thf(fact_326_finite__Diff2,axiom,
! [B6: set_a,A3: set_a] :
( ( finite_finite_a @ B6 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A3 @ B6 ) )
= ( finite_finite_a @ A3 ) ) ) ).
% finite_Diff2
thf(fact_327_a__l__coset__subset__G,axiom,
! [H2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_328_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_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__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_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__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_333_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_334_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_335_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_336_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_337_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_338_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_339_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_340_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_341_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_342_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_343_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_344_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_345_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_346_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_347_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_348_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_349_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_350_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_351_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_352_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_353_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_354_card__le__sym__Diff,axiom,
! [A3: set_set_a,B6: set_set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( finite_finite_set_a @ B6 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ A3 ) @ ( finite_card_set_a @ B6 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A3 @ B6 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B6 @ A3 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_355_card__le__sym__Diff,axiom,
! [A3: set_list_a,B6: set_list_a] :
( ( finite_finite_list_a @ A3 )
=> ( ( finite_finite_list_a @ B6 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_a @ A3 ) @ ( finite_card_list_a @ B6 ) )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A3 @ B6 ) ) @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ B6 @ A3 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_356_card__le__sym__Diff,axiom,
! [A3: set_nat,B6: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ B6 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B6 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B6 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B6 @ A3 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_357_card__le__sym__Diff,axiom,
! [A3: set_a,B6: set_a] :
( ( finite_finite_a @ A3 )
=> ( ( finite_finite_a @ B6 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ ( finite_card_a @ B6 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A3 @ B6 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B6 @ A3 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_358_Diff__infinite__finite,axiom,
! [T: set_list_a,S: set_list_a] :
( ( finite_finite_list_a @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_359_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_360_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_361_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_362_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_363_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_364_Suc__le__D,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
=> ? [M4: nat] :
( M3
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_365_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_366_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_367_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_368_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_369_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_370_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z2: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z2 )
=> ( R @ X4 @ Z2 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_371_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_372_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_373_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_374_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_375_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_376_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_377_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_378_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_379_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_380_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_381_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_382_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_383_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_384_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_385_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_386_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_387_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_388_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_389_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_390_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_391_mult__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 @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_392_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_393_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_394_diff__card__le__card__Diff,axiom,
! [B6: set_set_a,A3: set_set_a] :
( ( finite_finite_set_a @ B6 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_a @ A3 ) @ ( finite_card_set_a @ B6 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A3 @ B6 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_395_diff__card__le__card__Diff,axiom,
! [B6: set_list_a,A3: set_list_a] :
( ( finite_finite_list_a @ B6 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_list_a @ A3 ) @ ( finite_card_list_a @ B6 ) ) @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A3 @ B6 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_396_diff__card__le__card__Diff,axiom,
! [B6: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B6 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B6 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B6 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_397_diff__card__le__card__Diff,axiom,
! [B6: set_a,A3: set_a] :
( ( finite_finite_a @ B6 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A3 ) @ ( finite_card_a @ B6 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A3 @ B6 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_398_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_399_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_400_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_401_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_402_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_403_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_404_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_405_card__mono,axiom,
! [B6: set_set_a,A3: set_set_a] :
( ( finite_finite_set_a @ B6 )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ B6 )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A3 ) @ ( finite_card_set_a @ B6 ) ) ) ) ).
% card_mono
thf(fact_406_card__mono,axiom,
! [B6: set_list_a,A3: set_list_a] :
( ( finite_finite_list_a @ B6 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ B6 )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ A3 ) @ ( finite_card_list_a @ B6 ) ) ) ) ).
% card_mono
thf(fact_407_card__mono,axiom,
! [B6: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B6 )
=> ( ( ord_less_eq_set_nat @ A3 @ B6 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B6 ) ) ) ) ).
% card_mono
thf(fact_408_card__mono,axiom,
! [B6: set_a,A3: set_a] :
( ( finite_finite_a @ B6 )
=> ( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ ( finite_card_a @ B6 ) ) ) ) ).
% card_mono
thf(fact_409_card__seteq,axiom,
! [B6: set_set_a,A3: set_set_a] :
( ( finite_finite_set_a @ B6 )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ B6 ) @ ( finite_card_set_a @ A3 ) )
=> ( A3 = B6 ) ) ) ) ).
% card_seteq
thf(fact_410_card__seteq,axiom,
! [B6: set_list_a,A3: set_list_a] :
( ( finite_finite_list_a @ B6 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_a @ B6 ) @ ( finite_card_list_a @ A3 ) )
=> ( A3 = B6 ) ) ) ) ).
% card_seteq
thf(fact_411_card__seteq,axiom,
! [B6: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B6 )
=> ( ( ord_less_eq_set_nat @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B6 ) @ ( finite_card_nat @ A3 ) )
=> ( A3 = B6 ) ) ) ) ).
% card_seteq
thf(fact_412_card__seteq,axiom,
! [B6: set_a,A3: set_a] :
( ( finite_finite_a @ B6 )
=> ( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B6 ) @ ( finite_card_a @ A3 ) )
=> ( A3 = B6 ) ) ) ) ).
% card_seteq
thf(fact_413_exists__subset__between,axiom,
! [A3: set_set_a,N: nat,C3: set_set_a] :
( ( ord_less_eq_nat @ ( finite_card_set_a @ A3 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ C3 ) )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ C3 )
=> ( ( finite_finite_set_a @ C3 )
=> ? [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ C3 )
& ( ( finite_card_set_a @ B5 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_414_exists__subset__between,axiom,
! [A3: set_list_a,N: nat,C3: set_list_a] :
( ( ord_less_eq_nat @ ( finite_card_list_a @ A3 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_list_a @ C3 ) )
=> ( ( ord_le8861187494160871172list_a @ A3 @ C3 )
=> ( ( finite_finite_list_a @ C3 )
=> ? [B5: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B5 )
& ( ord_le8861187494160871172list_a @ B5 @ C3 )
& ( ( finite_card_list_a @ B5 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_415_exists__subset__between,axiom,
! [A3: set_nat,N: nat,C3: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C3 ) )
=> ( ( ord_less_eq_set_nat @ A3 @ C3 )
=> ( ( finite_finite_nat @ C3 )
=> ? [B5: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ C3 )
& ( ( finite_card_nat @ B5 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_416_exists__subset__between,axiom,
! [A3: set_a,N: nat,C3: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C3 ) )
=> ( ( ord_less_eq_set_a @ A3 @ C3 )
=> ( ( finite_finite_a @ C3 )
=> ? [B5: set_a] :
( ( ord_less_eq_set_a @ A3 @ B5 )
& ( ord_less_eq_set_a @ B5 @ C3 )
& ( ( finite_card_a @ B5 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_417_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ S ) )
=> ~ ! [T2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ T2 @ S )
=> ( ( ( finite_card_set_a @ T2 )
= N )
=> ~ ( finite_finite_set_a @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_418_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_list_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_list_a @ S ) )
=> ~ ! [T2: set_list_a] :
( ( ord_le8861187494160871172list_a @ T2 @ S )
=> ( ( ( finite_card_list_a @ T2 )
= N )
=> ~ ( finite_finite_list_a @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_419_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T2: set_nat] :
( ( ord_less_eq_set_nat @ T2 @ S )
=> ( ( ( finite_card_nat @ T2 )
= N )
=> ~ ( finite_finite_nat @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_420_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T2: set_a] :
( ( ord_less_eq_set_a @ T2 @ S )
=> ( ( ( finite_card_a @ T2 )
= N )
=> ~ ( finite_finite_a @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_421_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_set_a,C3: nat] :
( ! [G4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ G4 @ F2 )
=> ( ( finite_finite_set_a @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ G4 ) @ C3 ) ) )
=> ( ( finite_finite_set_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_set_a @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_422_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_list_a,C3: nat] :
( ! [G4: set_list_a] :
( ( ord_le8861187494160871172list_a @ G4 @ F2 )
=> ( ( finite_finite_list_a @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ G4 ) @ C3 ) ) )
=> ( ( finite_finite_list_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_list_a @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_423_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C3: nat] :
( ! [G4: set_nat] :
( ( ord_less_eq_set_nat @ G4 @ F2 )
=> ( ( finite_finite_nat @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G4 ) @ C3 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_424_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C3: nat] :
( ! [G4: set_a] :
( ( ord_less_eq_set_a @ G4 @ F2 )
=> ( ( finite_finite_a @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G4 ) @ C3 ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_425_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_426_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_427_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B7: int] : ( times_times_int @ B7 @ A4 ) ) ) ).
% mult.commute
thf(fact_428_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B7: nat] : ( times_times_nat @ B7 @ A4 ) ) ) ).
% mult.commute
thf(fact_429_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_430_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_431_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_432_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_433_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_434_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_435_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_436_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_437_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_438_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_439_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B7: nat] : ( plus_plus_nat @ B7 @ A4 ) ) ) ).
% add.commute
thf(fact_440_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B7: int] : ( plus_plus_int @ B7 @ A4 ) ) ) ).
% add.commute
thf(fact_441_group__add__class_Oadd_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% group_add_class.add.right_cancel
thf(fact_442_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_443_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_444_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_445_group__cancel_Oadd2,axiom,
! [B6: nat,K: nat,B: nat,A: nat] :
( ( B6
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B6 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_446_group__cancel_Oadd2,axiom,
! [B6: int,K: int,B: int,A: int] :
( ( B6
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B6 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_447_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_448_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_449_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_450_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_451_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_452_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_453_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_454_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_455_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_456_card__Diff__subset,axiom,
! [B6: set_set_a,A3: set_set_a] :
( ( finite_finite_set_a @ B6 )
=> ( ( ord_le3724670747650509150_set_a @ B6 @ A3 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A3 @ B6 ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A3 ) @ ( finite_card_set_a @ B6 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_457_card__Diff__subset,axiom,
! [B6: set_list_a,A3: set_list_a] :
( ( finite_finite_list_a @ B6 )
=> ( ( ord_le8861187494160871172list_a @ B6 @ A3 )
=> ( ( finite_card_list_a @ ( minus_646659088055828811list_a @ A3 @ B6 ) )
= ( minus_minus_nat @ ( finite_card_list_a @ A3 ) @ ( finite_card_list_a @ B6 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_458_card__Diff__subset,axiom,
! [B6: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B6 )
=> ( ( ord_less_eq_set_nat @ B6 @ A3 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B6 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B6 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_459_card__Diff__subset,axiom,
! [B6: set_a,A3: set_a] :
( ( finite_finite_a @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ A3 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A3 @ B6 ) )
= ( minus_minus_nat @ ( finite_card_a @ A3 ) @ ( finite_card_a @ B6 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_460_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_461_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_462_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_463_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_464_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_465_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_466_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_467_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_468_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_469_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_470_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_471_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_472_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_473_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B7: nat] :
? [C4: nat] :
( B7
= ( plus_plus_nat @ A4 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_474_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_475_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_476_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_477_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_478_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_479_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_480_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_481_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_482_combine__common__factor,axiom,
! [A: int,E2: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_483_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_484_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_485_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_486_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_487_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_488_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_489_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_490_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_491_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_492_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_493_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_494_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_495_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_496_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_497_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_498_group__cancel_Osub1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_499_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_500_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_501_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_502_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_503_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_504_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_505_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_506_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_507_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_508_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_509_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_510_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_511_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_512_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_513_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_514_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_515_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_516_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_517_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_518_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_519_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_520_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_521_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_522_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_523_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_524_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_525_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_526_eq__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_527_eq__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_528_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_529_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_530_abelian__group_Oa__card__cosets__equal,axiom,
! [G: partia2175431115845679010xt_a_b,C: set_a,H2: set_a] :
( ( abelian_group_a_b @ G )
=> ( ( member_set_a @ C @ ( a_RCOSETS_a_b @ G @ H2 ) )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( finite_card_a @ C )
= ( finite_card_a @ H2 ) ) ) ) ) ) ).
% abelian_group.a_card_cosets_equal
thf(fact_531_abelian__group_Oa__lcos__m__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,M2: set_a,G3: a,H: a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ G3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_l_coset_a_b @ G @ G3 @ ( a_l_coset_a_b @ G @ H @ M2 ) )
= ( a_l_coset_a_b @ G @ ( add_a_b @ G @ G3 @ H ) @ M2 ) ) ) ) ) ) ).
% abelian_group.a_lcos_m_assoc
thf(fact_532_abelian__group_Oa__lcos__mult__one,axiom,
! [G: partia2175431115845679010xt_a_b,M2: set_a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_l_coset_a_b @ G @ ( zero_a_b @ G ) @ M2 )
= M2 ) ) ) ).
% abelian_group.a_lcos_mult_one
thf(fact_533_carrier__is__subalgebra,axiom,
! [K2: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_534_subalgebra__in__carrier,axiom,
! [K2: set_a,V: set_a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V @ r )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_535_abelian__group_Oa__l__coset__subset__G,axiom,
! [G: partia2175431115845679010xt_a_b,H2: set_a,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ G @ X @ H2 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_group.a_l_coset_subset_G
thf(fact_536_subalgebra_Osmult__closed,axiom,
! [K2: set_a,V: set_a,R: partia2175431115845679010xt_a_b,K: a,V2: a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V @ R )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V2 @ V )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ V2 ) @ V ) ) ) ) ).
% subalgebra.smult_closed
thf(fact_537_ring_Ocarrier__is__subalgebra,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_538_ring_Osubalgebra__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,V: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V @ R )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_539_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_540_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_541_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_542_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_543_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_544_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_545_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_546_genideal__self,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ r @ S ) ) ) ).
% genideal_self
thf(fact_547_subset__Idl__subset,axiom,
! [I2: set_a,H2: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H2 @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H2 ) @ ( genideal_a_b @ r @ I2 ) ) ) ) ).
% subset_Idl_subset
thf(fact_548_line__extension__smult__closed,axiom,
! [K2: set_a,E: set_a,A: a,K: a,U: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ! [K4: a,V3: a] :
( ( member_a @ K4 @ K2 )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K4 @ V3 ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ U ) @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_549_a__rcosetsI,axiom,
! [H2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_set_a @ ( a_r_coset_a_b @ r @ H2 @ X ) @ ( a_RCOSETS_a_b @ r @ H2 ) ) ) ) ).
% a_rcosetsI
thf(fact_550_card__le__if__inj__on__rel,axiom,
! [B6: set_set_a,A3: set_a,R3: a > set_a > $o] :
( ( finite_finite_set_a @ B6 )
=> ( ! [A2: a] :
( ( member_a @ A2 @ A3 )
=> ? [B4: set_a] :
( ( member_set_a @ B4 @ B6 )
& ( R3 @ A2 @ B4 ) ) )
=> ( ! [A1: a,A22: a,B2: set_a] :
( ( member_a @ A1 @ A3 )
=> ( ( member_a @ A22 @ A3 )
=> ( ( member_set_a @ B2 @ B6 )
=> ( ( R3 @ A1 @ B2 )
=> ( ( R3 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ ( finite_card_set_a @ B6 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_551_card__le__if__inj__on__rel,axiom,
! [B6: set_set_a,A3: set_set_a,R3: set_a > set_a > $o] :
( ( finite_finite_set_a @ B6 )
=> ( ! [A2: set_a] :
( ( member_set_a @ A2 @ A3 )
=> ? [B4: set_a] :
( ( member_set_a @ B4 @ B6 )
& ( R3 @ A2 @ B4 ) ) )
=> ( ! [A1: set_a,A22: set_a,B2: set_a] :
( ( member_set_a @ A1 @ A3 )
=> ( ( member_set_a @ A22 @ A3 )
=> ( ( member_set_a @ B2 @ B6 )
=> ( ( R3 @ A1 @ B2 )
=> ( ( R3 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A3 ) @ ( finite_card_set_a @ B6 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_552_card__le__if__inj__on__rel,axiom,
! [B6: set_list_a,A3: set_a,R3: a > list_a > $o] :
( ( finite_finite_list_a @ B6 )
=> ( ! [A2: a] :
( ( member_a @ A2 @ A3 )
=> ? [B4: list_a] :
( ( member_list_a @ B4 @ B6 )
& ( R3 @ A2 @ B4 ) ) )
=> ( ! [A1: a,A22: a,B2: list_a] :
( ( member_a @ A1 @ A3 )
=> ( ( member_a @ A22 @ A3 )
=> ( ( member_list_a @ B2 @ B6 )
=> ( ( R3 @ A1 @ B2 )
=> ( ( R3 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ ( finite_card_list_a @ B6 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_553_card__le__if__inj__on__rel,axiom,
! [B6: set_list_a,A3: set_set_a,R3: set_a > list_a > $o] :
( ( finite_finite_list_a @ B6 )
=> ( ! [A2: set_a] :
( ( member_set_a @ A2 @ A3 )
=> ? [B4: list_a] :
( ( member_list_a @ B4 @ B6 )
& ( R3 @ A2 @ B4 ) ) )
=> ( ! [A1: set_a,A22: set_a,B2: list_a] :
( ( member_set_a @ A1 @ A3 )
=> ( ( member_set_a @ A22 @ A3 )
=> ( ( member_list_a @ B2 @ B6 )
=> ( ( R3 @ A1 @ B2 )
=> ( ( R3 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A3 ) @ ( finite_card_list_a @ B6 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_554_card__le__if__inj__on__rel,axiom,
! [B6: set_a,A3: set_a,R3: a > a > $o] :
( ( finite_finite_a @ B6 )
=> ( ! [A2: a] :
( ( member_a @ A2 @ A3 )
=> ? [B4: a] :
( ( member_a @ B4 @ B6 )
& ( R3 @ A2 @ B4 ) ) )
=> ( ! [A1: a,A22: a,B2: a] :
( ( member_a @ A1 @ A3 )
=> ( ( member_a @ A22 @ A3 )
=> ( ( member_a @ B2 @ B6 )
=> ( ( R3 @ A1 @ B2 )
=> ( ( R3 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ ( finite_card_a @ B6 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_555_card__le__if__inj__on__rel,axiom,
! [B6: set_a,A3: set_set_a,R3: set_a > a > $o] :
( ( finite_finite_a @ B6 )
=> ( ! [A2: set_a] :
( ( member_set_a @ A2 @ A3 )
=> ? [B4: a] :
( ( member_a @ B4 @ B6 )
& ( R3 @ A2 @ B4 ) ) )
=> ( ! [A1: set_a,A22: set_a,B2: a] :
( ( member_set_a @ A1 @ A3 )
=> ( ( member_set_a @ A22 @ A3 )
=> ( ( member_a @ B2 @ B6 )
=> ( ( R3 @ A1 @ B2 )
=> ( ( R3 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A3 ) @ ( finite_card_a @ B6 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_556_card__le__if__inj__on__rel,axiom,
! [B6: set_nat,A3: set_a,R3: a > nat > $o] :
( ( finite_finite_nat @ B6 )
=> ( ! [A2: a] :
( ( member_a @ A2 @ A3 )
=> ? [B4: nat] :
( ( member_nat @ B4 @ B6 )
& ( R3 @ A2 @ B4 ) ) )
=> ( ! [A1: a,A22: a,B2: nat] :
( ( member_a @ A1 @ A3 )
=> ( ( member_a @ A22 @ A3 )
=> ( ( member_nat @ B2 @ B6 )
=> ( ( R3 @ A1 @ B2 )
=> ( ( R3 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ ( finite_card_nat @ B6 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_557_card__le__if__inj__on__rel,axiom,
! [B6: set_nat,A3: set_set_a,R3: set_a > nat > $o] :
( ( finite_finite_nat @ B6 )
=> ( ! [A2: set_a] :
( ( member_set_a @ A2 @ A3 )
=> ? [B4: nat] :
( ( member_nat @ B4 @ B6 )
& ( R3 @ A2 @ B4 ) ) )
=> ( ! [A1: set_a,A22: set_a,B2: nat] :
( ( member_set_a @ A1 @ A3 )
=> ( ( member_set_a @ A22 @ A3 )
=> ( ( member_nat @ B2 @ B6 )
=> ( ( R3 @ A1 @ B2 )
=> ( ( R3 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A3 ) @ ( finite_card_nat @ B6 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_558_subring__props_I7_J,axiom,
! [K2: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ K2 ) ) ) ) ).
% subring_props(7)
thf(fact_559_subring__props_I2_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K2 ) ) ).
% subring_props(2)
thf(fact_560_subring__props_I6_J,axiom,
! [K2: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ K2 ) ) ) ) ).
% subring_props(6)
thf(fact_561_subring__props_I3_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K2 ) ) ).
% subring_props(3)
thf(fact_562_subring__props_I1_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_563_a__r__coset__subset__G,axiom,
! [H2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_r_coset_a_b @ r @ H2 @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_r_coset_subset_G
thf(fact_564_a__coset__add__assoc,axiom,
! [M2: set_a,G3: a,H: a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ ( a_r_coset_a_b @ r @ M2 @ G3 ) @ H )
= ( a_r_coset_a_b @ r @ M2 @ ( add_a_b @ r @ G3 @ H ) ) ) ) ) ) ).
% a_coset_add_assoc
thf(fact_565_a__rcosI,axiom,
! [H: a,H2: set_a,X: a] :
( ( member_a @ H @ H2 )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ H @ X ) @ ( a_r_coset_a_b @ r @ H2 @ X ) ) ) ) ) ).
% a_rcosI
thf(fact_566_a__coset__add__zero,axiom,
! [M2: set_a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ M2 @ ( zero_a_b @ r ) )
= M2 ) ) ).
% a_coset_add_zero
thf(fact_567_ring_Osubring__props_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( member_a @ ( zero_a_b @ R ) @ K2 ) ) ) ).
% ring.subring_props(2)
thf(fact_568_ring_Osubring__props_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,H1: a,H22: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( add_a_b @ R @ H1 @ H22 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(7)
thf(fact_569_ring_Osubring__props_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,H1: a,H22: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ H1 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H22 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(6)
thf(fact_570_ring_Osubring__props_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ K2 ) ) ) ).
% ring.subring_props(3)
thf(fact_571_ring_Osubring__props_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.subring_props(1)
thf(fact_572_abelian__monoid_Oa__r__coset__subset__G,axiom,
! [G: partia2175431115845679010xt_a_b,H2: set_a,X: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ord_less_eq_set_a @ ( a_r_coset_a_b @ G @ H2 @ X ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_monoid.a_r_coset_subset_G
thf(fact_573_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M6: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_eq_nat @ X3 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_574_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M2: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_575_abelian__group_Oa__coset__add__zero,axiom,
! [G: partia2175431115845679010xt_a_b,M2: set_a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_r_coset_a_b @ G @ M2 @ ( zero_a_b @ G ) )
= M2 ) ) ) ).
% abelian_group.a_coset_add_zero
thf(fact_576_abelian__group_Oa__rcosI,axiom,
! [G: partia2175431115845679010xt_a_b,H: a,H2: set_a,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ H @ H2 )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( add_a_b @ G @ H @ X ) @ ( a_r_coset_a_b @ G @ H2 @ X ) ) ) ) ) ) ).
% abelian_group.a_rcosI
thf(fact_577_abelian__group_Oa__coset__add__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,M2: set_a,G3: a,H: a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ G3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_r_coset_a_b @ G @ ( a_r_coset_a_b @ G @ M2 @ G3 ) @ H )
= ( a_r_coset_a_b @ G @ M2 @ ( add_a_b @ G @ G3 @ H ) ) ) ) ) ) ) ).
% abelian_group.a_coset_add_assoc
thf(fact_578_ring_Ogenideal__self,axiom,
! [R: partia2175431115845679010xt_a_b,S: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ R @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_579_ring_Osubset__Idl__subset,axiom,
! [R: partia2175431115845679010xt_a_b,I2: set_a,H2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ H2 @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ R @ H2 ) @ ( genideal_a_b @ R @ I2 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_580_abelian__group_Oa__rcosetsI,axiom,
! [G: partia2175431115845679010xt_a_b,H2: set_a,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_set_a @ ( a_r_coset_a_b @ G @ H2 @ X ) @ ( a_RCOSETS_a_b @ G @ H2 ) ) ) ) ) ).
% abelian_group.a_rcosetsI
thf(fact_581_ring_Oline__extension__smult__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a,A: a,K: a,U: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ! [K4: a,V3: a] :
( ( member_a @ K4 @ K2 )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K4 @ V3 ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ U ) @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_582_subalbegra__incl__imp__finite__dimension,axiom,
! [K2: set_a,E: set_a,V: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V @ r )
=> ( ( ord_less_eq_set_a @ V @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ V ) ) ) ) ) ).
% subalbegra_incl_imp_finite_dimension
thf(fact_583_a__rcos__assoc__lcos,axiom,
! [H2: set_a,K2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ ( a_r_coset_a_b @ r @ H2 @ X ) @ K2 )
= ( set_add_a_b @ r @ H2 @ ( a_l_coset_a_b @ r @ X @ K2 ) ) ) ) ) ) ).
% a_rcos_assoc_lcos
thf(fact_584_finite__dimension__imp__subalgebra,axiom,
! [K2: set_a,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ( embedd9027525575939734154ra_a_b @ K2 @ E @ r ) ) ) ).
% finite_dimension_imp_subalgebra
thf(fact_585_a__setmult__rcos__assoc,axiom,
! [H2: set_a,K2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ H2 @ ( a_r_coset_a_b @ r @ K2 @ X ) )
= ( a_r_coset_a_b @ r @ ( set_add_a_b @ r @ H2 @ K2 ) @ X ) ) ) ) ) ).
% a_setmult_rcos_assoc
thf(fact_586_telescopic__base__dim_I1_J,axiom,
! [K2: set_a,F2: set_a,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( subfield_a_b @ F2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ F2 )
=> ( ( embedd8708762675212832759on_a_b @ r @ F2 @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ E ) ) ) ) ) ).
% telescopic_base_dim(1)
thf(fact_587_set__add__closed,axiom,
! [A3: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A3 @ B6 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_add_closed
thf(fact_588_set__add__comm,axiom,
! [I2: set_a,J2: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ J2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ I2 @ J2 )
= ( set_add_a_b @ r @ J2 @ I2 ) ) ) ) ).
% set_add_comm
thf(fact_589_setadd__subset__G,axiom,
! [H2: set_a,K2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H2 @ K2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% setadd_subset_G
thf(fact_590_sum__space__dim_I1_J,axiom,
! [K2: set_a,E: set_a,F2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ F2 )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ ( set_add_a_b @ r @ E @ F2 ) ) ) ) ) ).
% sum_space_dim(1)
thf(fact_591_ring_Osum__space__dim_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a,F2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ F2 )
=> ( embedd8708762675212832759on_a_b @ R @ K2 @ ( set_add_a_b @ R @ E @ F2 ) ) ) ) ) ) ).
% ring.sum_space_dim(1)
thf(fact_592_ring_Ofinite__dimension_Ocong,axiom,
embedd8708762675212832759on_a_b = embedd8708762675212832759on_a_b ).
% ring.finite_dimension.cong
thf(fact_593_ring_Otelescopic__base__dim_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,F2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( subfield_a_b @ F2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ F2 )
=> ( ( embedd8708762675212832759on_a_b @ R @ F2 @ E )
=> ( embedd8708762675212832759on_a_b @ R @ K2 @ E ) ) ) ) ) ) ).
% ring.telescopic_base_dim(1)
thf(fact_594_ring_Oset__add__comm,axiom,
! [R: partia2175431115845679010xt_a_b,I2: set_a,J2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ J2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( set_add_a_b @ R @ I2 @ J2 )
= ( set_add_a_b @ R @ J2 @ I2 ) ) ) ) ) ).
% ring.set_add_comm
thf(fact_595_abelian__group_Osetadd__subset__G,axiom,
! [G: partia2175431115845679010xt_a_b,H2: set_a,K2: set_a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ G @ H2 @ K2 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_group.setadd_subset_G
thf(fact_596_abelian__monoid_Oset__add__closed,axiom,
! [G: partia2175431115845679010xt_a_b,A3: set_a,B6: set_a] :
( ( abelian_monoid_a_b @ G )
=> ( ( ord_less_eq_set_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ B6 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ G @ A3 @ B6 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_monoid.set_add_closed
thf(fact_597_ring_Ofinite__dimension__imp__subalgebra,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E )
=> ( embedd9027525575939734154ra_a_b @ K2 @ E @ R ) ) ) ) ).
% ring.finite_dimension_imp_subalgebra
thf(fact_598_abelian__group_Oa__setmult__rcos__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,H2: set_a,K2: set_a,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( set_add_a_b @ G @ H2 @ ( a_r_coset_a_b @ G @ K2 @ X ) )
= ( a_r_coset_a_b @ G @ ( set_add_a_b @ G @ H2 @ K2 ) @ X ) ) ) ) ) ) ).
% abelian_group.a_setmult_rcos_assoc
thf(fact_599_ring_Osubalbegra__incl__imp__finite__dimension,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a,V: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V @ R )
=> ( ( ord_less_eq_set_a @ V @ E )
=> ( embedd8708762675212832759on_a_b @ R @ K2 @ V ) ) ) ) ) ) ).
% ring.subalbegra_incl_imp_finite_dimension
thf(fact_600_abelian__group_Oa__rcos__assoc__lcos,axiom,
! [G: partia2175431115845679010xt_a_b,H2: set_a,K2: set_a,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( set_add_a_b @ G @ ( a_r_coset_a_b @ G @ H2 @ X ) @ K2 )
= ( set_add_a_b @ G @ H2 @ ( a_l_coset_a_b @ G @ X @ K2 ) ) ) ) ) ) ) ).
% abelian_group.a_rcos_assoc_lcos
thf(fact_601_add__additive__subgroups,axiom,
! [H2: set_a,K2: set_a] :
( ( additi2834746164131130830up_a_b @ H2 @ r )
=> ( ( additi2834746164131130830up_a_b @ K2 @ r )
=> ( additi2834746164131130830up_a_b @ ( set_add_a_b @ r @ H2 @ K2 ) @ r ) ) ) ).
% add_additive_subgroups
thf(fact_602_a__coset__add__inv2,axiom,
! [M2: set_a,X: a,Y: a] :
( ( ( a_r_coset_a_b @ r @ M2 @ X )
= ( a_r_coset_a_b @ r @ M2 @ Y ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ M2 @ ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) )
= M2 ) ) ) ) ) ).
% a_coset_add_inv2
thf(fact_603_a__coset__add__inv1,axiom,
! [M2: set_a,X: a,Y: a] :
( ( ( a_r_coset_a_b @ r @ M2 @ ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) )
= M2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ M2 @ X )
= ( a_r_coset_a_b @ r @ M2 @ Y ) ) ) ) ) ) ).
% a_coset_add_inv1
thf(fact_604_dimension__backwards,axiom,
! [K2: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ ( suc @ N ) @ K2 @ E )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ? [E3: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E3 )
& ~ ( member_a @ X4 @ E3 )
& ( E
= ( embedd971793762689825387on_a_b @ r @ K2 @ X4 @ E3 ) ) ) ) ) ) ).
% dimension_backwards
thf(fact_605_subring__props_I5_J,axiom,
! [K2: set_a,H: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H @ K2 )
=> ( member_a @ ( a_inv_a_b @ r @ H ) @ K2 ) ) ) ).
% subring_props(5)
thf(fact_606_dimension__is__inj,axiom,
! [K2: set_a,N: nat,E: set_a,M: nat] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M @ K2 @ E )
=> ( N = M ) ) ) ) ).
% dimension_is_inj
thf(fact_607_finite__dimensionE_H,axiom,
! [K2: set_a,E: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ~ ! [N2: nat] :
~ ( embedd2795209813406577254on_a_b @ r @ N2 @ K2 @ E ) ) ).
% finite_dimensionE'
thf(fact_608_finite__dimensionI,axiom,
! [N: nat,K2: set_a,E: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ E ) ) ).
% finite_dimensionI
thf(fact_609_finite__dimension__def,axiom,
! [K2: set_a,E: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
= ( ? [N4: nat] : ( embedd2795209813406577254on_a_b @ r @ N4 @ K2 @ E ) ) ) ).
% finite_dimension_def
thf(fact_610_r__neg2,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_611_r__neg1,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_612_local_Ominus__add,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_613_a__transpose__inv,axiom,
! [X: a,Y: a,Z: a] :
( ( ( add_a_b @ r @ X @ Y )
= Z )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_614_add_Oinv__solve__right_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
= A )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right'
thf(fact_615_add_Oinv__solve__right,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right
thf(fact_616_add_Oinv__solve__left_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
= A )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left'
thf(fact_617_add_Oinv__solve__left,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left
thf(fact_618_add_Oinv__mult__group,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_619_r__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% r_minus
thf(fact_620_l__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% l_minus
thf(fact_621_add_Onat__pow__inv,axiom,
! [X: a,I: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ I @ ( a_inv_a_b @ r @ X ) )
= ( a_inv_a_b @ r @ ( add_pow_a_b_nat @ r @ I @ X ) ) ) ) ).
% add.nat_pow_inv
thf(fact_622_space__subgroup__props_I3_J,axiom,
! [K2: set_a,N: nat,E: set_a,V1: a,V22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( member_a @ V1 @ E )
=> ( ( member_a @ V22 @ E )
=> ( member_a @ ( add_a_b @ r @ V1 @ V22 ) @ E ) ) ) ) ) ).
% space_subgroup_props(3)
thf(fact_623_space__subgroup__props_I2_J,axiom,
! [K2: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( member_a @ ( zero_a_b @ r ) @ E ) ) ) ).
% space_subgroup_props(2)
thf(fact_624_add_Oint__pow__inv,axiom,
! [X: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( a_inv_a_b @ r @ X ) )
= ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% add.int_pow_inv
thf(fact_625_telescopic__base,axiom,
! [K2: set_a,F2: set_a,N: nat,M: nat,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( subfield_a_b @ F2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ F2 )
=> ( ( embedd2795209813406577254on_a_b @ r @ M @ F2 @ E )
=> ( embedd2795209813406577254on_a_b @ r @ ( times_times_nat @ N @ M ) @ K2 @ E ) ) ) ) ) ).
% telescopic_base
thf(fact_626_space__subgroup__props_I5_J,axiom,
! [K2: set_a,N: nat,E: set_a,K: a,V2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V2 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ V2 ) @ E ) ) ) ) ) ).
% space_subgroup_props(5)
thf(fact_627_space__subgroup__props_I4_J,axiom,
! [K2: set_a,N: nat,E: set_a,V2: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( member_a @ V2 @ E )
=> ( member_a @ ( a_inv_a_b @ r @ V2 ) @ E ) ) ) ) ).
% space_subgroup_props(4)
thf(fact_628_unique__dimension,axiom,
! [K2: set_a,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ? [X4: nat] :
( ( embedd2795209813406577254on_a_b @ r @ X4 @ K2 @ E )
& ! [Y5: nat] :
( ( embedd2795209813406577254on_a_b @ r @ Y5 @ K2 @ E )
=> ( Y5 = X4 ) ) ) ) ) ).
% unique_dimension
thf(fact_629_l__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_630_minus__equality,axiom,
! [Y: a,X: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X )
= Y ) ) ) ) ).
% minus_equality
thf(fact_631_r__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_632_space__subgroup__props_I1_J,axiom,
! [K2: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% space_subgroup_props(1)
thf(fact_633_add_Oint__pow__diff,axiom,
! [X: a,N: int,M: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ ( minus_minus_int @ N @ M ) @ X )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ N @ X ) @ ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ M @ X ) ) ) ) ) ).
% add.int_pow_diff
thf(fact_634_Suc__dim,axiom,
! [V2: a,E: set_a,N: nat,K2: set_a] :
( ( member_a @ V2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ V2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( embedd2795209813406577254on_a_b @ r @ ( suc @ N ) @ K2 @ ( embedd971793762689825387on_a_b @ r @ K2 @ V2 @ E ) ) ) ) ) ).
% Suc_dim
thf(fact_635_local_Ominus__minus,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
= X ) ) ).
% local.minus_minus
thf(fact_636_a__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_637_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_638_add_Oinv__eq__1__iff,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X )
= ( zero_a_b @ r ) )
= ( X
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_639_Units__minus__one__closed,axiom,
member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_minus_one_closed
thf(fact_640_ring_Ospace__subgroup__props_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,V2: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( member_a @ V2 @ E )
=> ( member_a @ ( a_inv_a_b @ R @ V2 ) @ E ) ) ) ) ) ).
% ring.space_subgroup_props(4)
thf(fact_641_ring_Odimension_Ocong,axiom,
embedd2795209813406577254on_a_b = embedd2795209813406577254on_a_b ).
% ring.dimension.cong
thf(fact_642_additive__subgroup_Oa__Hcarr,axiom,
! [H2: set_a,G: partia2175431115845679010xt_a_b,H: a] :
( ( additi2834746164131130830up_a_b @ H2 @ G )
=> ( ( member_a @ H @ H2 )
=> ( member_a @ H @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% additive_subgroup.a_Hcarr
thf(fact_643_additive__subgroup_Ozero__closed,axiom,
! [H2: set_a,G: partia2175431115845679010xt_a_b] :
( ( additi2834746164131130830up_a_b @ H2 @ G )
=> ( member_a @ ( zero_a_b @ G ) @ H2 ) ) ).
% additive_subgroup.zero_closed
thf(fact_644_additive__subgroup_Oa__closed,axiom,
! [H2: set_a,G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( additi2834746164131130830up_a_b @ H2 @ G )
=> ( ( member_a @ X @ H2 )
=> ( ( member_a @ Y @ H2 )
=> ( member_a @ ( add_a_b @ G @ X @ Y ) @ H2 ) ) ) ) ).
% additive_subgroup.a_closed
thf(fact_645_ring_Oring__simprules_I20_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( a_inv_a_b @ R @ X ) )
= X ) ) ) ).
% ring.ring_simprules(20)
thf(fact_646_ring_Oring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_inv_a_b @ R @ X ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_647_ring_Ominus__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( a_inv_a_b @ R @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ).
% ring.minus_zero
thf(fact_648_ring_Osubring__props_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,H: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ H @ K2 )
=> ( member_a @ ( a_inv_a_b @ R @ H ) @ K2 ) ) ) ) ).
% ring.subring_props(5)
thf(fact_649_abelian__group_Ominus__minus,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_inv_a_b @ G @ ( a_inv_a_b @ G @ X ) )
= X ) ) ) ).
% abelian_group.minus_minus
thf(fact_650_abelian__group_Oa__inv__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( a_inv_a_b @ G @ X ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% abelian_group.a_inv_closed
thf(fact_651_ring_Odimension__is__inj,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,M: nat] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ R @ M @ K2 @ E )
=> ( N = M ) ) ) ) ) ).
% ring.dimension_is_inj
thf(fact_652_ring_Ofinite__dimension__def,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E )
= ( ? [N4: nat] : ( embedd2795209813406577254on_a_b @ R @ N4 @ K2 @ E ) ) ) ) ).
% ring.finite_dimension_def
thf(fact_653_ring_Ofinite__dimensionE_H,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E )
=> ~ ! [N2: nat] :
~ ( embedd2795209813406577254on_a_b @ R @ N2 @ K2 @ E ) ) ) ).
% ring.finite_dimensionE'
thf(fact_654_ring_Ofinite__dimensionI,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( embedd8708762675212832759on_a_b @ R @ K2 @ E ) ) ) ).
% ring.finite_dimensionI
thf(fact_655_additive__subgroup_Oa__subset,axiom,
! [H2: set_a,G: partia2175431115845679010xt_a_b] :
( ( additi2834746164131130830up_a_b @ H2 @ G )
=> ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ G ) ) ) ).
% additive_subgroup.a_subset
thf(fact_656_ring_Oring__simprules_I19_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ ( a_inv_a_b @ R @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_657_ring_Oring__simprules_I18_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ ( add_a_b @ R @ X @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_658_ring_Oring__simprules_I17_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_659_ring_Ol__minus,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( a_inv_a_b @ R @ X ) @ Y )
= ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_660_ring_Or__minus,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( a_inv_a_b @ R @ Y ) )
= ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_661_abelian__group_Oa__transpose__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( abelian_group_a_b @ G )
=> ( ( ( add_a_b @ G @ X @ Y )
= Z )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ Z )
= Y ) ) ) ) ) ) ).
% abelian_group.a_transpose_inv
thf(fact_662_abelian__group_Or__neg1,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ ( add_a_b @ G @ X @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg1
thf(fact_663_abelian__group_Or__neg2,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X @ ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg2
thf(fact_664_abelian__group_Ominus__add,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_inv_a_b @ G @ ( add_a_b @ G @ X @ Y ) )
= ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ ( a_inv_a_b @ G @ Y ) ) ) ) ) ) ).
% abelian_group.minus_add
thf(fact_665_ring_OUnits__minus__one__closed,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) @ ( units_a_ring_ext_a_b @ R ) ) ) ).
% ring.Units_minus_one_closed
thf(fact_666_abelian__group_Oadd__additive__subgroups,axiom,
! [G: partia2175431115845679010xt_a_b,H2: set_a,K2: set_a] :
( ( abelian_group_a_b @ G )
=> ( ( additi2834746164131130830up_a_b @ H2 @ G )
=> ( ( additi2834746164131130830up_a_b @ K2 @ G )
=> ( additi2834746164131130830up_a_b @ ( set_add_a_b @ G @ H2 @ K2 ) @ G ) ) ) ) ).
% abelian_group.add_additive_subgroups
thf(fact_667_ring_Ospace__subgroup__props_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( member_a @ ( zero_a_b @ R ) @ E ) ) ) ) ).
% ring.space_subgroup_props(2)
thf(fact_668_ring_Ospace__subgroup__props_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,V1: a,V22: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( member_a @ V1 @ E )
=> ( ( member_a @ V22 @ E )
=> ( member_a @ ( add_a_b @ R @ V1 @ V22 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(3)
thf(fact_669_ring_Ospace__subgroup__props_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,K: a,V2: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V2 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ V2 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(5)
thf(fact_670_ring_Otelescopic__base,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,F2: set_a,N: nat,M: nat,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( subfield_a_b @ F2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ F2 )
=> ( ( embedd2795209813406577254on_a_b @ R @ M @ F2 @ E )
=> ( embedd2795209813406577254on_a_b @ R @ ( times_times_nat @ N @ M ) @ K2 @ E ) ) ) ) ) ) ).
% ring.telescopic_base
thf(fact_671_ring_Ounique__dimension,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E )
=> ? [X4: nat] :
( ( embedd2795209813406577254on_a_b @ R @ X4 @ K2 @ E )
& ! [Y5: nat] :
( ( embedd2795209813406577254on_a_b @ R @ Y5 @ K2 @ E )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% ring.unique_dimension
thf(fact_672_ring_Oring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_673_ring_Oring__simprules_I16_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( a_inv_a_b @ R @ X ) )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_674_abelian__group_Ol__neg,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ X )
= ( zero_a_b @ G ) ) ) ) ).
% abelian_group.l_neg
thf(fact_675_abelian__group_Or__neg,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X @ ( a_inv_a_b @ G @ X ) )
= ( zero_a_b @ G ) ) ) ) ).
% abelian_group.r_neg
thf(fact_676_abelian__group_Ominus__equality,axiom,
! [G: partia2175431115845679010xt_a_b,Y: a,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( ( add_a_b @ G @ Y @ X )
= ( zero_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_inv_a_b @ G @ X )
= Y ) ) ) ) ) ).
% abelian_group.minus_equality
thf(fact_677_ring_Ospace__subgroup__props_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.space_subgroup_props(1)
thf(fact_678_ring_OSuc__dim,axiom,
! [R: partia2175431115845679010xt_a_b,V2: a,E: set_a,N: nat,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( member_a @ V2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ~ ( member_a @ V2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( embedd2795209813406577254on_a_b @ R @ ( suc @ N ) @ K2 @ ( embedd971793762689825387on_a_b @ R @ K2 @ V2 @ E ) ) ) ) ) ) ).
% ring.Suc_dim
thf(fact_679_abelian__group_Oa__coset__add__inv1,axiom,
! [G: partia2175431115845679010xt_a_b,M2: set_a,X: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( ( a_r_coset_a_b @ G @ M2 @ ( add_a_b @ G @ X @ ( a_inv_a_b @ G @ Y ) ) )
= M2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_r_coset_a_b @ G @ M2 @ X )
= ( a_r_coset_a_b @ G @ M2 @ Y ) ) ) ) ) ) ) ).
% abelian_group.a_coset_add_inv1
thf(fact_680_abelian__group_Oa__coset__add__inv2,axiom,
! [G: partia2175431115845679010xt_a_b,M2: set_a,X: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( ( a_r_coset_a_b @ G @ M2 @ X )
= ( a_r_coset_a_b @ G @ M2 @ Y ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_r_coset_a_b @ G @ M2 @ ( add_a_b @ G @ X @ ( a_inv_a_b @ G @ Y ) ) )
= M2 ) ) ) ) ) ) ).
% abelian_group.a_coset_add_inv2
thf(fact_681_ring_Odimension__backwards,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ ( suc @ N ) @ K2 @ E )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
& ? [E3: set_a] :
( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E3 )
& ~ ( member_a @ X4 @ E3 )
& ( E
= ( embedd971793762689825387on_a_b @ R @ K2 @ X4 @ E3 ) ) ) ) ) ) ) ).
% ring.dimension_backwards
thf(fact_682_a__lagrange,axiom,
! [H2: set_a] :
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( additi2834746164131130830up_a_b @ H2 @ r )
=> ( ( times_times_nat @ ( finite_card_set_a @ ( a_RCOSETS_a_b @ r @ H2 ) ) @ ( finite_card_a @ H2 ) )
= ( order_a_ring_ext_a_b @ r ) ) ) ) ).
% a_lagrange
thf(fact_683_minus__eq,axiom,
! [X: a,Y: a] :
( ( a_minus_a_b @ r @ X @ Y )
= ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).
% minus_eq
thf(fact_684_add_Oone__in__subset,axiom,
! [H2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H2 != bot_bot_set_a )
=> ( ! [X4: a] :
( ( member_a @ X4 @ H2 )
=> ( member_a @ ( a_inv_a_b @ r @ X4 ) @ H2 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ H2 )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ H2 )
=> ( member_a @ ( add_a_b @ r @ X4 @ Xa2 ) @ H2 ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H2 ) ) ) ) ) ).
% add.one_in_subset
thf(fact_685_dimension__sum__space,axiom,
! [K2: set_a,N: nat,E: set_a,M: nat,F2: set_a,K: nat] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M @ K2 @ F2 )
=> ( ( embedd2795209813406577254on_a_b @ r @ K @ K2 @ ( inf_inf_set_a @ E @ F2 ) )
=> ( embedd2795209813406577254on_a_b @ r @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) @ K2 @ ( set_add_a_b @ r @ E @ F2 ) ) ) ) ) ) ).
% dimension_sum_space
thf(fact_686_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_687_subring__props_I4_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( K2 != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_688_subalgebra__inter,axiom,
! [K2: set_a,V: set_a,V4: set_a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V4 @ r )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( inf_inf_set_a @ V @ V4 ) @ r ) ) ) ).
% subalgebra_inter
thf(fact_689_finite__Int,axiom,
! [F2: set_list_a,G: set_list_a] :
( ( ( finite_finite_list_a @ F2 )
| ( finite_finite_list_a @ G ) )
=> ( finite_finite_list_a @ ( inf_inf_set_list_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_690_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_691_finite__Int,axiom,
! [F2: set_a,G: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_692_minus__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_693_r__right__minus__eq,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( A = B ) ) ) ) ).
% r_right_minus_eq
thf(fact_694_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_695_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_696_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_697_infinite__imp__nonempty,axiom,
! [S: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ( S != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_698_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_699_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_700_finite__has__maximal,axiom,
! [A3: set_set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_701_finite__has__maximal,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_702_finite__has__minimal,axiom,
! [A3: set_set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_703_finite__has__minimal,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_704_ring_Osubalgebra__inter,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,V: set_a,V4: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V @ R )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V4 @ R )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( inf_inf_set_a @ V @ V4 ) @ R ) ) ) ) ).
% ring.subalgebra_inter
thf(fact_705_monoid_Ocarrier__not__empty,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( partia707051561876973205xt_a_b @ G )
!= bot_bot_set_a ) ) ).
% monoid.carrier_not_empty
thf(fact_706_ring_Osubring__props_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( K2 != bot_bot_set_a ) ) ) ).
% ring.subring_props(4)
thf(fact_707_ring_Oring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_minus_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_708_a__minus__def,axiom,
( a_minus_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,X3: a,Y4: a] : ( add_a_b @ R2 @ X3 @ ( a_inv_a_b @ R2 @ Y4 ) ) ) ) ).
% a_minus_def
thf(fact_709_abelian__group_Ominus__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( a_minus_a_b @ G @ X @ Y ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_group.minus_closed
thf(fact_710_card__Diff__subset__Int,axiom,
! [A3: set_set_a,B6: set_set_a] :
( ( finite_finite_set_a @ ( inf_inf_set_set_a @ A3 @ B6 ) )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A3 @ B6 ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A3 ) @ ( finite_card_set_a @ ( inf_inf_set_set_a @ A3 @ B6 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_711_card__Diff__subset__Int,axiom,
! [A3: set_list_a,B6: set_list_a] :
( ( finite_finite_list_a @ ( inf_inf_set_list_a @ A3 @ B6 ) )
=> ( ( finite_card_list_a @ ( minus_646659088055828811list_a @ A3 @ B6 ) )
= ( minus_minus_nat @ ( finite_card_list_a @ A3 ) @ ( finite_card_list_a @ ( inf_inf_set_list_a @ A3 @ B6 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_712_card__Diff__subset__Int,axiom,
! [A3: set_nat,B6: set_nat] :
( ( finite_finite_nat @ ( inf_inf_set_nat @ A3 @ B6 ) )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B6 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A3 @ B6 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_713_card__Diff__subset__Int,axiom,
! [A3: set_a,B6: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A3 @ B6 ) )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A3 @ B6 ) )
= ( minus_minus_nat @ ( finite_card_a @ A3 ) @ ( finite_card_a @ ( inf_inf_set_a @ A3 @ B6 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_714_ring_Oring__simprules_I14_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( a_minus_a_b @ R @ X @ Y )
= ( add_a_b @ R @ X @ ( a_inv_a_b @ R @ Y ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_715_abelian__group_Ominus__eq,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( a_minus_a_b @ G @ X @ Y )
= ( add_a_b @ G @ X @ ( a_inv_a_b @ G @ Y ) ) ) ) ).
% abelian_group.minus_eq
thf(fact_716_ring_Odimension__sum__space,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,M: nat,F2: set_a,K: nat] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ R @ M @ K2 @ F2 )
=> ( ( embedd2795209813406577254on_a_b @ R @ K @ K2 @ ( inf_inf_set_a @ E @ F2 ) )
=> ( embedd2795209813406577254on_a_b @ R @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) @ K2 @ ( set_add_a_b @ R @ E @ F2 ) ) ) ) ) ) ) ).
% ring.dimension_sum_space
thf(fact_717_abelian__group_Oa__lagrange,axiom,
! [G: partia2175431115845679010xt_a_b,H2: set_a] :
( ( abelian_group_a_b @ G )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( additi2834746164131130830up_a_b @ H2 @ G )
=> ( ( times_times_nat @ ( finite_card_set_a @ ( a_RCOSETS_a_b @ G @ H2 ) ) @ ( finite_card_a @ H2 ) )
= ( order_a_ring_ext_a_b @ G ) ) ) ) ) ).
% abelian_group.a_lagrange
thf(fact_718_Diff__disjoint,axiom,
! [A3: set_a,B6: set_a] :
( ( inf_inf_set_a @ A3 @ ( minus_minus_set_a @ B6 @ A3 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_719_Diff__eq__empty__iff,axiom,
! [A3: set_a,B6: set_a] :
( ( ( minus_minus_set_a @ A3 @ B6 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A3 @ B6 ) ) ).
% Diff_eq_empty_iff
thf(fact_720_dimension__direct__sum__space,axiom,
! [K2: set_a,N: nat,E: set_a,M: nat,F2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M @ K2 @ F2 )
=> ( ( ( inf_inf_set_a @ E @ F2 )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( embedd2795209813406577254on_a_b @ r @ ( plus_plus_nat @ N @ M ) @ K2 @ ( set_add_a_b @ r @ E @ F2 ) ) ) ) ) ) ).
% dimension_direct_sum_space
thf(fact_721_subsetI,axiom,
! [A3: set_set_a,B6: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A3 )
=> ( member_set_a @ X4 @ B6 ) )
=> ( ord_le3724670747650509150_set_a @ A3 @ B6 ) ) ).
% subsetI
thf(fact_722_subsetI,axiom,
! [A3: set_a,B6: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( member_a @ X4 @ B6 ) )
=> ( ord_less_eq_set_a @ A3 @ B6 ) ) ).
% subsetI
thf(fact_723_subset__antisym,axiom,
! [A3: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ A3 )
=> ( A3 = B6 ) ) ) ).
% subset_antisym
thf(fact_724_DiffI,axiom,
! [C: set_a,A3: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ A3 )
=> ( ~ ( member_set_a @ C @ B6 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A3 @ B6 ) ) ) ) ).
% DiffI
thf(fact_725_DiffI,axiom,
! [C: a,A3: set_a,B6: set_a] :
( ( member_a @ C @ A3 )
=> ( ~ ( member_a @ C @ B6 )
=> ( member_a @ C @ ( minus_minus_set_a @ A3 @ B6 ) ) ) ) ).
% DiffI
thf(fact_726_Diff__iff,axiom,
! [C: set_a,A3: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A3 @ B6 ) )
= ( ( member_set_a @ C @ A3 )
& ~ ( member_set_a @ C @ B6 ) ) ) ).
% Diff_iff
thf(fact_727_Diff__iff,axiom,
! [C: a,A3: set_a,B6: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B6 ) )
= ( ( member_a @ C @ A3 )
& ~ ( member_a @ C @ B6 ) ) ) ).
% Diff_iff
thf(fact_728_Diff__idemp,axiom,
! [A3: set_a,B6: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ B6 ) @ B6 )
= ( minus_minus_set_a @ A3 @ B6 ) ) ).
% Diff_idemp
thf(fact_729_genideal__self_H,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_730_genideal__zero,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% genideal_zero
thf(fact_731_carrier__one__not__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) ) ) ).
% carrier_one_not_zero
thf(fact_732_carrier__one__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% carrier_one_zero
thf(fact_733_one__zeroD,axiom,
( ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) )
=> ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% one_zeroD
thf(fact_734_one__zeroI,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% one_zeroI
thf(fact_735_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_736_Idl__subset__ideal_H,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_737_genideal__one,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) )
= ( partia707051561876973205xt_a_b @ r ) ) ).
% genideal_one
thf(fact_738_subfield__m__inv__simprule,axiom,
! [K2: set_a,K: a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ A ) @ K2 )
=> ( member_a @ A @ K2 ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_739_subset__empty,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_740_empty__subsetI,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% empty_subsetI
thf(fact_741_insert__subset,axiom,
! [X: set_a,A3: set_set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A3 ) @ B6 )
= ( ( member_set_a @ X @ B6 )
& ( ord_le3724670747650509150_set_a @ A3 @ B6 ) ) ) ).
% insert_subset
thf(fact_742_insert__subset,axiom,
! [X: a,A3: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A3 ) @ B6 )
= ( ( member_a @ X @ B6 )
& ( ord_less_eq_set_a @ A3 @ B6 ) ) ) ).
% insert_subset
thf(fact_743_finite__insert,axiom,
! [A: list_a,A3: set_list_a] :
( ( finite_finite_list_a @ ( insert_list_a @ A @ A3 ) )
= ( finite_finite_list_a @ A3 ) ) ).
% finite_insert
thf(fact_744_finite__insert,axiom,
! [A: a,A3: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A3 ) )
= ( finite_finite_a @ A3 ) ) ).
% finite_insert
thf(fact_745_finite__insert,axiom,
! [A: nat,A3: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A3 ) )
= ( finite_finite_nat @ A3 ) ) ).
% finite_insert
thf(fact_746_Int__subset__iff,axiom,
! [C3: set_a,A3: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A3 @ B6 ) )
= ( ( ord_less_eq_set_a @ C3 @ A3 )
& ( ord_less_eq_set_a @ C3 @ B6 ) ) ) ).
% Int_subset_iff
thf(fact_747_Diff__cancel,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ A3 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_748_empty__Diff,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A3 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_749_Diff__empty,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% Diff_empty
thf(fact_750_Diff__insert0,axiom,
! [X: set_a,A3: set_set_a,B6: set_set_a] :
( ~ ( member_set_a @ X @ A3 )
=> ( ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X @ B6 ) )
= ( minus_5736297505244876581_set_a @ A3 @ B6 ) ) ) ).
% Diff_insert0
thf(fact_751_Diff__insert0,axiom,
! [X: a,A3: set_a,B6: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( minus_minus_set_a @ A3 @ ( insert_a @ X @ B6 ) )
= ( minus_minus_set_a @ A3 @ B6 ) ) ) ).
% Diff_insert0
thf(fact_752_insert__Diff1,axiom,
! [X: set_a,B6: set_set_a,A3: set_set_a] :
( ( member_set_a @ X @ B6 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A3 ) @ B6 )
= ( minus_5736297505244876581_set_a @ A3 @ B6 ) ) ) ).
% insert_Diff1
thf(fact_753_insert__Diff1,axiom,
! [X: a,B6: set_a,A3: set_a] :
( ( member_a @ X @ B6 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B6 )
= ( minus_minus_set_a @ A3 @ B6 ) ) ) ).
% insert_Diff1
thf(fact_754_space__subgroup__props_I6_J,axiom,
! [K2: set_a,N: nat,E: set_a,K: a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ A ) @ E )
=> ( member_a @ A @ E ) ) ) ) ) ) ).
% space_subgroup_props(6)
thf(fact_755_singleton__insert__inj__eq_H,axiom,
! [A: a,A3: set_a,B: a] :
( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_756_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A3: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_757_insert__Diff__single,axiom,
! [A: a,A3: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_758_finite__Diff__insert,axiom,
! [A3: set_list_a,A: list_a,B6: set_list_a] :
( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ A @ B6 ) ) )
= ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A3 @ B6 ) ) ) ).
% finite_Diff_insert
thf(fact_759_finite__Diff__insert,axiom,
! [A3: set_nat,A: nat,B6: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ B6 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B6 ) ) ) ).
% finite_Diff_insert
thf(fact_760_finite__Diff__insert,axiom,
! [A3: set_a,A: a,B6: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ B6 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A3 @ B6 ) ) ) ).
% finite_Diff_insert
thf(fact_761_card__insert__disjoint,axiom,
! [A3: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ~ ( member_set_a @ X @ A3 )
=> ( ( finite_card_set_a @ ( insert_set_a @ X @ A3 ) )
= ( suc @ ( finite_card_set_a @ A3 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_762_card__insert__disjoint,axiom,
! [A3: set_list_a,X: list_a] :
( ( finite_finite_list_a @ A3 )
=> ( ~ ( member_list_a @ X @ A3 )
=> ( ( finite_card_list_a @ ( insert_list_a @ X @ A3 ) )
= ( suc @ ( finite_card_list_a @ A3 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_763_card__insert__disjoint,axiom,
! [A3: set_a,X: a] :
( ( finite_finite_a @ A3 )
=> ( ~ ( member_a @ X @ A3 )
=> ( ( finite_card_a @ ( insert_a @ X @ A3 ) )
= ( suc @ ( finite_card_a @ A3 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_764_card__insert__disjoint,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ~ ( member_nat @ X @ A3 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A3 ) )
= ( suc @ ( finite_card_nat @ A3 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_765_Diff__insert,axiom,
! [A3: set_a,A: a,B6: set_a] :
( ( minus_minus_set_a @ A3 @ ( insert_a @ A @ B6 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ B6 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_766_insert__Diff,axiom,
! [A: set_a,A3: set_set_a] :
( ( member_set_a @ A @ A3 )
=> ( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_767_insert__Diff,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_768_Diff__insert2,axiom,
! [A3: set_a,A: a,B6: set_a] :
( ( minus_minus_set_a @ A3 @ ( insert_a @ A @ B6 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B6 ) ) ).
% Diff_insert2
thf(fact_769_insert__Diff__if,axiom,
! [X: set_a,B6: set_set_a,A3: set_set_a] :
( ( ( member_set_a @ X @ B6 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A3 ) @ B6 )
= ( minus_5736297505244876581_set_a @ A3 @ B6 ) ) )
& ( ~ ( member_set_a @ X @ B6 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A3 ) @ B6 )
= ( insert_set_a @ X @ ( minus_5736297505244876581_set_a @ A3 @ B6 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_770_insert__Diff__if,axiom,
! [X: a,B6: set_a,A3: set_a] :
( ( ( member_a @ X @ B6 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B6 )
= ( minus_minus_set_a @ A3 @ B6 ) ) )
& ( ~ ( member_a @ X @ B6 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B6 )
= ( insert_a @ X @ ( minus_minus_set_a @ A3 @ B6 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_771_Diff__insert__absorb,axiom,
! [X: set_a,A3: set_set_a] :
( ~ ( member_set_a @ X @ A3 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A3 ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_772_Diff__insert__absorb,axiom,
! [X: a,A3: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_773_subset__singleton__iff,axiom,
! [X6: set_a,A: a] :
( ( ord_less_eq_set_a @ X6 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X6 = bot_bot_set_a )
| ( X6
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_774_subset__singletonD,axiom,
! [A3: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A3 = bot_bot_set_a )
| ( A3
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_775_subset__Diff__insert,axiom,
! [A3: set_set_a,B6: set_set_a,X: set_a,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( minus_5736297505244876581_set_a @ B6 @ ( insert_set_a @ X @ C3 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A3 @ ( minus_5736297505244876581_set_a @ B6 @ C3 ) )
& ~ ( member_set_a @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_776_subset__Diff__insert,axiom,
! [A3: set_a,B6: set_a,X: a,C3: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B6 @ ( insert_a @ X @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B6 @ C3 ) )
& ~ ( member_a @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_777_finite_OinsertI,axiom,
! [A3: set_list_a,A: list_a] :
( ( finite_finite_list_a @ A3 )
=> ( finite_finite_list_a @ ( insert_list_a @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_778_finite_OinsertI,axiom,
! [A3: set_a,A: a] :
( ( finite_finite_a @ A3 )
=> ( finite_finite_a @ ( insert_a @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_779_finite_OinsertI,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_780_insert__mono,axiom,
! [C3: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C3 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_781_subset__insert,axiom,
! [X: set_a,A3: set_set_a,B6: set_set_a] :
( ~ ( member_set_a @ X @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ A3 @ ( insert_set_a @ X @ B6 ) )
= ( ord_le3724670747650509150_set_a @ A3 @ B6 ) ) ) ).
% subset_insert
thf(fact_782_subset__insert,axiom,
! [X: a,A3: set_a,B6: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B6 ) )
= ( ord_less_eq_set_a @ A3 @ B6 ) ) ) ).
% subset_insert
thf(fact_783_subset__insertI,axiom,
! [B6: set_a,A: a] : ( ord_less_eq_set_a @ B6 @ ( insert_a @ A @ B6 ) ) ).
% subset_insertI
thf(fact_784_subset__insertI2,axiom,
! [A3: set_a,B6: set_a,B: a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ B6 ) ) ) ).
% subset_insertI2
thf(fact_785_Diff__single__insert,axiom,
! [A3: set_a,X: a,B6: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B6 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B6 ) ) ) ).
% Diff_single_insert
thf(fact_786_subset__insert__iff,axiom,
! [A3: set_set_a,X: set_a,B6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ ( insert_set_a @ X @ B6 ) )
= ( ( ( member_set_a @ X @ A3 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B6 ) )
& ( ~ ( member_set_a @ X @ A3 )
=> ( ord_le3724670747650509150_set_a @ A3 @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_787_subset__insert__iff,axiom,
! [A3: set_a,X: a,B6: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B6 ) )
= ( ( ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B6 ) )
& ( ~ ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ A3 @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_788_finite_Ocases,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( ( A != bot_bot_set_list_a )
=> ~ ! [A6: set_list_a] :
( ? [A2: list_a] :
( A
= ( insert_list_a @ A2 @ A6 ) )
=> ~ ( finite_finite_list_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_789_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A2: nat] :
( A
= ( insert_nat @ A2 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_790_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A6: set_a] :
( ? [A2: a] :
( A
= ( insert_a @ A2 @ A6 ) )
=> ~ ( finite_finite_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_791_finite_Osimps,axiom,
( finite_finite_list_a
= ( ^ [A4: set_list_a] :
( ( A4 = bot_bot_set_list_a )
| ? [A7: set_list_a,B7: list_a] :
( ( A4
= ( insert_list_a @ B7 @ A7 ) )
& ( finite_finite_list_a @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_792_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A7: set_nat,B7: nat] :
( ( A4
= ( insert_nat @ B7 @ A7 ) )
& ( finite_finite_nat @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_793_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A7: set_a,B7: a] :
( ( A4
= ( insert_a @ B7 @ A7 ) )
& ( finite_finite_a @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_794_finite__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X4: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X4 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_795_finite__induct,axiom,
! [F2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X4: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ~ ( member_list_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a @ X4 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_796_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X4 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_797_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X4 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_798_finite__ne__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( F2 != bot_bot_set_set_a )
=> ( ! [X4: set_a] : ( P @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) )
=> ( ! [X4: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( F3 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_799_finite__ne__induct,axiom,
! [F2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( F2 != bot_bot_set_list_a )
=> ( ! [X4: list_a] : ( P @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) )
=> ( ! [X4: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( F3 != bot_bot_set_list_a )
=> ( ~ ( member_list_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a @ X4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_800_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X4: nat] : ( P @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
=> ( ! [X4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_801_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X4: a] : ( P @ ( insert_a @ X4 @ bot_bot_set_a ) )
=> ( ! [X4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_802_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A3: set_set_a] :
( ! [A6: set_set_a] :
( ~ ( finite_finite_set_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X4: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X4 @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_803_infinite__finite__induct,axiom,
! [P: set_list_a > $o,A3: set_list_a] :
( ! [A6: set_list_a] :
( ~ ( finite_finite_list_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X4: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ~ ( member_list_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a @ X4 @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_804_infinite__finite__induct,axiom,
! [P: set_nat > $o,A3: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X4 @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_805_infinite__finite__induct,axiom,
! [P: set_a > $o,A3: set_a] :
( ! [A6: set_a] :
( ~ ( finite_finite_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X4 @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_806_card__insert__le,axiom,
! [A3: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ ( finite_card_a @ ( insert_a @ X @ A3 ) ) ) ).
% card_insert_le
thf(fact_807_card__insert__le,axiom,
! [A3: set_set_a,X: set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ A3 ) @ ( finite_card_set_a @ ( insert_set_a @ X @ A3 ) ) ) ).
% card_insert_le
thf(fact_808_finite__subset__induct_H,axiom,
! [F2: set_set_a,A3: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A3 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A2: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A2 @ A3 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A3 )
=> ( ~ ( member_set_a @ A2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A2 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_809_finite__subset__induct_H,axiom,
! [F2: set_list_a,A3: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( ord_le8861187494160871172list_a @ F2 @ A3 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A2: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( member_list_a @ A2 @ A3 )
=> ( ( ord_le8861187494160871172list_a @ F3 @ A3 )
=> ( ~ ( member_list_a @ A2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a @ A2 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_810_finite__subset__induct_H,axiom,
! [F2: set_nat,A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A2 @ A3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A3 )
=> ( ~ ( member_nat @ A2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A2 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_811_finite__subset__induct_H,axiom,
! [F2: set_a,A3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A2 @ A3 )
=> ( ( ord_less_eq_set_a @ F3 @ A3 )
=> ( ~ ( member_a @ A2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A2 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_812_finite__subset__induct,axiom,
! [F2: set_set_a,A3: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A3 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A2: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A2 @ A3 )
=> ( ~ ( member_set_a @ A2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_813_finite__subset__induct,axiom,
! [F2: set_list_a,A3: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( ord_le8861187494160871172list_a @ F2 @ A3 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A2: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( member_list_a @ A2 @ A3 )
=> ( ~ ( member_list_a @ A2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a @ A2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_814_finite__subset__induct,axiom,
! [F2: set_nat,A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A2 @ A3 )
=> ( ~ ( member_nat @ A2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_815_finite__subset__induct,axiom,
! [F2: set_a,A3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A2 @ A3 )
=> ( ~ ( member_a @ A2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_816_finite__empty__induct,axiom,
! [A3: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ A3 )
=> ( ( P @ A3 )
=> ( ! [A2: set_a,A6: set_set_a] :
( ( finite_finite_set_a @ A6 )
=> ( ( member_set_a @ A2 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_5736297505244876581_set_a @ A6 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_817_finite__empty__induct,axiom,
! [A3: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ A3 )
=> ( ( P @ A3 )
=> ( ! [A2: list_a,A6: set_list_a] :
( ( finite_finite_list_a @ A6 )
=> ( ( member_list_a @ A2 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ) ) ) )
=> ( P @ bot_bot_set_list_a ) ) ) ) ).
% finite_empty_induct
thf(fact_818_finite__empty__induct,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ( P @ A3 )
=> ( ! [A2: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat @ A2 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_819_finite__empty__induct,axiom,
! [A3: set_a,P: set_a > $o] :
( ( finite_finite_a @ A3 )
=> ( ( P @ A3 )
=> ( ! [A2: a,A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( member_a @ A2 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_820_infinite__coinduct,axiom,
! [X6: set_list_a > $o,A3: set_list_a] :
( ( X6 @ A3 )
=> ( ! [A6: set_list_a] :
( ( X6 @ A6 )
=> ? [X5: list_a] :
( ( member_list_a @ X5 @ A6 )
& ( ( X6 @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) )
| ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) ) ) )
=> ~ ( finite_finite_list_a @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_821_infinite__coinduct,axiom,
! [X6: set_nat > $o,A3: set_nat] :
( ( X6 @ A3 )
=> ( ! [A6: set_nat] :
( ( X6 @ A6 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A6 )
& ( ( X6 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_822_infinite__coinduct,axiom,
! [X6: set_a > $o,A3: set_a] :
( ( X6 @ A3 )
=> ( ! [A6: set_a] :
( ( X6 @ A6 )
=> ? [X5: a] :
( ( member_a @ X5 @ A6 )
& ( ( X6 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A6 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_823_infinite__remove,axiom,
! [S: set_list_a,A: list_a] :
( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% infinite_remove
thf(fact_824_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_825_infinite__remove,axiom,
! [S: set_a,A: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_826_card__Suc__eq__finite,axiom,
! [A3: set_set_a,K: nat] :
( ( ( finite_card_set_a @ A3 )
= ( suc @ K ) )
= ( ? [B7: set_a,B8: set_set_a] :
( ( A3
= ( insert_set_a @ B7 @ B8 ) )
& ~ ( member_set_a @ B7 @ B8 )
& ( ( finite_card_set_a @ B8 )
= K )
& ( finite_finite_set_a @ B8 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_827_card__Suc__eq__finite,axiom,
! [A3: set_list_a,K: nat] :
( ( ( finite_card_list_a @ A3 )
= ( suc @ K ) )
= ( ? [B7: list_a,B8: set_list_a] :
( ( A3
= ( insert_list_a @ B7 @ B8 ) )
& ~ ( member_list_a @ B7 @ B8 )
& ( ( finite_card_list_a @ B8 )
= K )
& ( finite_finite_list_a @ B8 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_828_card__Suc__eq__finite,axiom,
! [A3: set_a,K: nat] :
( ( ( finite_card_a @ A3 )
= ( suc @ K ) )
= ( ? [B7: a,B8: set_a] :
( ( A3
= ( insert_a @ B7 @ B8 ) )
& ~ ( member_a @ B7 @ B8 )
& ( ( finite_card_a @ B8 )
= K )
& ( finite_finite_a @ B8 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_829_card__Suc__eq__finite,axiom,
! [A3: set_nat,K: nat] :
( ( ( finite_card_nat @ A3 )
= ( suc @ K ) )
= ( ? [B7: nat,B8: set_nat] :
( ( A3
= ( insert_nat @ B7 @ B8 ) )
& ~ ( member_nat @ B7 @ B8 )
& ( ( finite_card_nat @ B8 )
= K )
& ( finite_finite_nat @ B8 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_830_card__insert__if,axiom,
! [A3: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( ( member_set_a @ X @ A3 )
=> ( ( finite_card_set_a @ ( insert_set_a @ X @ A3 ) )
= ( finite_card_set_a @ A3 ) ) )
& ( ~ ( member_set_a @ X @ A3 )
=> ( ( finite_card_set_a @ ( insert_set_a @ X @ A3 ) )
= ( suc @ ( finite_card_set_a @ A3 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_831_card__insert__if,axiom,
! [A3: set_list_a,X: list_a] :
( ( finite_finite_list_a @ A3 )
=> ( ( ( member_list_a @ X @ A3 )
=> ( ( finite_card_list_a @ ( insert_list_a @ X @ A3 ) )
= ( finite_card_list_a @ A3 ) ) )
& ( ~ ( member_list_a @ X @ A3 )
=> ( ( finite_card_list_a @ ( insert_list_a @ X @ A3 ) )
= ( suc @ ( finite_card_list_a @ A3 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_832_card__insert__if,axiom,
! [A3: set_a,X: a] :
( ( finite_finite_a @ A3 )
=> ( ( ( member_a @ X @ A3 )
=> ( ( finite_card_a @ ( insert_a @ X @ A3 ) )
= ( finite_card_a @ A3 ) ) )
& ( ~ ( member_a @ X @ A3 )
=> ( ( finite_card_a @ ( insert_a @ X @ A3 ) )
= ( suc @ ( finite_card_a @ A3 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_833_card__insert__if,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ( member_nat @ X @ A3 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A3 ) )
= ( finite_card_nat @ A3 ) ) )
& ( ~ ( member_nat @ X @ A3 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A3 ) )
= ( suc @ ( finite_card_nat @ A3 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_834_in__mono,axiom,
! [A3: set_set_a,B6: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B6 )
=> ( ( member_set_a @ X @ A3 )
=> ( member_set_a @ X @ B6 ) ) ) ).
% in_mono
thf(fact_835_in__mono,axiom,
! [A3: set_a,B6: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ X @ B6 ) ) ) ).
% in_mono
thf(fact_836_subsetD,axiom,
! [A3: set_set_a,B6: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B6 )
=> ( ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ B6 ) ) ) ).
% subsetD
thf(fact_837_subsetD,axiom,
! [A3: set_a,B6: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B6 ) ) ) ).
% subsetD
thf(fact_838_equalityE,axiom,
! [A3: set_a,B6: set_a] :
( ( A3 = B6 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B6 )
=> ~ ( ord_less_eq_set_a @ B6 @ A3 ) ) ) ).
% equalityE
thf(fact_839_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A7: set_set_a,B8: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A7 )
=> ( member_set_a @ X3 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_840_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B8: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A7 )
=> ( member_a @ X3 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_841_equalityD1,axiom,
! [A3: set_a,B6: set_a] :
( ( A3 = B6 )
=> ( ord_less_eq_set_a @ A3 @ B6 ) ) ).
% equalityD1
thf(fact_842_equalityD2,axiom,
! [A3: set_a,B6: set_a] :
( ( A3 = B6 )
=> ( ord_less_eq_set_a @ B6 @ A3 ) ) ).
% equalityD2
thf(fact_843_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A7: set_set_a,B8: set_set_a] :
! [T3: set_a] :
( ( member_set_a @ T3 @ A7 )
=> ( member_set_a @ T3 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_844_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B8: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A7 )
=> ( member_a @ T3 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_845_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_846_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_847_subset__trans,axiom,
! [A3: set_a,B6: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ C3 )
=> ( ord_less_eq_set_a @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_848_set__eq__subset,axiom,
( ( ^ [Y6: set_a,Z4: set_a] : ( Y6 = Z4 ) )
= ( ^ [A7: set_a,B8: set_a] :
( ( ord_less_eq_set_a @ A7 @ B8 )
& ( ord_less_eq_set_a @ B8 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_849_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_850_DiffE,axiom,
! [C: set_a,A3: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A3 @ B6 ) )
=> ~ ( ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ B6 ) ) ) ).
% DiffE
thf(fact_851_DiffE,axiom,
! [C: a,A3: set_a,B6: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B6 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B6 ) ) ) ).
% DiffE
thf(fact_852_DiffD1,axiom,
! [C: set_a,A3: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A3 @ B6 ) )
=> ( member_set_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_853_DiffD1,axiom,
! [C: a,A3: set_a,B6: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B6 ) )
=> ( member_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_854_DiffD2,axiom,
! [C: set_a,A3: set_set_a,B6: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A3 @ B6 ) )
=> ~ ( member_set_a @ C @ B6 ) ) ).
% DiffD2
thf(fact_855_DiffD2,axiom,
! [C: a,A3: set_a,B6: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B6 ) )
=> ~ ( member_a @ C @ B6 ) ) ).
% DiffD2
thf(fact_856_remove__induct,axiom,
! [P: set_set_a > $o,B6: set_set_a] :
( ( P @ bot_bot_set_set_a )
=> ( ( ~ ( finite_finite_set_a @ B6 )
=> ( P @ B6 ) )
=> ( ! [A6: set_set_a] :
( ( finite_finite_set_a @ A6 )
=> ( ( A6 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A6 @ B6 )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A6 )
=> ( P @ ( minus_5736297505244876581_set_a @ A6 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B6 ) ) ) ) ).
% remove_induct
thf(fact_857_remove__induct,axiom,
! [P: set_list_a > $o,B6: set_list_a] :
( ( P @ bot_bot_set_list_a )
=> ( ( ~ ( finite_finite_list_a @ B6 )
=> ( P @ B6 ) )
=> ( ! [A6: set_list_a] :
( ( finite_finite_list_a @ A6 )
=> ( ( A6 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A6 @ B6 )
=> ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A6 )
=> ( P @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B6 ) ) ) ) ).
% remove_induct
thf(fact_858_remove__induct,axiom,
! [P: set_nat > $o,B6: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B6 )
=> ( P @ B6 ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B6 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B6 ) ) ) ) ).
% remove_induct
thf(fact_859_remove__induct,axiom,
! [P: set_a > $o,B6: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B6 )
=> ( P @ B6 ) )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B6 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B6 ) ) ) ) ).
% remove_induct
thf(fact_860_finite__remove__induct,axiom,
! [B6: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ B6 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A6: set_set_a] :
( ( finite_finite_set_a @ A6 )
=> ( ( A6 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A6 @ B6 )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A6 )
=> ( P @ ( minus_5736297505244876581_set_a @ A6 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B6 ) ) ) ) ).
% finite_remove_induct
thf(fact_861_finite__remove__induct,axiom,
! [B6: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ B6 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A6: set_list_a] :
( ( finite_finite_list_a @ A6 )
=> ( ( A6 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A6 @ B6 )
=> ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A6 )
=> ( P @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B6 ) ) ) ) ).
% finite_remove_induct
thf(fact_862_finite__remove__induct,axiom,
! [B6: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B6 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B6 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B6 ) ) ) ) ).
% finite_remove_induct
thf(fact_863_finite__remove__induct,axiom,
! [B6: set_a,P: set_a > $o] :
( ( finite_finite_a @ B6 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B6 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B6 ) ) ) ) ).
% finite_remove_induct
thf(fact_864_card__le__Suc__iff,axiom,
! [N: nat,A3: set_set_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_set_a @ A3 ) )
= ( ? [A4: set_a,B8: set_set_a] :
( ( A3
= ( insert_set_a @ A4 @ B8 ) )
& ~ ( member_set_a @ A4 @ B8 )
& ( ord_less_eq_nat @ N @ ( finite_card_set_a @ B8 ) )
& ( finite_finite_set_a @ B8 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_865_card__le__Suc__iff,axiom,
! [N: nat,A3: set_list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_list_a @ A3 ) )
= ( ? [A4: list_a,B8: set_list_a] :
( ( A3
= ( insert_list_a @ A4 @ B8 ) )
& ~ ( member_list_a @ A4 @ B8 )
& ( ord_less_eq_nat @ N @ ( finite_card_list_a @ B8 ) )
& ( finite_finite_list_a @ B8 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_866_card__le__Suc__iff,axiom,
! [N: nat,A3: set_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_a @ A3 ) )
= ( ? [A4: a,B8: set_a] :
( ( A3
= ( insert_a @ A4 @ B8 ) )
& ~ ( member_a @ A4 @ B8 )
& ( ord_less_eq_nat @ N @ ( finite_card_a @ B8 ) )
& ( finite_finite_a @ B8 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_867_card__le__Suc__iff,axiom,
! [N: nat,A3: set_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A3 ) )
= ( ? [A4: nat,B8: set_nat] :
( ( A3
= ( insert_nat @ A4 @ B8 ) )
& ~ ( member_nat @ A4 @ B8 )
& ( ord_less_eq_nat @ N @ ( finite_card_nat @ B8 ) )
& ( finite_finite_nat @ B8 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_868_card__Diff1__le,axiom,
! [A3: set_set_a,X: set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A3 ) ) ).
% card_Diff1_le
thf(fact_869_card__Diff1__le,axiom,
! [A3: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A3 ) ) ).
% card_Diff1_le
thf(fact_870_ring_Ogenideal__self_H,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ I @ ( genideal_a_b @ R @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_871_ring_Ogenideal__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( genideal_a_b @ R @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ).
% ring.genideal_zero
thf(fact_872_ring_Ozeropideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( principalideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ).
% ring.zeropideal
thf(fact_873_principalideal_Ogenerate,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
& ( I2
= ( genideal_a_b @ R @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_874_card_Oremove,axiom,
! [A3: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ X @ A3 )
=> ( ( finite_card_set_a @ A3 )
= ( suc @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ).
% card.remove
thf(fact_875_card_Oremove,axiom,
! [A3: set_list_a,X: list_a] :
( ( finite_finite_list_a @ A3 )
=> ( ( member_list_a @ X @ A3 )
=> ( ( finite_card_list_a @ A3 )
= ( suc @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ) ) ) ) ).
% card.remove
thf(fact_876_card_Oremove,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ( finite_card_nat @ A3 )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% card.remove
thf(fact_877_card_Oremove,axiom,
! [A3: set_a,X: a] :
( ( finite_finite_a @ A3 )
=> ( ( member_a @ X @ A3 )
=> ( ( finite_card_a @ A3 )
= ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ) ).
% card.remove
thf(fact_878_card_Oinsert__remove,axiom,
! [A3: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( finite_card_set_a @ ( insert_set_a @ X @ A3 ) )
= ( suc @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_879_card_Oinsert__remove,axiom,
! [A3: set_list_a,X: list_a] :
( ( finite_finite_list_a @ A3 )
=> ( ( finite_card_list_a @ ( insert_list_a @ X @ A3 ) )
= ( suc @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_880_card_Oinsert__remove,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A3 ) )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_881_card_Oinsert__remove,axiom,
! [A3: set_a,X: a] :
( ( finite_finite_a @ A3 )
=> ( ( finite_card_a @ ( insert_a @ X @ A3 ) )
= ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_882_card__Suc__Diff1,axiom,
! [A3: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ X @ A3 )
=> ( ( suc @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) )
= ( finite_card_set_a @ A3 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_883_card__Suc__Diff1,axiom,
! [A3: set_list_a,X: list_a] :
( ( finite_finite_list_a @ A3 )
=> ( ( member_list_a @ X @ A3 )
=> ( ( suc @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) )
= ( finite_card_list_a @ A3 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_884_card__Suc__Diff1,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
= ( finite_card_nat @ A3 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_885_card__Suc__Diff1,axiom,
! [A3: set_a,X: a] :
( ( finite_finite_a @ A3 )
=> ( ( member_a @ X @ A3 )
=> ( ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
= ( finite_card_a @ A3 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_886_ring_OIdl__subset__ideal_H,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ R @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ R @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ R @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_887_ring_Ogenideal__one,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( genideal_a_b @ R @ ( insert_a @ ( one_a_ring_ext_a_b @ R ) @ bot_bot_set_a ) )
= ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.genideal_one
thf(fact_888_semiring_Oone__zeroD,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) )
=> ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_889_semiring_Oone__zeroI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_890_semiring_Ocarrier__one__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_891_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
!= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_892_Int__mono,axiom,
! [A3: set_a,C3: set_a,B6: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A3 @ C3 )
=> ( ( ord_less_eq_set_a @ B6 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B6 ) @ ( inf_inf_set_a @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_893_Int__lower1,axiom,
! [A3: set_a,B6: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B6 ) @ A3 ) ).
% Int_lower1
thf(fact_894_Int__lower2,axiom,
! [A3: set_a,B6: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B6 ) @ B6 ) ).
% Int_lower2
thf(fact_895_Int__absorb1,axiom,
! [B6: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B6 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B6 )
= B6 ) ) ).
% Int_absorb1
thf(fact_896_Int__absorb2,axiom,
! [A3: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( inf_inf_set_a @ A3 @ B6 )
= A3 ) ) ).
% Int_absorb2
thf(fact_897_Int__greatest,axiom,
! [C3: set_a,A3: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ C3 @ A3 )
=> ( ( ord_less_eq_set_a @ C3 @ B6 )
=> ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A3 @ B6 ) ) ) ) ).
% Int_greatest
thf(fact_898_Int__Collect__mono,axiom,
! [A3: set_set_a,B6: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A3 @ B6 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A3 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A3 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B6 @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_899_Int__Collect__mono,axiom,
! [A3: set_a,B6: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B6 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_900_double__diff,axiom,
! [A3: set_a,B6: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ C3 )
=> ( ( minus_minus_set_a @ B6 @ ( minus_minus_set_a @ C3 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_901_Diff__subset,axiom,
! [A3: set_a,B6: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B6 ) @ A3 ) ).
% Diff_subset
thf(fact_902_Diff__mono,axiom,
! [A3: set_a,C3: set_a,D2: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A3 @ C3 )
=> ( ( ord_less_eq_set_a @ D2 @ B6 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B6 ) @ ( minus_minus_set_a @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_903_Diff__Int__distrib2,axiom,
! [A3: set_a,B6: set_a,C3: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A3 @ B6 ) @ C3 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A3 @ C3 ) @ ( inf_inf_set_a @ B6 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_904_Diff__Int__distrib,axiom,
! [C3: set_a,A3: set_a,B6: set_a] :
( ( inf_inf_set_a @ C3 @ ( minus_minus_set_a @ A3 @ B6 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C3 @ A3 ) @ ( inf_inf_set_a @ C3 @ B6 ) ) ) ).
% Diff_Int_distrib
thf(fact_905_Diff__Diff__Int,axiom,
! [A3: set_a,B6: set_a] :
( ( minus_minus_set_a @ A3 @ ( minus_minus_set_a @ A3 @ B6 ) )
= ( inf_inf_set_a @ A3 @ B6 ) ) ).
% Diff_Diff_Int
thf(fact_906_Diff__Int2,axiom,
! [A3: set_a,C3: set_a,B6: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A3 @ C3 ) @ ( inf_inf_set_a @ B6 @ C3 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A3 @ C3 ) @ B6 ) ) ).
% Diff_Int2
thf(fact_907_Int__Diff,axiom,
! [A3: set_a,B6: set_a,C3: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A3 @ B6 ) @ C3 )
= ( inf_inf_set_a @ A3 @ ( minus_minus_set_a @ B6 @ C3 ) ) ) ).
% Int_Diff
thf(fact_908_ring_Ospace__subgroup__props_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,K: a,A: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ A ) @ E )
=> ( member_a @ A @ E ) ) ) ) ) ) ) ).
% ring.space_subgroup_props(6)
thf(fact_909_ring_Odimension__direct__sum__space,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,M: nat,F2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ R @ M @ K2 @ F2 )
=> ( ( ( inf_inf_set_a @ E @ F2 )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
=> ( embedd2795209813406577254on_a_b @ R @ ( plus_plus_nat @ N @ M ) @ K2 @ ( set_add_a_b @ R @ E @ F2 ) ) ) ) ) ) ) ).
% ring.dimension_direct_sum_space
thf(fact_910_Int__Diff__disjoint,axiom,
! [A3: set_a,B6: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B6 ) @ ( minus_minus_set_a @ A3 @ B6 ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_911_Diff__triv,axiom,
! [A3: set_a,B6: set_a] :
( ( ( inf_inf_set_a @ A3 @ B6 )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A3 @ B6 )
= A3 ) ) ).
% Diff_triv
thf(fact_912_dimension_Ocases,axiom,
! [A12: nat,A23: set_a,A32: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ A12 @ A23 @ A32 )
=> ( ( ( A12 = zero_zero_nat )
=> ( A32
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ~ ! [V3: a,E4: set_a,N2: nat] :
( ( A12
= ( suc @ N2 ) )
=> ( ( A32
= ( embedd971793762689825387on_a_b @ r @ A23 @ V3 @ E4 ) )
=> ( ( member_a @ V3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ V3 @ E4 )
=> ~ ( embedd2795209813406577254on_a_b @ r @ N2 @ A23 @ E4 ) ) ) ) ) ) ) ).
% dimension.cases
thf(fact_913_dimension_Osimps,axiom,
! [A12: nat,A23: set_a,A32: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ A12 @ A23 @ A32 )
= ( ? [K5: set_a] :
( ( A12 = zero_zero_nat )
& ( A23 = K5 )
& ( A32
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
| ? [V5: a,E5: set_a,N4: nat,K5: set_a] :
( ( A12
= ( suc @ N4 ) )
& ( A23 = K5 )
& ( A32
= ( embedd971793762689825387on_a_b @ r @ K5 @ V5 @ E5 ) )
& ( member_a @ V5 @ ( partia707051561876973205xt_a_b @ r ) )
& ~ ( member_a @ V5 @ E5 )
& ( embedd2795209813406577254on_a_b @ r @ N4 @ K5 @ E5 ) ) ) ) ).
% dimension.simps
thf(fact_914_zero__dim,axiom,
! [K2: set_a] : ( embedd2795209813406577254on_a_b @ r @ zero_zero_nat @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% zero_dim
thf(fact_915_dimension__zero,axiom,
! [K2: set_a,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ zero_zero_nat @ K2 @ E )
=> ( E
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% dimension_zero
thf(fact_916_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_917_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_918_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_919_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_920_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_921_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_922_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_923_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_924_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_925_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_926_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_927_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_928_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_929_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_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_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_938_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_939_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_940_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_941_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_942_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_943_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_944_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_945_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_946_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_947_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_948_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_949_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_950_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_951_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_952_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_953_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_954_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_955_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_956_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_957_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_958_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_959_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_960_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_961_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_962_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_963_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_964_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_965_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_966_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_967_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_968_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_969_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_970_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_971_card_Oempty,axiom,
( ( finite_card_set_a @ bot_bot_set_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_972_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_973_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_974_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_975_card_Oinfinite,axiom,
! [A3: set_set_a] :
( ~ ( finite_finite_set_a @ A3 )
=> ( ( finite_card_set_a @ A3 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_976_card_Oinfinite,axiom,
! [A3: set_list_a] :
( ~ ( finite_finite_list_a @ A3 )
=> ( ( finite_card_list_a @ A3 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_977_card_Oinfinite,axiom,
! [A3: set_a] :
( ~ ( finite_finite_a @ A3 )
=> ( ( finite_card_a @ A3 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_978_card_Oinfinite,axiom,
! [A3: set_nat] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( finite_card_nat @ A3 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_979_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_980_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_981_card__0__eq,axiom,
! [A3: set_set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( ( finite_card_set_a @ A3 )
= zero_zero_nat )
= ( A3 = bot_bot_set_set_a ) ) ) ).
% card_0_eq
thf(fact_982_card__0__eq,axiom,
! [A3: set_list_a] :
( ( finite_finite_list_a @ A3 )
=> ( ( ( finite_card_list_a @ A3 )
= zero_zero_nat )
= ( A3 = bot_bot_set_list_a ) ) ) ).
% card_0_eq
thf(fact_983_card__0__eq,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ( finite_card_nat @ A3 )
= zero_zero_nat )
= ( A3 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_984_card__0__eq,axiom,
! [A3: set_a] :
( ( finite_finite_a @ A3 )
=> ( ( ( finite_card_a @ A3 )
= zero_zero_nat )
= ( A3 = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_985_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_986_add_Onat__pow__0,axiom,
! [X: a] :
( ( add_pow_a_b_nat @ r @ zero_zero_nat @ X )
= ( zero_a_b @ r ) ) ).
% add.nat_pow_0
thf(fact_987_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_988_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_989_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_990_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_991_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_992_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_993_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_994_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_995_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_996_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_997_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_998_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_999_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_1000_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1001_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_1002_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1003_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_1004_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_1005_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1006_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_1007_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_1008_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1009_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1010_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1011_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1012_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1013_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1014_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P @ X4 @ Y3 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1015_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1016_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1017_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1018_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1019_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1020_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1021_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1022_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1023_eq__iff__diff__eq__0,axiom,
( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
= ( ^ [A4: int,B7: int] :
( ( minus_minus_int @ A4 @ B7 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1024_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_1025_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1026_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1027_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1028_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1029_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1030_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_1031_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_1032_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1033_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1034_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1035_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1036_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1037_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1038_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1039_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_1040_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_1041_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_1042_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1043_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1044_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1045_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1046_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1047_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1048_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1049_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1050_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1051_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1052_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1053_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_1054_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_1055_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_1056_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_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_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_1068_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_1069_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B7: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B7 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1070_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_1071_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_1072_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_1073_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_1074_card__eq__0__iff,axiom,
! [A3: set_set_a] :
( ( ( finite_card_set_a @ A3 )
= zero_zero_nat )
= ( ( A3 = bot_bot_set_set_a )
| ~ ( finite_finite_set_a @ A3 ) ) ) ).
% card_eq_0_iff
thf(fact_1075_card__eq__0__iff,axiom,
! [A3: set_list_a] :
( ( ( finite_card_list_a @ A3 )
= zero_zero_nat )
= ( ( A3 = bot_bot_set_list_a )
| ~ ( finite_finite_list_a @ A3 ) ) ) ).
% card_eq_0_iff
thf(fact_1076_card__eq__0__iff,axiom,
! [A3: set_nat] :
( ( ( finite_card_nat @ A3 )
= zero_zero_nat )
= ( ( A3 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A3 ) ) ) ).
% card_eq_0_iff
thf(fact_1077_card__eq__0__iff,axiom,
! [A3: set_a] :
( ( ( finite_card_a @ A3 )
= zero_zero_nat )
= ( ( A3 = bot_bot_set_a )
| ~ ( finite_finite_a @ A3 ) ) ) ).
% card_eq_0_iff
thf(fact_1078_card__Suc__eq,axiom,
! [A3: set_set_a,K: nat] :
( ( ( finite_card_set_a @ A3 )
= ( suc @ K ) )
= ( ? [B7: set_a,B8: set_set_a] :
( ( A3
= ( insert_set_a @ B7 @ B8 ) )
& ~ ( member_set_a @ B7 @ B8 )
& ( ( finite_card_set_a @ B8 )
= K )
& ( ( K = zero_zero_nat )
=> ( B8 = bot_bot_set_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1079_card__Suc__eq,axiom,
! [A3: set_a,K: nat] :
( ( ( finite_card_a @ A3 )
= ( suc @ K ) )
= ( ? [B7: a,B8: set_a] :
( ( A3
= ( insert_a @ B7 @ B8 ) )
& ~ ( member_a @ B7 @ B8 )
& ( ( finite_card_a @ B8 )
= K )
& ( ( K = zero_zero_nat )
=> ( B8 = bot_bot_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1080_card__eq__SucD,axiom,
! [A3: set_set_a,K: nat] :
( ( ( finite_card_set_a @ A3 )
= ( suc @ K ) )
=> ? [B2: set_a,B5: set_set_a] :
( ( A3
= ( insert_set_a @ B2 @ B5 ) )
& ~ ( member_set_a @ B2 @ B5 )
& ( ( finite_card_set_a @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bot_set_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_1081_card__eq__SucD,axiom,
! [A3: set_a,K: nat] :
( ( ( finite_card_a @ A3 )
= ( suc @ K ) )
=> ? [B2: a,B5: set_a] :
( ( A3
= ( insert_a @ B2 @ B5 ) )
& ~ ( member_a @ B2 @ B5 )
& ( ( finite_card_a @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bot_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_1082_card__1__singleton__iff,axiom,
! [A3: set_set_a] :
( ( ( finite_card_set_a @ A3 )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: set_a] :
( A3
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1083_card__1__singleton__iff,axiom,
! [A3: set_a] :
( ( ( finite_card_a @ A3 )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: a] :
( A3
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1084_card__le__Suc0__iff__eq,axiom,
! [A3: set_set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ A3 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ A3 )
=> ( X3 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1085_card__le__Suc0__iff__eq,axiom,
! [A3: set_list_a] :
( ( finite_finite_list_a @ A3 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_a @ A3 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A3 )
=> ! [Y4: list_a] :
( ( member_list_a @ Y4 @ A3 )
=> ( X3 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1086_card__le__Suc0__iff__eq,axiom,
! [A3: set_a] :
( ( finite_finite_a @ A3 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ A3 )
=> ( X3 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1087_card__le__Suc0__iff__eq,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ A3 )
=> ( X3 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1088_ring_Ozero__dim,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( embedd2795209813406577254on_a_b @ R @ zero_zero_nat @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ).
% ring.zero_dim
thf(fact_1089_ring_Odimension__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ zero_zero_nat @ K2 @ E )
=> ( E
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ) ).
% ring.dimension_zero
thf(fact_1090_ring_Odimension_Ocases,axiom,
! [R: partia2175431115845679010xt_a_b,A12: nat,A23: set_a,A32: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ A12 @ A23 @ A32 )
=> ( ( ( A12 = zero_zero_nat )
=> ( A32
!= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ~ ! [V3: a,E4: set_a,N2: nat] :
( ( A12
= ( suc @ N2 ) )
=> ( ( A32
= ( embedd971793762689825387on_a_b @ R @ A23 @ V3 @ E4 ) )
=> ( ( member_a @ V3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ~ ( member_a @ V3 @ E4 )
=> ~ ( embedd2795209813406577254on_a_b @ R @ N2 @ A23 @ E4 ) ) ) ) ) ) ) ) ).
% ring.dimension.cases
thf(fact_1091_ring_Odimension_Osimps,axiom,
! [R: partia2175431115845679010xt_a_b,A12: nat,A23: set_a,A32: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ A12 @ A23 @ A32 )
= ( ? [K5: set_a] :
( ( A12 = zero_zero_nat )
& ( A23 = K5 )
& ( A32
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
| ? [V5: a,E5: set_a,N4: nat,K5: set_a] :
( ( A12
= ( suc @ N4 ) )
& ( A23 = K5 )
& ( A32
= ( embedd971793762689825387on_a_b @ R @ K5 @ V5 @ E5 ) )
& ( member_a @ V5 @ ( partia707051561876973205xt_a_b @ R ) )
& ~ ( member_a @ V5 @ E5 )
& ( embedd2795209813406577254on_a_b @ R @ N4 @ K5 @ E5 ) ) ) ) ) ).
% ring.dimension.simps
thf(fact_1092_ring_Osubfield__m__inv__simprule,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,K: a,A: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ A ) @ K2 )
=> ( member_a @ A @ K2 ) ) ) ) ) ) ).
% ring.subfield_m_inv_simprule
thf(fact_1093_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1094_subfield__m__inv_I3_J,axiom,
! [K2: set_a,K: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ K ) @ K )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% subfield_m_inv(3)
thf(fact_1095_inv__eq__imp__eq,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( m_inv_a_ring_ext_a_b @ r @ Y ) )
=> ( X = Y ) ) ) ) ).
% inv_eq_imp_eq
thf(fact_1096_inv__eq__one__eq,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( one_a_ring_ext_a_b @ r ) )
= ( X
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% inv_eq_one_eq
thf(fact_1097_inv__char,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ X )
= Y ) ) ) ) ) ).
% inv_char
thf(fact_1098_inv__unique_H,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( Y
= ( m_inv_a_ring_ext_a_b @ r @ X ) ) ) ) ) ) ).
% inv_unique'
thf(fact_1099_inv__eq__neg__one__eq,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) )
= ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% inv_eq_neg_one_eq
thf(fact_1100_subfield__m__inv_I1_J,axiom,
! [K2: set_a,K: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ K ) @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ).
% subfield_m_inv(1)
thf(fact_1101_subfield__m__inv_I2_J,axiom,
! [K2: set_a,K: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ K @ ( m_inv_a_ring_ext_a_b @ r @ K ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% subfield_m_inv(2)
thf(fact_1102_group_Oinv__inv,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( m_inv_a_Product_unit @ G @ ( m_inv_a_Product_unit @ G @ X ) )
= X ) ) ) ).
% group.inv_inv
thf(fact_1103_group_Oinv__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( m_inv_a_ring_ext_a_b @ G @ ( m_inv_a_ring_ext_a_b @ G @ X ) )
= X ) ) ) ).
% group.inv_inv
thf(fact_1104_inv__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) )
= ( one_a_ring_ext_a_b @ r ) ) ).
% inv_one
thf(fact_1105_Units__inv__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ X ) )
= X ) ) ).
% Units_inv_inv
thf(fact_1106_Units__inv__Units,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_inv_Units
thf(fact_1107_Units__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_inv_closed
thf(fact_1108_inv__neg__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) )
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ).
% inv_neg_one
thf(fact_1109_Units__l__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% Units_l_inv
thf(fact_1110_Units__r__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( m_inv_a_ring_ext_a_b @ r @ X ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% Units_r_inv
thf(fact_1111_group_Oinv__closed,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ ( m_inv_a_Product_unit @ G @ X ) @ ( partia6735698275553448452t_unit @ G ) ) ) ) ).
% group.inv_closed
thf(fact_1112_group_Oinv__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ G @ X ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% group.inv_closed
thf(fact_1113_monoid_Oinv__one,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( m_inv_a_Product_unit @ G @ ( one_a_Product_unit @ G ) )
= ( one_a_Product_unit @ G ) ) ) ).
% monoid.inv_one
thf(fact_1114_monoid_Oinv__one,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( m_inv_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) )
= ( one_a_ring_ext_a_b @ G ) ) ) ).
% monoid.inv_one
thf(fact_1115_monoid_Oinv__eq__imp__eq,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ Y @ ( units_a_Product_unit @ G ) )
=> ( ( ( m_inv_a_Product_unit @ G @ X )
= ( m_inv_a_Product_unit @ G @ Y ) )
=> ( X = Y ) ) ) ) ) ).
% monoid.inv_eq_imp_eq
thf(fact_1116_monoid_Oinv__eq__imp__eq,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ G @ X )
= ( m_inv_a_ring_ext_a_b @ G @ Y ) )
=> ( X = Y ) ) ) ) ) ).
% monoid.inv_eq_imp_eq
thf(fact_1117_monoid_OUnits__inv__inv,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( ( m_inv_a_Product_unit @ G @ ( m_inv_a_Product_unit @ G @ X ) )
= X ) ) ) ).
% monoid.Units_inv_inv
thf(fact_1118_monoid_OUnits__inv__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( m_inv_a_ring_ext_a_b @ G @ ( m_inv_a_ring_ext_a_b @ G @ X ) )
= X ) ) ) ).
% monoid.Units_inv_inv
thf(fact_1119_monoid_OUnits__inv__Units,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( member_a @ ( m_inv_a_Product_unit @ G @ X ) @ ( units_a_Product_unit @ G ) ) ) ) ).
% monoid.Units_inv_Units
thf(fact_1120_monoid_OUnits__inv__Units,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ G @ X ) @ ( units_a_ring_ext_a_b @ G ) ) ) ) ).
% monoid.Units_inv_Units
thf(fact_1121_monoid_Ounits__of__inv,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( ( m_inv_a_Product_unit @ ( units_7501539392726747778t_unit @ G ) @ X )
= ( m_inv_a_Product_unit @ G @ X ) ) ) ) ).
% monoid.units_of_inv
thf(fact_1122_monoid_Ounits__of__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( m_inv_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G ) @ X )
= ( m_inv_a_ring_ext_a_b @ G @ X ) ) ) ) ).
% monoid.units_of_inv
thf(fact_1123_group_Oinv__mult__group,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( m_inv_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X @ Y ) )
= ( mult_a_Product_unit @ G @ ( m_inv_a_Product_unit @ G @ Y ) @ ( m_inv_a_Product_unit @ G @ X ) ) ) ) ) ) ).
% group.inv_mult_group
thf(fact_1124_group_Oinv__mult__group,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( m_inv_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ G @ ( m_inv_a_ring_ext_a_b @ G @ Y ) @ ( m_inv_a_ring_ext_a_b @ G @ X ) ) ) ) ) ) ).
% group.inv_mult_group
thf(fact_1125_group_Oinv__solve__left,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( A
= ( mult_a_Product_unit @ G @ ( m_inv_a_Product_unit @ G @ B ) @ C ) )
= ( C
= ( mult_a_Product_unit @ G @ B @ A ) ) ) ) ) ) ) ).
% group.inv_solve_left
thf(fact_1126_group_Oinv__solve__left,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ G @ ( m_inv_a_ring_ext_a_b @ G @ B ) @ C ) )
= ( C
= ( mult_a_ring_ext_a_b @ G @ B @ A ) ) ) ) ) ) ) ).
% group.inv_solve_left
thf(fact_1127_group_Oinv__solve__left_H,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ ( m_inv_a_Product_unit @ G @ B ) @ C )
= A )
= ( C
= ( mult_a_Product_unit @ G @ B @ A ) ) ) ) ) ) ) ).
% group.inv_solve_left'
thf(fact_1128_group_Oinv__solve__left_H,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ ( m_inv_a_ring_ext_a_b @ G @ B ) @ C )
= A )
= ( C
= ( mult_a_ring_ext_a_b @ G @ B @ A ) ) ) ) ) ) ) ).
% group.inv_solve_left'
thf(fact_1129_group_Oinv__solve__right,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( A
= ( mult_a_Product_unit @ G @ B @ ( m_inv_a_Product_unit @ G @ C ) ) )
= ( B
= ( mult_a_Product_unit @ G @ A @ C ) ) ) ) ) ) ) ).
% group.inv_solve_right
thf(fact_1130_group_Oinv__solve__right,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ G @ B @ ( m_inv_a_ring_ext_a_b @ G @ C ) ) )
= ( B
= ( mult_a_ring_ext_a_b @ G @ A @ C ) ) ) ) ) ) ) ).
% group.inv_solve_right
thf(fact_1131_group_Oinv__solve__right_H,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ B @ ( m_inv_a_Product_unit @ G @ C ) )
= A )
= ( B
= ( mult_a_Product_unit @ G @ A @ C ) ) ) ) ) ) ) ).
% group.inv_solve_right'
thf(fact_1132_group_Oinv__solve__right_H,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ B @ ( m_inv_a_ring_ext_a_b @ G @ C ) )
= A )
= ( B
= ( mult_a_ring_ext_a_b @ G @ A @ C ) ) ) ) ) ) ) ).
% group.inv_solve_right'
thf(fact_1133_group_Oinv__eq__1__iff,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( m_inv_a_Product_unit @ G @ X )
= ( one_a_Product_unit @ G ) )
= ( X
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% group.inv_eq_1_iff
thf(fact_1134_group_Oinv__eq__1__iff,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ G @ X )
= ( one_a_ring_ext_a_b @ G ) )
= ( X
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% group.inv_eq_1_iff
thf(fact_1135_monoid_OUnits__inv__closed,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( member_a @ ( m_inv_a_Product_unit @ G @ X ) @ ( partia6735698275553448452t_unit @ G ) ) ) ) ).
% monoid.Units_inv_closed
thf(fact_1136_monoid_OUnits__inv__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ G @ X ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% monoid.Units_inv_closed
thf(fact_1137_ring_Oinv__neg__one,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( m_inv_a_ring_ext_a_b @ R @ ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) )
= ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) ) ) ).
% ring.inv_neg_one
thf(fact_1138_monoid_Oinv__eq__one__eq,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( ( ( m_inv_a_Product_unit @ G @ X )
= ( one_a_Product_unit @ G ) )
= ( X
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% monoid.inv_eq_one_eq
thf(fact_1139_monoid_Oinv__eq__one__eq,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ G @ X )
= ( one_a_ring_ext_a_b @ G ) )
= ( X
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.inv_eq_one_eq
thf(fact_1140_group_Oinv__equality,axiom,
! [G: partia8223610829204095565t_unit,Y: a,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ Y @ X )
= ( one_a_Product_unit @ G ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( m_inv_a_Product_unit @ G @ X )
= Y ) ) ) ) ) ).
% group.inv_equality
thf(fact_1141_group_Oinv__equality,axiom,
! [G: partia2175431115845679010xt_a_b,Y: a,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( m_inv_a_ring_ext_a_b @ G @ X )
= Y ) ) ) ) ) ).
% group.inv_equality
thf(fact_1142_group_Or__inv,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ X @ ( m_inv_a_Product_unit @ G @ X ) )
= ( one_a_Product_unit @ G ) ) ) ) ).
% group.r_inv
thf(fact_1143_group_Or__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X @ ( m_inv_a_ring_ext_a_b @ G @ X ) )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ).
% group.r_inv
thf(fact_1144_group_Ol__inv,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( m_inv_a_Product_unit @ G @ X ) @ X )
= ( one_a_Product_unit @ G ) ) ) ) ).
% group.l_inv
thf(fact_1145_group_Ol__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( m_inv_a_ring_ext_a_b @ G @ X ) @ X )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ).
% group.l_inv
thf(fact_1146_monoid_Oinv__unique_H,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ X @ Y )
= ( one_a_Product_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ Y @ X )
= ( one_a_Product_unit @ G ) )
=> ( Y
= ( m_inv_a_Product_unit @ G @ X ) ) ) ) ) ) ) ).
% monoid.inv_unique'
thf(fact_1147_monoid_Oinv__unique_H,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
= ( one_a_ring_ext_a_b @ G ) )
=> ( Y
= ( m_inv_a_ring_ext_a_b @ G @ X ) ) ) ) ) ) ) ).
% monoid.inv_unique'
thf(fact_1148_monoid_Oinv__char,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ X @ Y )
= ( one_a_Product_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ Y @ X )
= ( one_a_Product_unit @ G ) )
=> ( ( m_inv_a_Product_unit @ G @ X )
= Y ) ) ) ) ) ) ).
% monoid.inv_char
thf(fact_1149_monoid_Oinv__char,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( m_inv_a_ring_ext_a_b @ G @ X )
= Y ) ) ) ) ) ) ).
% monoid.inv_char
thf(fact_1150_monoid_OUnits__r__inv,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ X @ ( m_inv_a_Product_unit @ G @ X ) )
= ( one_a_Product_unit @ G ) ) ) ) ).
% monoid.Units_r_inv
thf(fact_1151_monoid_OUnits__r__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X @ ( m_inv_a_ring_ext_a_b @ G @ X ) )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ).
% monoid.Units_r_inv
thf(fact_1152_monoid_OUnits__l__inv,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( m_inv_a_Product_unit @ G @ X ) @ X )
= ( one_a_Product_unit @ G ) ) ) ) ).
% monoid.Units_l_inv
thf(fact_1153_monoid_OUnits__l__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( m_inv_a_ring_ext_a_b @ G @ X ) @ X )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ).
% monoid.Units_l_inv
thf(fact_1154_ring_Oinv__eq__neg__one__eq,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ R ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ R @ X )
= ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) )
= ( X
= ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) ) ) ) ) ).
% ring.inv_eq_neg_one_eq
thf(fact_1155_ring_Osubfield__m__inv_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,K: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ R @ K ) @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ) ) ).
% ring.subfield_m_inv(1)
thf(fact_1156_group_Oone__in__subset,axiom,
! [G: partia8223610829204095565t_unit,H2: set_a] :
( ( group_a_Product_unit @ G )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( H2 != bot_bot_set_a )
=> ( ! [X4: a] :
( ( member_a @ X4 @ H2 )
=> ( member_a @ ( m_inv_a_Product_unit @ G @ X4 ) @ H2 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ H2 )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ H2 )
=> ( member_a @ ( mult_a_Product_unit @ G @ X4 @ Xa2 ) @ H2 ) ) )
=> ( member_a @ ( one_a_Product_unit @ G ) @ H2 ) ) ) ) ) ) ).
% group.one_in_subset
thf(fact_1157_group_Oone__in__subset,axiom,
! [G: partia2175431115845679010xt_a_b,H2: set_a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( H2 != bot_bot_set_a )
=> ( ! [X4: a] :
( ( member_a @ X4 @ H2 )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ G @ X4 ) @ H2 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ H2 )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X4 @ Xa2 ) @ H2 ) ) )
=> ( member_a @ ( one_a_ring_ext_a_b @ G ) @ H2 ) ) ) ) ) ) ).
% group.one_in_subset
thf(fact_1158_ring_Osubfield__m__inv_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,K: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ R @ K @ ( m_inv_a_ring_ext_a_b @ R @ K ) )
= ( one_a_ring_ext_a_b @ R ) ) ) ) ) ).
% ring.subfield_m_inv(2)
thf(fact_1159_ring_Osubfield__m__inv_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,K: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( m_inv_a_ring_ext_a_b @ R @ K ) @ K )
= ( one_a_ring_ext_a_b @ R ) ) ) ) ) ).
% ring.subfield_m_inv(3)
thf(fact_1160_subfieldE_I4_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K1 @ K2 )
=> ( ( member_a @ K22 @ K2 )
=> ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
= ( mult_a_ring_ext_a_b @ R @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_1161_subfieldE_I3_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K2 @ R )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subfieldE(3)
thf(fact_1162_subfieldE_I5_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K1 @ K2 )
=> ( ( member_a @ K22 @ K2 )
=> ( ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
= ( zero_a_b @ R ) )
=> ( ( K1
= ( zero_a_b @ R ) )
| ( K22
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_1163_subfieldE_I6_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K2 @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% subfieldE(6)
thf(fact_1164_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1165_order__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_a_ring_ext_a_b @ r ) )
= ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% order_gt_0_iff_finite
thf(fact_1166_subringI,axiom,
! [H2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ H2 )
=> ( ! [H3: a] :
( ( member_a @ H3 @ H2 )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ H2 ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H23 ) @ H2 ) ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( add_a_b @ r @ H12 @ H23 ) @ H2 ) ) )
=> ( subring_a_b @ H2 @ r ) ) ) ) ) ) ).
% subringI
thf(fact_1167_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_1168_subring__inter,axiom,
! [I2: set_a,J2: set_a] :
( ( subring_a_b @ I2 @ r )
=> ( ( subring_a_b @ J2 @ r )
=> ( subring_a_b @ ( inf_inf_set_a @ I2 @ J2 ) @ r ) ) ) ).
% subring_inter
thf(fact_1169_units__of__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_Product_unit @ ( units_8174867845824275201xt_a_b @ r ) @ X )
= ( m_inv_a_ring_ext_a_b @ r @ X ) ) ) ).
% units_of_inv
thf(fact_1170_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1171_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1172_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1173_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1174_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_1175_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1176_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1177_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1178_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1179_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_1180_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_1181_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_1182_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1183_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_1184_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1185_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_1186_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_1187_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_1188_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_1189_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1190_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1191_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1192_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1193_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1194_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_1195_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_1196_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1197_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1198_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1199_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_1200_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1201_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1202_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_1203_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_1204_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_1205_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_1206_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_1207_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_1208_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_1209_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1210_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1211_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_1212_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_1213_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1214_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1215_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1216_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1217_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1218_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1219_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1220_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1221_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1222_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1223_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1224_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1225_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1226_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1227_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K4 )
=> ~ ( P @ I4 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1228_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1229_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_1230_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1231_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1232_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_1233_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_1234_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_1235_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1236_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_1237_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1238_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1239_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1240_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1241_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1242_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1243_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1244_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M6: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1245_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K4: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1246_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_1247_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_1248_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1249_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1250_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1251_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1252_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1253_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1254_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_1255_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_1256_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_1257_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I5: nat] :
( ( J
= ( suc @ I5 ) )
=> ( P @ I5 ) )
=> ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ J )
=> ( ( P @ ( suc @ I5 ) )
=> ( P @ I5 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1258_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I5: nat] : ( P @ I5 @ ( suc @ I5 ) )
=> ( ! [I5: nat,J4: nat,K4: nat] :
( ( ord_less_nat @ I5 @ J4 )
=> ( ( ord_less_nat @ J4 @ K4 )
=> ( ( P @ I5 @ J4 )
=> ( ( P @ J4 @ K4 )
=> ( P @ I5 @ K4 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1259_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_1260_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_1261_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1262_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_1263_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_1264_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1265_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_1266_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1267_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1268_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1269_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1270_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J4: nat] :
( ( ord_less_nat @ I @ J4 )
=> ( K
!= ( suc @ J4 ) ) ) ) ).
% Suc_lessE
thf(fact_1271_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1272_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J4: nat] :
( ( ord_less_nat @ I @ J4 )
=> ( K
!= ( suc @ J4 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1273_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I5: nat,J4: nat] :
( ( ord_less_nat @ I5 @ J4 )
=> ( ord_less_nat @ ( F @ I5 ) @ ( F @ J4 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1274_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
% Conjectures (1)
thf(conj_0,conjecture,
finite_finite_list_a @ ( bounde2262800523058855161ls_a_b @ r @ n ) ).
%------------------------------------------------------------------------------