TPTP Problem File: ITP107^1.p
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%------------------------------------------------------------------------------
% File : ITP107^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Localization problem prob_542__8966104_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Localization/prob_542__8966104_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 440 ( 123 unt; 84 typ; 0 def)
% Number of atoms : 1153 ( 410 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 3789 ( 92 ~; 13 |; 67 &;3050 @)
% ( 0 <=>; 567 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 123 ( 123 >; 0 *; 0 +; 0 <<)
% Number of symbols : 73 ( 72 usr; 11 con; 0-3 aty)
% Number of variables : 833 ( 32 ^; 767 !; 34 ?; 833 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:38:32.674
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Ounit_J_J,type,
partia539144763t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Congruence__Oeq____object__Oeq____object____ext_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Ounit_J_J,type,
partia1993116613t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J_J,type,
partia1833973666xt_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,
partia96731725t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
set_se1596668135od_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_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_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (72)
thf(sy_c_AbelCoset_OA__RCOSETS_001tf__a_001tf__b,type,
a_RCOSETS_a_b: partia1833973666xt_a_b > set_a > set_set_a ).
thf(sy_c_AbelCoset_Oa__l__coset_001tf__a_001tf__b,type,
a_l_coset_a_b: partia1833973666xt_a_b > a > set_a > set_a ).
thf(sy_c_AbelCoset_Oa__r__coset_001tf__a_001tf__b,type,
a_r_coset_a_b: partia1833973666xt_a_b > set_a > a > set_a ).
thf(sy_c_AbelCoset_Oadditive__subgroup_001tf__a_001tf__b,type,
additi2104487374up_a_b: set_a > partia1833973666xt_a_b > $o ).
thf(sy_c_AbelCoset_Oset__add_001tf__a_001tf__b,type,
set_add_a_b: partia1833973666xt_a_b > set_a > set_a > set_a ).
thf(sy_c_Congruence_Oequivalence_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Ounit,type,
equiva2136214035t_unit: partia1993116613t_unit > $o ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Congruence__Oeq____object__Oeq____object____ext_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Ounit_J,type,
partia206007992t_unit: partia1993116613t_unit > set_Product_prod_a_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Ounit_J,type,
partia616129906t_unit: partia539144763t_unit > set_se1596668135od_a_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Product____Type__Ounit_J,type,
partia1955795460t_unit: partia96731725t_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,
partia1066395285xt_a_b: partia1833973666xt_a_b > set_a ).
thf(sy_c_Group_OUnits_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Ounit,type,
units_1734235042t_unit: partia539144763t_unit > set_se1596668135od_a_a ).
thf(sy_c_Group_OUnits_001tf__a_001t__Product____Type__Ounit,type,
units_a_Product_unit: partia96731725t_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: partia1833973666xt_a_b > set_a ).
thf(sy_c_Group_Ogroup_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Ounit,type,
group_52730354t_unit: partia539144763t_unit > $o ).
thf(sy_c_Group_Ogroup_001tf__a_001t__Product____Type__Ounit,type,
group_a_Product_unit: partia96731725t_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: partia1833973666xt_a_b > $o ).
thf(sy_c_Group_Omonoid_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Ounit,type,
monoid1242292471t_unit: partia539144763t_unit > $o ).
thf(sy_c_Group_Omonoid_001tf__a_001t__Product____Type__Ounit,type,
monoid1447254976t_unit: partia96731725t_unit > $o ).
thf(sy_c_Group_Omonoid_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
monoid1022105667xt_a_b: partia1833973666xt_a_b > $o ).
thf(sy_c_Group_Omonoid_Omult_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Ounit,type,
mult_s977248631t_unit: partia539144763t_unit > set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Product____Type__Ounit,type,
mult_a_Product_unit: partia96731725t_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: partia1833973666xt_a_b > a > a > a ).
thf(sy_c_Group_Omonoid_Oone_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Ounit,type,
one_se1488596747t_unit: partia539144763t_unit > set_Product_prod_a_a ).
thf(sy_c_Group_Omonoid_Oone_001tf__a_001t__Product____Type__Ounit,type,
one_a_Product_unit: partia96731725t_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: partia1833973666xt_a_b > a ).
thf(sy_c_Group_Ounits__of_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Ounit,type,
units_390215097t_unit: partia539144763t_unit > partia539144763t_unit ).
thf(sy_c_Group_Ounits__of_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
units_1411277569xt_a_b: partia1833973666xt_a_b > partia96731725t_unit ).
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_Localization__Mirabelle__afvchqjmgj_Oeq__class__of__rng__of__frac_001tf__a_001tf__a_001t__Product____Type__Ounit,type,
locali1170246543t_unit: partia1993116613t_unit > a > a > set_Product_prod_a_a ).
thf(sy_c_Localization__Mirabelle__afvchqjmgj_Oeq__obj__rng__of__frac_001tf__a_001tf__b,type,
locali1648887798ac_a_b: partia1833973666xt_a_b > set_a > $o ).
thf(sy_c_Localization__Mirabelle__afvchqjmgj_Oeq__obj__rng__of__frac_Orec__monoid__rng__of__frac_001tf__a_001tf__b,type,
locali1944243560ac_a_b: partia1833973666xt_a_b > set_a > partia539144763t_unit ).
thf(sy_c_Localization__Mirabelle__afvchqjmgj_Omult__submonoid__of__crng_001tf__a_001tf__b,type,
locali807230110ng_a_b: partia1833973666xt_a_b > set_a > $o ).
thf(sy_c_Localization__Mirabelle__afvchqjmgj_Omult__submonoid__of__rng_001tf__a_001tf__b,type,
locali880295127ng_a_b: partia1833973666xt_a_b > set_a > $o ).
thf(sy_c_Localization__Mirabelle__afvchqjmgj_Oset__eq__class__of__rng__of__frac_001tf__a_001tf__a_001t__Product____Type__Ounit,type,
locali1538946157t_unit: partia1993116613t_unit > set_se1596668135od_a_a ).
thf(sy_c_Localization__Mirabelle__afvchqjmgj_Osubmonoid_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Ounit,type,
locali1488796916t_unit: partia539144763t_unit > set_se1596668135od_a_a > $o ).
thf(sy_c_Localization__Mirabelle__afvchqjmgj_Osubmonoid_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
locali425460998xt_a_b: partia1833973666xt_a_b > set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
bot_bo2131659635od_a_a: set_Product_prod_a_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
bot_bo1424703315od_a_a: set_se1596668135od_a_a ).
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__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
ord_le391835783od_a_a: set_se1596668135od_a_a > set_se1596668135od_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le318720350_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_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Ring_Oabelian__monoid_001tf__a_001tf__b,type,
abelian_monoid_a_b: partia1833973666xt_a_b > $o ).
thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Int__Oint,type,
add_pow_a_b_int: partia1833973666xt_a_b > int > a > a ).
thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Nat__Onat,type,
add_pow_a_b_nat: partia1833973666xt_a_b > nat > a > a ).
thf(sy_c_Ring_Ofield_001tf__a_001tf__b,type,
field_a_b: partia1833973666xt_a_b > $o ).
thf(sy_c_Ring_Oring_001tf__a_001tf__b,type,
ring_a_b: partia1833973666xt_a_b > $o ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia1833973666xt_a_b > a ).
thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
semiring_a_b: partia1833973666xt_a_b > $o ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
collec183727474od_a_a: ( set_Product_prod_a_a > $o ) > set_se1596668135od_a_a ).
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_OPow_001tf__a,type,
pow_a: set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
insert1116662519od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
insert1738756823od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > set_se1596668135od_a_a ).
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_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $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: partia1833973666xt_a_b ).
thf(sy_v_S,type,
s: set_a ).
thf(sy_v_r,type,
r2: a ).
thf(sy_v_r_H,type,
r3: a ).
thf(sy_v_rel,type,
rel: partia1993116613t_unit ).
thf(sy_v_s,type,
s2: a ).
thf(sy_v_s_H,type,
s3: a ).
% Relevant facts (355)
thf(fact_0_f3,axiom,
( ( mult_a_ring_ext_a_b @ r @ r2 @ r3 )
= ( mult_a_ring_ext_a_b @ r @ r3 @ r2 ) ) ).
% f3
thf(fact_1_f4,axiom,
( ( mult_a_ring_ext_a_b @ r @ s2 @ s3 )
= ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ) ).
% f4
thf(fact_2_eq__obj__rng__of__frac__axioms,axiom,
locali1648887798ac_a_b @ r @ s ).
% eq_obj_rng_of_frac_axioms
thf(fact_3_f1,axiom,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
= ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r2 @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ) ).
% f1
thf(fact_4_f2,axiom,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) )
= ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r3 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ) ) ).
% f2
thf(fact_5_mult__submonoid__of__crng__axioms,axiom,
locali807230110ng_a_b @ r @ s ).
% mult_submonoid_of_crng_axioms
thf(fact_6_mult__submonoid__of__rng__axioms,axiom,
locali880295127ng_a_b @ r @ s ).
% mult_submonoid_of_rng_axioms
thf(fact_7_eq__obj__rng__of__frac_Orec__monoid__rng__of__frac_Ocong,axiom,
locali1944243560ac_a_b = locali1944243560ac_a_b ).
% eq_obj_rng_of_frac.rec_monoid_rng_of_frac.cong
thf(fact_8_submonoid__axioms,axiom,
locali425460998xt_a_b @ r @ s ).
% submonoid_axioms
thf(fact_9_m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ s )
=> ( ( member_a @ Y @ s )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ s ) ) ) ).
% m_closed
thf(fact_10_monoid__rng__of__frac,axiom,
monoid1242292471t_unit @ ( locali1944243560ac_a_b @ r @ s ) ).
% monoid_rng_of_frac
thf(fact_11_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_12_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ s ).
% one_closed
thf(fact_13_equiv__obj__rng__of__frac,axiom,
equiva2136214035t_unit @ rel ).
% equiv_obj_rng_of_frac
thf(fact_14_assms_I2_J,axiom,
member449909584od_a_a @ ( product_Pair_a_a @ r3 @ s3 ) @ ( partia206007992t_unit @ rel ) ).
% assms(2)
thf(fact_15_assms_I1_J,axiom,
member449909584od_a_a @ ( product_Pair_a_a @ r2 @ s2 ) @ ( partia206007992t_unit @ rel ) ).
% assms(1)
thf(fact_16_assoc__mult__rng__of__frac,axiom,
! [R: a,S: a,T: a,U: a,V: a,W: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ T @ U ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ V @ W ) @ ( partia206007992t_unit @ rel ) )
=> ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ R @ S ) @ ( locali1170246543t_unit @ rel @ T @ U ) ) @ ( locali1170246543t_unit @ rel @ V @ W ) )
= ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ R @ S ) @ ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ T @ U ) @ ( locali1170246543t_unit @ rel @ V @ W ) ) ) ) ) ) ) ).
% assoc_mult_rng_of_frac
thf(fact_17_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_18_mult__rng__of__frac__fundamental__lemma,axiom,
! [R: a,S: a,R2: a,S2: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S2 ) @ ( partia206007992t_unit @ rel ) )
=> ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ R @ S ) @ ( locali1170246543t_unit @ rel @ R2 @ S2 ) )
= ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ R @ R2 ) @ ( mult_a_ring_ext_a_b @ r @ S @ S2 ) ) ) ) ) ).
% mult_rng_of_frac_fundamental_lemma
thf(fact_19_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_20_closed__mult__rng__of__frac,axiom,
! [R: a,S: a,T: a,U: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ T @ U ) @ ( partia206007992t_unit @ rel ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ R @ S ) @ ( locali1170246543t_unit @ rel @ T @ U ) ) @ ( locali1538946157t_unit @ rel ) ) ) ) ).
% closed_mult_rng_of_frac
thf(fact_21_submonoid_Oaxioms_I1_J,axiom,
! [M: partia539144763t_unit,S3: set_se1596668135od_a_a] :
( ( locali1488796916t_unit @ M @ S3 )
=> ( monoid1242292471t_unit @ M ) ) ).
% submonoid.axioms(1)
thf(fact_22_submonoid_Oaxioms_I1_J,axiom,
! [M: partia1833973666xt_a_b,S3: set_a] :
( ( locali425460998xt_a_b @ M @ S3 )
=> ( monoid1022105667xt_a_b @ M ) ) ).
% submonoid.axioms(1)
thf(fact_23_eq__obj__rng__of__frac_Oaxioms_I2_J,axiom,
! [R3: partia1833973666xt_a_b,S3: set_a] :
( ( locali1648887798ac_a_b @ R3 @ S3 )
=> ( locali807230110ng_a_b @ R3 @ S3 ) ) ).
% eq_obj_rng_of_frac.axioms(2)
thf(fact_24_mult__submonoid__of__rng_Oaxioms_I2_J,axiom,
! [R3: partia1833973666xt_a_b,S3: set_a] :
( ( locali880295127ng_a_b @ R3 @ S3 )
=> ( locali425460998xt_a_b @ R3 @ S3 ) ) ).
% mult_submonoid_of_rng.axioms(2)
thf(fact_25_mult__submonoid__of__rng_Oaxioms_I1_J,axiom,
! [R3: partia1833973666xt_a_b,S3: set_a] :
( ( locali880295127ng_a_b @ R3 @ S3 )
=> ( ring_a_b @ R3 ) ) ).
% mult_submonoid_of_rng.axioms(1)
thf(fact_26_mult__submonoid__of__crng_Oaxioms_I2_J,axiom,
! [R3: partia1833973666xt_a_b,S3: set_a] :
( ( locali807230110ng_a_b @ R3 @ S3 )
=> ( locali880295127ng_a_b @ R3 @ S3 ) ) ).
% mult_submonoid_of_crng.axioms(2)
thf(fact_27_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Oone__closed,axiom,
! [M: partia539144763t_unit,S3: set_se1596668135od_a_a] :
( ( locali1488796916t_unit @ M @ S3 )
=> ( member1838126896od_a_a @ ( one_se1488596747t_unit @ M ) @ S3 ) ) ).
% Localization_Mirabelle_afvchqjmgj.submonoid.one_closed
thf(fact_28_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Oone__closed,axiom,
! [M: partia1833973666xt_a_b,S3: set_a] :
( ( locali425460998xt_a_b @ M @ S3 )
=> ( member_a @ ( one_a_ring_ext_a_b @ M ) @ S3 ) ) ).
% Localization_Mirabelle_afvchqjmgj.submonoid.one_closed
thf(fact_29_mult__submonoid__of__rng__def,axiom,
( locali880295127ng_a_b
= ( ^ [R4: partia1833973666xt_a_b,S4: set_a] :
( ( ring_a_b @ R4 )
& ( locali425460998xt_a_b @ R4 @ S4 ) ) ) ) ).
% mult_submonoid_of_rng_def
thf(fact_30_mult__submonoid__of__rng_Ointro,axiom,
! [R3: partia1833973666xt_a_b,S3: set_a] :
( ( ring_a_b @ R3 )
=> ( ( locali425460998xt_a_b @ R3 @ S3 )
=> ( locali880295127ng_a_b @ R3 @ S3 ) ) ) ).
% mult_submonoid_of_rng.intro
thf(fact_31_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Ois__submonoid,axiom,
! [M: partia1833973666xt_a_b,S3: set_a] :
( ( locali425460998xt_a_b @ M @ S3 )
=> ( locali425460998xt_a_b @ M @ S3 ) ) ).
% Localization_Mirabelle_afvchqjmgj.submonoid.is_submonoid
thf(fact_32_eq__obj__rng__of__frac_Omonoid__rng__of__frac,axiom,
! [R3: partia1833973666xt_a_b,S3: set_a] :
( ( locali1648887798ac_a_b @ R3 @ S3 )
=> ( monoid1242292471t_unit @ ( locali1944243560ac_a_b @ R3 @ S3 ) ) ) ).
% eq_obj_rng_of_frac.monoid_rng_of_frac
thf(fact_33_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Om__closed,axiom,
! [M: partia539144763t_unit,S3: set_se1596668135od_a_a,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( locali1488796916t_unit @ M @ S3 )
=> ( ( member1838126896od_a_a @ X @ S3 )
=> ( ( member1838126896od_a_a @ Y @ S3 )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ M @ X @ Y ) @ S3 ) ) ) ) ).
% Localization_Mirabelle_afvchqjmgj.submonoid.m_closed
thf(fact_34_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Om__closed,axiom,
! [M: partia1833973666xt_a_b,S3: set_a,X: a,Y: a] :
( ( locali425460998xt_a_b @ M @ S3 )
=> ( ( member_a @ X @ S3 )
=> ( ( member_a @ Y @ S3 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ M @ X @ Y ) @ S3 ) ) ) ) ).
% Localization_Mirabelle_afvchqjmgj.submonoid.m_closed
thf(fact_35_right__unit__mult__rng__of__frac,axiom,
! [R: a,S: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
=> ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ R @ S ) @ ( one_se1488596747t_unit @ ( locali1944243560ac_a_b @ r @ s ) ) )
= ( locali1170246543t_unit @ rel @ R @ S ) ) ) ).
% right_unit_mult_rng_of_frac
thf(fact_36_left__unit__mult__rng__of__frac,axiom,
! [R: a,S: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
=> ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( one_se1488596747t_unit @ ( locali1944243560ac_a_b @ r @ s ) ) @ ( locali1170246543t_unit @ rel @ R @ S ) )
= ( locali1170246543t_unit @ rel @ R @ S ) ) ) ).
% left_unit_mult_rng_of_frac
thf(fact_37_zero__in__mult__submonoid,axiom,
! [R: a,S: a,R2: a,S2: a] :
( ( member_a @ ( zero_a_b @ r ) @ s )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S2 ) @ ( partia206007992t_unit @ rel ) )
=> ( ( locali1170246543t_unit @ rel @ R @ S )
= ( locali1170246543t_unit @ rel @ R2 @ S2 ) ) ) ) ) ).
% zero_in_mult_submonoid
thf(fact_38_non__empty__class,axiom,
! [R: a,S: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
=> ( ( locali1170246543t_unit @ rel @ R @ S )
!= bot_bo2131659635od_a_a ) ) ).
% non_empty_class
thf(fact_39_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia1066395285xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_40_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 @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia1066395285xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% inv_unique
thf(fact_41_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_42_prod_Oinject,axiom,
! [X1: a,X22: a,Y1: a,Y22: a] :
( ( ( product_Pair_a_a @ X1 @ X22 )
= ( product_Pair_a_a @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_43_old_Oprod_Oinject,axiom,
! [A: a,B: a,A2: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_44_semiring_Oaxioms_I1_J,axiom,
! [R3: partia1833973666xt_a_b] :
( ( semiring_a_b @ R3 )
=> ( abelian_monoid_a_b @ R3 ) ) ).
% semiring.axioms(1)
thf(fact_45_monoid__axioms,axiom,
monoid1022105667xt_a_b @ r ).
% monoid_axioms
thf(fact_46_m__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia1066395285xt_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_47_m__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% m_comm
thf(fact_48_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_50_mem__Collect__eq,axiom,
! [A: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( member1838126896od_a_a @ A @ ( collec183727474od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_51_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_52_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
! [A3: set_Product_prod_a_a] :
( ( collec645855634od_a_a
@ ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A3: set_se1596668135od_a_a] :
( ( collec183727474od_a_a
@ ^ [X3: set_Product_prod_a_a] : ( member1838126896od_a_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_55_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_56_m__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_57_Units__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X @ ( partia1066395285xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_58_Units__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X2 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_59_Units__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_60_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) ).
% zero_closed
thf(fact_61_semiring__simprules_I3_J,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% semiring_simprules(3)
thf(fact_62_semiring__simprules_I4_J,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) ).
% semiring_simprules(4)
thf(fact_63_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_64_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_65_l__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_66_r__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_67_l__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
= X ) ) ).
% l_one
thf(fact_68_r__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
= X ) ) ).
% r_one
thf(fact_69_Units__l__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia1066395285xt_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_70_abelian__monoidE_I2_J,axiom,
! [R3: partia1833973666xt_a_b] :
( ( abelian_monoid_a_b @ R3 )
=> ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% abelian_monoidE(2)
thf(fact_71_ring_Oring__simprules_I2_J,axiom,
! [R3: partia1833973666xt_a_b] :
( ( ring_a_b @ R3 )
=> ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% ring.ring_simprules(2)
thf(fact_72_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia1833973666xt_a_b] :
( ( semiring_a_b @ R3 )
=> ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_73_abelian__monoid_Ozero__closed,axiom,
! [G: partia1833973666xt_a_b] :
( ( abelian_monoid_a_b @ G )
=> ( member_a @ ( zero_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_74_ring_Ois__monoid,axiom,
! [R3: partia1833973666xt_a_b] :
( ( ring_a_b @ R3 )
=> ( monoid1022105667xt_a_b @ R3 ) ) ).
% ring.is_monoid
thf(fact_75_semiring_Oaxioms_I2_J,axiom,
! [R3: partia1833973666xt_a_b] :
( ( semiring_a_b @ R3 )
=> ( monoid1022105667xt_a_b @ R3 ) ) ).
% semiring.axioms(2)
thf(fact_76_ring_Oring__simprules_I24_J,axiom,
! [R3: partia1833973666xt_a_b,X: a] :
( ( ring_a_b @ R3 )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ ( zero_a_b @ R3 ) @ X )
= ( zero_a_b @ R3 ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_77_ring_Oring__simprules_I25_J,axiom,
! [R3: partia1833973666xt_a_b,X: a] :
( ( ring_a_b @ R3 )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ X @ ( zero_a_b @ R3 ) )
= ( zero_a_b @ R3 ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_78_semiring_Or__null,axiom,
! [R3: partia1833973666xt_a_b,X: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ X @ ( zero_a_b @ R3 ) )
= ( zero_a_b @ R3 ) ) ) ) ).
% semiring.r_null
thf(fact_79_semiring_Ol__null,axiom,
! [R3: partia1833973666xt_a_b,X: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ ( zero_a_b @ R3 ) @ X )
= ( zero_a_b @ R3 ) ) ) ) ).
% semiring.l_null
thf(fact_80_ring_Oring__simprules_I5_J,axiom,
! [R3: partia1833973666xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R3 )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_81_ring_Oring__simprules_I11_J,axiom,
! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R3 )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R3 @ X @ ( mult_a_ring_ext_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_82_semiring_Osemiring__simprules_I3_J,axiom,
! [R3: partia1833973666xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_83_semiring_Osemiring__simprules_I8_J,axiom,
! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R3 @ X @ ( mult_a_ring_ext_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_84_ring_Oring__simprules_I6_J,axiom,
! [R3: partia1833973666xt_a_b] :
( ( ring_a_b @ R3 )
=> ( member_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% ring.ring_simprules(6)
thf(fact_85_semiring_Osemiring__simprules_I4_J,axiom,
! [R3: partia1833973666xt_a_b] :
( ( semiring_a_b @ R3 )
=> ( member_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_86_ring_Oring__simprules_I12_J,axiom,
! [R3: partia1833973666xt_a_b,X: a] :
( ( ring_a_b @ R3 )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ ( one_a_ring_ext_a_b @ R3 ) @ X )
= X ) ) ) ).
% ring.ring_simprules(12)
thf(fact_87_semiring_Osemiring__simprules_I9_J,axiom,
! [R3: partia1833973666xt_a_b,X: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ ( one_a_ring_ext_a_b @ R3 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_88_old_Oprod_Oinducts,axiom,
! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
( ! [A4: a,B3: a] : ( P @ ( product_Pair_a_a @ A4 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_89_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A4: a,B3: a] :
( Y
!= ( product_Pair_a_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_90_Pair__inject,axiom,
! [A: a,B: a,A2: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_91_prod__cases,axiom,
! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
( ! [A4: a,B3: a] : ( P @ ( product_Pair_a_a @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_92_surj__pair,axiom,
! [P2: product_prod_a_a] :
? [X2: a,Y3: a] :
( P2
= ( product_Pair_a_a @ X2 @ Y3 ) ) ).
% surj_pair
thf(fact_93_ring_Ois__ring,axiom,
! [R3: partia1833973666xt_a_b] :
( ( ring_a_b @ R3 )
=> ( ring_a_b @ R3 ) ) ).
% ring.is_ring
thf(fact_94_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( partia1066395285xt_a_b @ r ) )
=> ( ( A4
!= ( zero_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia1066395285xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A4 @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_95_monoid_Or__one,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ G )
=> ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
=> ( ( mult_s977248631t_unit @ G @ X @ ( one_se1488596747t_unit @ G ) )
= X ) ) ) ).
% monoid.r_one
thf(fact_96_monoid_Or__one,axiom,
! [G: partia1833973666xt_a_b,X: a] :
( ( monoid1022105667xt_a_b @ G )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X @ ( one_a_ring_ext_a_b @ G ) )
= X ) ) ) ).
% monoid.r_one
thf(fact_97_monoid_Ol__one,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ G )
=> ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
=> ( ( mult_s977248631t_unit @ G @ ( one_se1488596747t_unit @ G ) @ X )
= X ) ) ) ).
% monoid.l_one
thf(fact_98_monoid_Ol__one,axiom,
! [G: partia1833973666xt_a_b,X: a] :
( ( monoid1022105667xt_a_b @ G )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X )
= X ) ) ) ).
% monoid.l_one
thf(fact_99_monoid_Oone__closed,axiom,
! [G: partia539144763t_unit] :
( ( monoid1242292471t_unit @ G )
=> ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G ) @ ( partia616129906t_unit @ G ) ) ) ).
% monoid.one_closed
thf(fact_100_monoid_Oone__closed,axiom,
! [G: partia1833973666xt_a_b] :
( ( monoid1022105667xt_a_b @ G )
=> ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) ) ) ).
% monoid.one_closed
thf(fact_101_monoid_OUnits__r__inv__ex,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ G )
=> ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
=> ? [X2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
& ( ( mult_s977248631t_unit @ G @ X @ X2 )
= ( one_se1488596747t_unit @ G ) ) ) ) ) ).
% monoid.Units_r_inv_ex
thf(fact_102_monoid_OUnits__r__inv__ex,axiom,
! [G: partia1833973666xt_a_b,X: a] :
( ( monoid1022105667xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ X @ X2 )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.Units_r_inv_ex
thf(fact_103_monoid_OUnits__l__inv__ex,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ G )
=> ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
=> ? [X2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
& ( ( mult_s977248631t_unit @ G @ X2 @ X )
= ( one_se1488596747t_unit @ G ) ) ) ) ) ).
% monoid.Units_l_inv_ex
thf(fact_104_monoid_OUnits__l__inv__ex,axiom,
! [G: partia1833973666xt_a_b,X: a] :
( ( monoid1022105667xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ X2 @ X )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.Units_l_inv_ex
thf(fact_105_monoid_OUnits__inv__comm,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ G )
=> ( ( ( mult_s977248631t_unit @ G @ X @ Y )
= ( one_se1488596747t_unit @ G ) )
=> ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Y @ ( units_1734235042t_unit @ G ) )
=> ( ( mult_s977248631t_unit @ G @ Y @ X )
= ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% monoid.Units_inv_comm
thf(fact_106_monoid_OUnits__inv__comm,axiom,
! [G: partia1833973666xt_a_b,X: a,Y: a] :
( ( monoid1022105667xt_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_107_add__pow__ldistr__int,axiom,
! [A: a,B: a,K: int] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_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_108_add__pow__rdistr__int,axiom,
! [A: a,B: a,K: int] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_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_109_add_Oint__pow__closed,axiom,
! [X: a,I: int] :
( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
=> ( member_a @ ( add_pow_a_b_int @ r @ I @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% add.int_pow_closed
thf(fact_110_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_111_field_Ois__ring,axiom,
! [R3: partia1833973666xt_a_b] :
( ( field_a_b @ R3 )
=> ( ring_a_b @ R3 ) ) ).
% field.is_ring
thf(fact_112_one__not__zero,axiom,
! [R3: partia1833973666xt_a_b] :
( ( field_a_b @ R3 )
=> ( ( one_a_ring_ext_a_b @ R3 )
!= ( zero_a_b @ R3 ) ) ) ).
% one_not_zero
thf(fact_113_integral,axiom,
! [R3: partia1833973666xt_a_b,A: a,B: a] :
( ( field_a_b @ R3 )
=> ( ( ( mult_a_ring_ext_a_b @ R3 @ A @ B )
= ( zero_a_b @ R3 ) )
=> ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( A
= ( zero_a_b @ R3 ) )
| ( B
= ( zero_a_b @ R3 ) ) ) ) ) ) ) ).
% integral
thf(fact_114_ring_Oadd__pow__ldistr__int,axiom,
! [R3: partia1833973666xt_a_b,A: a,B: a,K: int] :
( ( ring_a_b @ R3 )
=> ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ ( add_pow_a_b_int @ R3 @ K @ A ) @ B )
= ( add_pow_a_b_int @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_ldistr_int
thf(fact_115_ring_Oadd__pow__rdistr__int,axiom,
! [R3: partia1833973666xt_a_b,A: a,B: a,K: int] :
( ( ring_a_b @ R3 )
=> ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ A @ ( add_pow_a_b_int @ R3 @ K @ B ) )
= ( add_pow_a_b_int @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_rdistr_int
thf(fact_116_monoid_Ocarrier__not__empty,axiom,
! [G: partia539144763t_unit] :
( ( monoid1242292471t_unit @ G )
=> ( ( partia616129906t_unit @ G )
!= bot_bo1424703315od_a_a ) ) ).
% monoid.carrier_not_empty
thf(fact_117_monoid_Ocarrier__not__empty,axiom,
! [G: partia1833973666xt_a_b] :
( ( monoid1022105667xt_a_b @ G )
=> ( ( partia1066395285xt_a_b @ G )
!= bot_bot_set_a ) ) ).
% monoid.carrier_not_empty
thf(fact_118_monoid_Om__assoc,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ G )
=> ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Z @ ( partia616129906t_unit @ G ) )
=> ( ( mult_s977248631t_unit @ G @ ( mult_s977248631t_unit @ G @ X @ Y ) @ Z )
= ( mult_s977248631t_unit @ G @ X @ ( mult_s977248631t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).
% monoid.m_assoc
thf(fact_119_monoid_Om__assoc,axiom,
! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
( ( monoid1022105667xt_a_b @ G )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia1066395285xt_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_120_monoid_Om__closed,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ G )
=> ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G @ X @ Y ) @ ( partia616129906t_unit @ G ) ) ) ) ) ).
% monoid.m_closed
thf(fact_121_monoid_Om__closed,axiom,
! [G: partia1833973666xt_a_b,X: a,Y: a] :
( ( monoid1022105667xt_a_b @ G )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ ( partia1066395285xt_a_b @ G ) ) ) ) ) ).
% monoid.m_closed
thf(fact_122_monoid_OUnits__closed,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ G )
=> ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
=> ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) ) ) ) ).
% monoid.Units_closed
thf(fact_123_monoid_OUnits__closed,axiom,
! [G: partia1833973666xt_a_b,X: a] :
( ( monoid1022105667xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( member_a @ X @ ( partia1066395285xt_a_b @ G ) ) ) ) ).
% monoid.Units_closed
thf(fact_124_monoid_OUnits__m__closed,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ G )
=> ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Y @ ( units_1734235042t_unit @ G ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G @ X @ Y ) @ ( units_1734235042t_unit @ G ) ) ) ) ) ).
% monoid.Units_m_closed
thf(fact_125_monoid_OUnits__m__closed,axiom,
! [G: partia1833973666xt_a_b,X: a,Y: a] :
( ( monoid1022105667xt_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_126_monoid_OUnits__one__closed,axiom,
! [G: partia1833973666xt_a_b] :
( ( monoid1022105667xt_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_127_monoid_OUnits__one__closed,axiom,
! [G: partia539144763t_unit] :
( ( monoid1242292471t_unit @ G )
=> ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G ) @ ( units_1734235042t_unit @ G ) ) ) ).
% monoid.Units_one_closed
thf(fact_128_monoidI,axiom,
! [G: partia539144763t_unit] :
( ! [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ ( partia616129906t_unit @ G ) ) ) )
=> ( ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G ) @ ( partia616129906t_unit @ G ) )
=> ( ! [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Z2 @ ( partia616129906t_unit @ G ) )
=> ( ( mult_s977248631t_unit @ G @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ Z2 )
= ( mult_s977248631t_unit @ G @ X2 @ ( mult_s977248631t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
=> ( ( mult_s977248631t_unit @ G @ ( one_se1488596747t_unit @ G ) @ X2 )
= X2 ) )
=> ( ! [X2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
=> ( ( mult_s977248631t_unit @ G @ X2 @ ( one_se1488596747t_unit @ G ) )
= X2 ) )
=> ( monoid1242292471t_unit @ G ) ) ) ) ) ) ).
% monoidI
thf(fact_129_monoidI,axiom,
! [G: partia1833973666xt_a_b] :
( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ ( partia1066395285xt_a_b @ G ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) )
=> ( ! [X2: a,Y3: a,Z2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ Z2 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ G @ X2 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X2 )
= X2 ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X2 @ ( one_a_ring_ext_a_b @ G ) )
= X2 ) )
=> ( monoid1022105667xt_a_b @ G ) ) ) ) ) ) ).
% monoidI
thf(fact_130_Group_Omonoid__def,axiom,
( monoid1242292471t_unit
= ( ^ [G2: partia539144763t_unit] :
( ! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X3 @ ( partia616129906t_unit @ G2 ) )
=> ( ( member1838126896od_a_a @ Y4 @ ( partia616129906t_unit @ G2 ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G2 @ X3 @ Y4 ) @ ( partia616129906t_unit @ G2 ) ) ) )
& ! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X3 @ ( partia616129906t_unit @ G2 ) )
=> ( ( member1838126896od_a_a @ Y4 @ ( partia616129906t_unit @ G2 ) )
=> ( ( member1838126896od_a_a @ Z3 @ ( partia616129906t_unit @ G2 ) )
=> ( ( mult_s977248631t_unit @ G2 @ ( mult_s977248631t_unit @ G2 @ X3 @ Y4 ) @ Z3 )
= ( mult_s977248631t_unit @ G2 @ X3 @ ( mult_s977248631t_unit @ G2 @ Y4 @ Z3 ) ) ) ) ) )
& ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G2 ) @ ( partia616129906t_unit @ G2 ) )
& ! [X3: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X3 @ ( partia616129906t_unit @ G2 ) )
=> ( ( mult_s977248631t_unit @ G2 @ ( one_se1488596747t_unit @ G2 ) @ X3 )
= X3 ) )
& ! [X3: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X3 @ ( partia616129906t_unit @ G2 ) )
=> ( ( mult_s977248631t_unit @ G2 @ X3 @ ( one_se1488596747t_unit @ G2 ) )
= X3 ) ) ) ) ) ).
% Group.monoid_def
thf(fact_131_Group_Omonoid__def,axiom,
( monoid1022105667xt_a_b
= ( ^ [G2: partia1833973666xt_a_b] :
( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G2 ) )
=> ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ G2 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G2 @ X3 @ Y4 ) @ ( partia1066395285xt_a_b @ G2 ) ) ) )
& ! [X3: a,Y4: a,Z3: a] :
( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G2 ) )
=> ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ G2 ) )
=> ( ( member_a @ Z3 @ ( partia1066395285xt_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 ) @ ( partia1066395285xt_a_b @ G2 ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G2 ) )
=> ( ( mult_a_ring_ext_a_b @ G2 @ ( one_a_ring_ext_a_b @ G2 ) @ X3 )
= X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G2 ) )
=> ( ( mult_a_ring_ext_a_b @ G2 @ X3 @ ( one_a_ring_ext_a_b @ G2 ) )
= X3 ) ) ) ) ) ).
% Group.monoid_def
thf(fact_132_Group_Omonoid_Ointro,axiom,
! [G: partia539144763t_unit] :
( ! [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ ( partia616129906t_unit @ G ) ) ) )
=> ( ! [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Z2 @ ( partia616129906t_unit @ G ) )
=> ( ( mult_s977248631t_unit @ G @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ Z2 )
= ( mult_s977248631t_unit @ G @ X2 @ ( mult_s977248631t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G ) @ ( partia616129906t_unit @ G ) )
=> ( ! [X2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
=> ( ( mult_s977248631t_unit @ G @ ( one_se1488596747t_unit @ G ) @ X2 )
= X2 ) )
=> ( ! [X2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
=> ( ( mult_s977248631t_unit @ G @ X2 @ ( one_se1488596747t_unit @ G ) )
= X2 ) )
=> ( monoid1242292471t_unit @ G ) ) ) ) ) ) ).
% Group.monoid.intro
thf(fact_133_Group_Omonoid_Ointro,axiom,
! [G: partia1833973666xt_a_b] :
( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ ( partia1066395285xt_a_b @ G ) ) ) )
=> ( ! [X2: a,Y3: a,Z2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ Z2 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ G @ X2 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X2 )
= X2 ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X2 @ ( one_a_ring_ext_a_b @ G ) )
= X2 ) )
=> ( monoid1022105667xt_a_b @ G ) ) ) ) ) ) ).
% Group.monoid.intro
thf(fact_134_monoid_Oinv__unique,axiom,
! [G: partia539144763t_unit,Y: set_Product_prod_a_a,X: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ G )
=> ( ( ( mult_s977248631t_unit @ G @ Y @ X )
= ( one_se1488596747t_unit @ G ) )
=> ( ( ( mult_s977248631t_unit @ G @ X @ Y2 )
= ( one_se1488596747t_unit @ G ) )
=> ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Y2 @ ( partia616129906t_unit @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% monoid.inv_unique
thf(fact_135_monoid_Oinv__unique,axiom,
! [G: partia1833973666xt_a_b,Y: a,X: a,Y2: a] :
( ( monoid1022105667xt_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 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia1066395285xt_a_b @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% monoid.inv_unique
thf(fact_136_monoid_Oone__unique,axiom,
! [G: partia539144763t_unit,U: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ G )
=> ( ( member1838126896od_a_a @ U @ ( partia616129906t_unit @ G ) )
=> ( ! [X2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
=> ( ( mult_s977248631t_unit @ G @ U @ X2 )
= X2 ) )
=> ( U
= ( one_se1488596747t_unit @ G ) ) ) ) ) ).
% monoid.one_unique
thf(fact_137_monoid_Oone__unique,axiom,
! [G: partia1833973666xt_a_b,U: a] :
( ( monoid1022105667xt_a_b @ G )
=> ( ( member_a @ U @ ( partia1066395285xt_a_b @ G ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ U @ X2 )
= X2 ) )
=> ( U
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.one_unique
thf(fact_138_monoid_OUnits__l__cancel,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( monoid1242292471t_unit @ G )
=> ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Z @ ( partia616129906t_unit @ G ) )
=> ( ( ( mult_s977248631t_unit @ G @ X @ Y )
= ( mult_s977248631t_unit @ G @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ).
% monoid.Units_l_cancel
thf(fact_139_monoid_OUnits__l__cancel,axiom,
! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
( ( monoid1022105667xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia1066395285xt_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_140_subset,axiom,
ord_less_eq_set_a @ s @ ( partia1066395285xt_a_b @ r ) ).
% subset
thf(fact_141_add__pow__rdistr,axiom,
! [A: a,B: a,K: nat] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_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_142_add__pow__ldistr,axiom,
! [A: a,B: a,K: nat] :
( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia1066395285xt_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_143_group__l__invI,axiom,
( ! [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ r ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia1066395285xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ Xa @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) )
=> ( group_a_ring_ext_a_b @ r ) ) ).
% group_l_invI
thf(fact_144_carrier__not__empty,axiom,
( ( partia1066395285xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_145_add_Onat__pow__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
=> ( member_a @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% add.nat_pow_closed
thf(fact_146_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_147_group_Ois__group,axiom,
! [G: partia1833973666xt_a_b] :
( ( group_a_ring_ext_a_b @ G )
=> ( group_a_ring_ext_a_b @ G ) ) ).
% group.is_group
thf(fact_148_group_Ois__group,axiom,
! [G: partia96731725t_unit] :
( ( group_a_Product_unit @ G )
=> ( group_a_Product_unit @ G ) ) ).
% group.is_group
thf(fact_149_group_OUnits,axiom,
! [G: partia96731725t_unit] :
( ( group_a_Product_unit @ G )
=> ( ord_less_eq_set_a @ ( partia1955795460t_unit @ G ) @ ( units_a_Product_unit @ G ) ) ) ).
% group.Units
thf(fact_150_group_OUnits,axiom,
! [G: partia1833973666xt_a_b] :
( ( group_a_ring_ext_a_b @ G )
=> ( ord_less_eq_set_a @ ( partia1066395285xt_a_b @ G ) @ ( units_a_ring_ext_a_b @ G ) ) ) ).
% group.Units
thf(fact_151_group_Ois__monoid,axiom,
! [G: partia96731725t_unit] :
( ( group_a_Product_unit @ G )
=> ( monoid1447254976t_unit @ G ) ) ).
% group.is_monoid
thf(fact_152_group_Ois__monoid,axiom,
! [G: partia539144763t_unit] :
( ( group_52730354t_unit @ G )
=> ( monoid1242292471t_unit @ G ) ) ).
% group.is_monoid
thf(fact_153_group_Ois__monoid,axiom,
! [G: partia1833973666xt_a_b] :
( ( group_a_ring_ext_a_b @ G )
=> ( monoid1022105667xt_a_b @ G ) ) ).
% group.is_monoid
thf(fact_154_Group_Ogroup_Oright__cancel,axiom,
! [G: partia96731725t_unit,X: a,Y: a,Z: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
=> ( ( member_a @ Z @ ( partia1955795460t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ Y @ X )
= ( mult_a_Product_unit @ G @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% Group.group.right_cancel
thf(fact_155_Group_Ogroup_Oright__cancel,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( group_52730354t_unit @ G )
=> ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Z @ ( partia616129906t_unit @ G ) )
=> ( ( ( mult_s977248631t_unit @ G @ Y @ X )
= ( mult_s977248631t_unit @ G @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% Group.group.right_cancel
thf(fact_156_Group_Ogroup_Oright__cancel,axiom,
! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia1066395285xt_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_157_group_OUnits__eq,axiom,
! [G: partia96731725t_unit] :
( ( group_a_Product_unit @ G )
=> ( ( units_a_Product_unit @ G )
= ( partia1955795460t_unit @ G ) ) ) ).
% group.Units_eq
thf(fact_158_group_OUnits__eq,axiom,
! [G: partia1833973666xt_a_b] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( units_a_ring_ext_a_b @ G )
= ( partia1066395285xt_a_b @ G ) ) ) ).
% group.Units_eq
thf(fact_159_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Osubset,axiom,
! [M: partia1833973666xt_a_b,S3: set_a] :
( ( locali425460998xt_a_b @ M @ S3 )
=> ( ord_less_eq_set_a @ S3 @ ( partia1066395285xt_a_b @ M ) ) ) ).
% Localization_Mirabelle_afvchqjmgj.submonoid.subset
thf(fact_160_group_Or__cancel__one_H,axiom,
! [G: partia96731725t_unit,X: a,A: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
=> ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
=> ( ( X
= ( mult_a_Product_unit @ G @ A @ X ) )
= ( A
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.r_cancel_one'
thf(fact_161_group_Or__cancel__one_H,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( group_52730354t_unit @ G )
=> ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ A @ ( partia616129906t_unit @ G ) )
=> ( ( X
= ( mult_s977248631t_unit @ G @ A @ X ) )
= ( A
= ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% group.r_cancel_one'
thf(fact_162_group_Or__cancel__one_H,axiom,
! [G: partia1833973666xt_a_b,X: a,A: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia1066395285xt_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_163_group_Ol__cancel__one_H,axiom,
! [G: partia96731725t_unit,X: a,A: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
=> ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
=> ( ( X
= ( mult_a_Product_unit @ G @ X @ A ) )
= ( A
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.l_cancel_one'
thf(fact_164_group_Ol__cancel__one_H,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( group_52730354t_unit @ G )
=> ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ A @ ( partia616129906t_unit @ G ) )
=> ( ( X
= ( mult_s977248631t_unit @ G @ X @ A ) )
= ( A
= ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% group.l_cancel_one'
thf(fact_165_group_Ol__cancel__one_H,axiom,
! [G: partia1833973666xt_a_b,X: a,A: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia1066395285xt_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_166_group_Or__cancel__one,axiom,
! [G: partia96731725t_unit,X: a,A: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
=> ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ A @ X )
= X )
= ( A
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.r_cancel_one
thf(fact_167_group_Or__cancel__one,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( group_52730354t_unit @ G )
=> ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ A @ ( partia616129906t_unit @ G ) )
=> ( ( ( mult_s977248631t_unit @ G @ A @ X )
= X )
= ( A
= ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% group.r_cancel_one
thf(fact_168_group_Or__cancel__one,axiom,
! [G: partia1833973666xt_a_b,X: a,A: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia1066395285xt_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_169_group_Ol__cancel__one,axiom,
! [G: partia96731725t_unit,X: a,A: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
=> ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ X @ A )
= X )
= ( A
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.l_cancel_one
thf(fact_170_group_Ol__cancel__one,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a,A: set_Product_prod_a_a] :
( ( group_52730354t_unit @ G )
=> ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ A @ ( partia616129906t_unit @ G ) )
=> ( ( ( mult_s977248631t_unit @ G @ X @ A )
= X )
= ( A
= ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% group.l_cancel_one
thf(fact_171_group_Ol__cancel__one,axiom,
! [G: partia1833973666xt_a_b,X: a,A: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia1066395285xt_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_172_group_Or__inv__ex,axiom,
! [G: partia96731725t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ X @ X2 )
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% group.r_inv_ex
thf(fact_173_group_Or__inv__ex,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
( ( group_52730354t_unit @ G )
=> ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
=> ? [X2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
& ( ( mult_s977248631t_unit @ G @ X @ X2 )
= ( one_se1488596747t_unit @ G ) ) ) ) ) ).
% group.r_inv_ex
thf(fact_174_group_Or__inv__ex,axiom,
! [G: partia1833973666xt_a_b,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ X @ X2 )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% group.r_inv_ex
thf(fact_175_group_Ol__inv__ex,axiom,
! [G: partia96731725t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ X2 @ X )
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% group.l_inv_ex
thf(fact_176_group_Ol__inv__ex,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
( ( group_52730354t_unit @ G )
=> ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
=> ? [X2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
& ( ( mult_s977248631t_unit @ G @ X2 @ X )
= ( one_se1488596747t_unit @ G ) ) ) ) ) ).
% group.l_inv_ex
thf(fact_177_group_Ol__inv__ex,axiom,
! [G: partia1833973666xt_a_b,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ X2 @ X )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% group.l_inv_ex
thf(fact_178_group_Oinv__comm,axiom,
! [G: partia96731725t_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 @ ( partia1955795460t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ Y @ X )
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.inv_comm
thf(fact_179_group_Oinv__comm,axiom,
! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( group_52730354t_unit @ G )
=> ( ( ( mult_s977248631t_unit @ G @ X @ Y )
= ( one_se1488596747t_unit @ G ) )
=> ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
=> ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
=> ( ( mult_s977248631t_unit @ G @ Y @ X )
= ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% group.inv_comm
thf(fact_180_group_Oinv__comm,axiom,
! [G: partia1833973666xt_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 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% group.inv_comm
thf(fact_181_groupI,axiom,
! [G: partia96731725t_unit] :
( ! [X2: a] :
( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia1955795460t_unit @ G ) )
=> ( member_a @ ( mult_a_Product_unit @ G @ X2 @ Y3 ) @ ( partia1955795460t_unit @ G ) ) ) )
=> ( ( member_a @ ( one_a_Product_unit @ G ) @ ( partia1955795460t_unit @ G ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia1955795460t_unit @ G ) )
=> ! [Z2: a] :
( ( member_a @ Z2 @ ( partia1955795460t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X2 @ Y3 ) @ Z2 )
= ( mult_a_Product_unit @ G @ X2 @ ( mult_a_Product_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( one_a_Product_unit @ G ) @ X2 )
= X2 ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia1955795460t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ Xa @ X2 )
= ( one_a_Product_unit @ G ) ) ) )
=> ( group_a_Product_unit @ G ) ) ) ) ) ) ).
% groupI
thf(fact_182_groupI,axiom,
! [G: partia539144763t_unit] :
( ! [X2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
=> ! [Y3: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
=> ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ ( partia616129906t_unit @ G ) ) ) )
=> ( ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G ) @ ( partia616129906t_unit @ G ) )
=> ( ! [X2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
=> ! [Y3: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
=> ! [Z2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ Z2 @ ( partia616129906t_unit @ G ) )
=> ( ( mult_s977248631t_unit @ G @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ Z2 )
= ( mult_s977248631t_unit @ G @ X2 @ ( mult_s977248631t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
=> ( ( mult_s977248631t_unit @ G @ ( one_se1488596747t_unit @ G ) @ X2 )
= X2 ) )
=> ( ! [X2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
=> ? [Xa: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ Xa @ ( partia616129906t_unit @ G ) )
& ( ( mult_s977248631t_unit @ G @ Xa @ X2 )
= ( one_se1488596747t_unit @ G ) ) ) )
=> ( group_52730354t_unit @ G ) ) ) ) ) ) ).
% groupI
thf(fact_183_groupI,axiom,
! [G: partia1833973666xt_a_b] :
( ! [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ ( partia1066395285xt_a_b @ G ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
=> ! [Z2: a] :
( ( member_a @ Z2 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ G @ X2 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X2 )
= X2 ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia1066395285xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ Xa @ X2 )
= ( one_a_ring_ext_a_b @ G ) ) ) )
=> ( group_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% groupI
thf(fact_184_monoid_Ogroup__l__invI,axiom,
! [G: partia96731725t_unit] :
( ( monoid1447254976t_unit @ G )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia1955795460t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ Xa @ X2 )
= ( one_a_Product_unit @ G ) ) ) )
=> ( group_a_Product_unit @ G ) ) ) ).
% monoid.group_l_invI
thf(fact_185_monoid_Ogroup__l__invI,axiom,
! [G: partia539144763t_unit] :
( ( monoid1242292471t_unit @ G )
=> ( ! [X2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
=> ? [Xa: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ Xa @ ( partia616129906t_unit @ G ) )
& ( ( mult_s977248631t_unit @ G @ Xa @ X2 )
= ( one_se1488596747t_unit @ G ) ) ) )
=> ( group_52730354t_unit @ G ) ) ) ).
% monoid.group_l_invI
thf(fact_186_monoid_Ogroup__l__invI,axiom,
! [G: partia1833973666xt_a_b] :
( ( monoid1022105667xt_a_b @ G )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia1066395285xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ Xa @ X2 )
= ( one_a_ring_ext_a_b @ G ) ) ) )
=> ( group_a_ring_ext_a_b @ G ) ) ) ).
% monoid.group_l_invI
thf(fact_187_semiring_Oadd__pow__rdistr,axiom,
! [R3: partia1833973666xt_a_b,A: a,B: a,K: nat] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ A @ ( add_pow_a_b_nat @ R3 @ K @ B ) )
= ( add_pow_a_b_nat @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_rdistr
thf(fact_188_semiring_Oadd__pow__ldistr,axiom,
! [R3: partia1833973666xt_a_b,A: a,B: a,K: nat] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ ( add_pow_a_b_nat @ R3 @ K @ A ) @ B )
= ( add_pow_a_b_nat @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_ldistr
thf(fact_189_a__lcos__mult__one,axiom,
! [M: set_a] :
( ( ord_less_eq_set_a @ M @ ( partia1066395285xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M )
= M ) ) ).
% a_lcos_mult_one
thf(fact_190_set__add__closed,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( partia1066395285xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B4 @ ( partia1066395285xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A3 @ B4 ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% set_add_closed
thf(fact_191_setadd__subset__G,axiom,
! [H: set_a,K2: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia1066395285xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H @ K2 ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% setadd_subset_G
thf(fact_192_subset__empty,axiom,
! [A3: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A3 @ bot_bo2131659635od_a_a )
= ( A3 = bot_bo2131659635od_a_a ) ) ).
% subset_empty
thf(fact_193_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_194_subset__empty,axiom,
! [A3: set_set_a] :
( ( ord_le318720350_set_a @ A3 @ bot_bot_set_set_a )
= ( A3 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_195_empty__iff,axiom,
! [C: set_Product_prod_a_a] :
~ ( member1838126896od_a_a @ C @ bot_bo1424703315od_a_a ) ).
% empty_iff
thf(fact_196_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_197_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member449909584od_a_a @ C @ bot_bo2131659635od_a_a ) ).
% empty_iff
thf(fact_198_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_199_all__not__in__conv,axiom,
! [A3: set_se1596668135od_a_a] :
( ( ! [X3: set_Product_prod_a_a] :
~ ( member1838126896od_a_a @ X3 @ A3 ) )
= ( A3 = bot_bo1424703315od_a_a ) ) ).
% all_not_in_conv
thf(fact_200_all__not__in__conv,axiom,
! [A3: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A3 ) )
= ( A3 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_201_all__not__in__conv,axiom,
! [A3: set_Product_prod_a_a] :
( ( ! [X3: product_prod_a_a] :
~ ( member449909584od_a_a @ X3 @ A3 ) )
= ( A3 = bot_bo2131659635od_a_a ) ) ).
% all_not_in_conv
thf(fact_202_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_203_Collect__empty__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( ( collec645855634od_a_a @ P )
= bot_bo2131659635od_a_a )
= ( ! [X3: product_prod_a_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_204_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_205_empty__Collect__eq,axiom,
! [P: product_prod_a_a > $o] :
( ( bot_bo2131659635od_a_a
= ( collec645855634od_a_a @ P ) )
= ( ! [X3: product_prod_a_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_206_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_207_subsetI,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ! [X2: product_prod_a_a] :
( ( member449909584od_a_a @ X2 @ A3 )
=> ( member449909584od_a_a @ X2 @ B4 ) )
=> ( ord_le1824328871od_a_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_208_subsetI,axiom,
! [A3: set_se1596668135od_a_a,B4: set_se1596668135od_a_a] :
( ! [X2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X2 @ A3 )
=> ( member1838126896od_a_a @ X2 @ B4 ) )
=> ( ord_le391835783od_a_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_209_subsetI,axiom,
! [A3: set_a,B4: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B4 ) )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_210_subsetI,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( member_set_a @ X2 @ B4 ) )
=> ( ord_le318720350_set_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_211_subset__antisym,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_212_subset__antisym,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( ord_le318720350_set_a @ A3 @ B4 )
=> ( ( ord_le318720350_set_a @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_213_a__l__coset__subset__G,axiom,
! [H: set_a,X: a] :
( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_214_empty__subsetI,axiom,
! [A3: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ bot_bo2131659635od_a_a @ A3 ) ).
% empty_subsetI
thf(fact_215_empty__subsetI,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% empty_subsetI
thf(fact_216_empty__subsetI,axiom,
! [A3: set_set_a] : ( ord_le318720350_set_a @ bot_bot_set_set_a @ A3 ) ).
% empty_subsetI
thf(fact_217_emptyE,axiom,
! [A: set_Product_prod_a_a] :
~ ( member1838126896od_a_a @ A @ bot_bo1424703315od_a_a ) ).
% emptyE
thf(fact_218_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_219_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ) ).
% emptyE
thf(fact_220_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_221_equals0D,axiom,
! [A3: set_se1596668135od_a_a,A: set_Product_prod_a_a] :
( ( A3 = bot_bo1424703315od_a_a )
=> ~ ( member1838126896od_a_a @ A @ A3 ) ) ).
% equals0D
thf(fact_222_equals0D,axiom,
! [A3: set_set_a,A: set_a] :
( ( A3 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A3 ) ) ).
% equals0D
thf(fact_223_equals0D,axiom,
! [A3: set_Product_prod_a_a,A: product_prod_a_a] :
( ( A3 = bot_bo2131659635od_a_a )
=> ~ ( member449909584od_a_a @ A @ A3 ) ) ).
% equals0D
thf(fact_224_equals0D,axiom,
! [A3: set_a,A: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A @ A3 ) ) ).
% equals0D
thf(fact_225_equals0I,axiom,
! [A3: set_se1596668135od_a_a] :
( ! [Y3: set_Product_prod_a_a] :
~ ( member1838126896od_a_a @ Y3 @ A3 )
=> ( A3 = bot_bo1424703315od_a_a ) ) ).
% equals0I
thf(fact_226_equals0I,axiom,
! [A3: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A3 )
=> ( A3 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_227_equals0I,axiom,
! [A3: set_Product_prod_a_a] :
( ! [Y3: product_prod_a_a] :
~ ( member449909584od_a_a @ Y3 @ A3 )
=> ( A3 = bot_bo2131659635od_a_a ) ) ).
% equals0I
thf(fact_228_equals0I,axiom,
! [A3: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_229_ex__in__conv,axiom,
! [A3: set_se1596668135od_a_a] :
( ( ? [X3: set_Product_prod_a_a] : ( member1838126896od_a_a @ X3 @ A3 ) )
= ( A3 != bot_bo1424703315od_a_a ) ) ).
% ex_in_conv
thf(fact_230_ex__in__conv,axiom,
! [A3: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A3 ) )
= ( A3 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_231_ex__in__conv,axiom,
! [A3: set_Product_prod_a_a] :
( ( ? [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A3 ) )
= ( A3 != bot_bo2131659635od_a_a ) ) ).
% ex_in_conv
thf(fact_232_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_233_in__mono,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A3 @ B4 )
=> ( ( member449909584od_a_a @ X @ A3 )
=> ( member449909584od_a_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_234_in__mono,axiom,
! [A3: set_se1596668135od_a_a,B4: set_se1596668135od_a_a,X: set_Product_prod_a_a] :
( ( ord_le391835783od_a_a @ A3 @ B4 )
=> ( ( member1838126896od_a_a @ X @ A3 )
=> ( member1838126896od_a_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_235_in__mono,axiom,
! [A3: set_a,B4: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_236_in__mono,axiom,
! [A3: set_set_a,B4: set_set_a,X: set_a] :
( ( ord_le318720350_set_a @ A3 @ B4 )
=> ( ( member_set_a @ X @ A3 )
=> ( member_set_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_237_subsetD,axiom,
! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A3 @ B4 )
=> ( ( member449909584od_a_a @ C @ A3 )
=> ( member449909584od_a_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_238_subsetD,axiom,
! [A3: set_se1596668135od_a_a,B4: set_se1596668135od_a_a,C: set_Product_prod_a_a] :
( ( ord_le391835783od_a_a @ A3 @ B4 )
=> ( ( member1838126896od_a_a @ C @ A3 )
=> ( member1838126896od_a_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_239_subsetD,axiom,
! [A3: set_a,B4: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_240_subsetD,axiom,
! [A3: set_set_a,B4: set_set_a,C: set_a] :
( ( ord_le318720350_set_a @ A3 @ B4 )
=> ( ( member_set_a @ C @ A3 )
=> ( member_set_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_241_equalityE,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ~ ( ord_less_eq_set_a @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_242_equalityE,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( A3 = B4 )
=> ~ ( ( ord_le318720350_set_a @ A3 @ B4 )
=> ~ ( ord_le318720350_set_a @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_243_subset__eq,axiom,
( ord_le1824328871od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
! [X3: product_prod_a_a] :
( ( member449909584od_a_a @ X3 @ A5 )
=> ( member449909584od_a_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_244_subset__eq,axiom,
( ord_le391835783od_a_a
= ( ^ [A5: set_se1596668135od_a_a,B5: set_se1596668135od_a_a] :
! [X3: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ X3 @ A5 )
=> ( member1838126896od_a_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_245_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_246_subset__eq,axiom,
( ord_le318720350_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( member_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_247_equalityD1,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_248_equalityD1,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( A3 = B4 )
=> ( ord_le318720350_set_a @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_249_equalityD2,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ( ord_less_eq_set_a @ B4 @ A3 ) ) ).
% equalityD2
thf(fact_250_equalityD2,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( A3 = B4 )
=> ( ord_le318720350_set_a @ B4 @ A3 ) ) ).
% equalityD2
thf(fact_251_subset__iff,axiom,
( ord_le1824328871od_a_a
= ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
! [T2: product_prod_a_a] :
( ( member449909584od_a_a @ T2 @ A5 )
=> ( member449909584od_a_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_252_subset__iff,axiom,
( ord_le391835783od_a_a
= ( ^ [A5: set_se1596668135od_a_a,B5: set_se1596668135od_a_a] :
! [T2: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ T2 @ A5 )
=> ( member1838126896od_a_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_253_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A5 )
=> ( member_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_254_subset__iff,axiom,
( ord_le318720350_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [T2: set_a] :
( ( member_set_a @ T2 @ A5 )
=> ( member_set_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_255_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_256_subset__refl,axiom,
! [A3: set_set_a] : ( ord_le318720350_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_257_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_258_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le318720350_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_259_subset__trans,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_260_subset__trans,axiom,
! [A3: set_set_a,B4: set_set_a,C2: set_set_a] :
( ( ord_le318720350_set_a @ A3 @ B4 )
=> ( ( ord_le318720350_set_a @ B4 @ C2 )
=> ( ord_le318720350_set_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_261_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z4: set_a] : Y5 = Z4 )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_262_set__eq__subset,axiom,
( ( ^ [Y5: set_set_a,Z4: set_set_a] : Y5 = Z4 )
= ( ^ [A5: set_set_a,B5: set_set_a] :
( ( ord_le318720350_set_a @ A5 @ B5 )
& ( ord_le318720350_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_263_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_264_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le318720350_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_265_units__group,axiom,
group_a_Product_unit @ ( units_1411277569xt_a_b @ r ) ).
% units_group
thf(fact_266_add__additive__subgroups,axiom,
! [H: set_a,K2: set_a] :
( ( additi2104487374up_a_b @ H @ r )
=> ( ( additi2104487374up_a_b @ K2 @ r )
=> ( additi2104487374up_a_b @ ( set_add_a_b @ r @ H @ K2 ) @ r ) ) ) ).
% add_additive_subgroups
thf(fact_267_a__rcos__assoc__lcos,axiom,
! [H: set_a,K2: set_a,X: a] :
( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ ( a_r_coset_a_b @ r @ H @ X ) @ K2 )
= ( set_add_a_b @ r @ H @ ( a_l_coset_a_b @ r @ X @ K2 ) ) ) ) ) ) ).
% a_rcos_assoc_lcos
thf(fact_268_a__r__coset__subset__G,axiom,
! [H: set_a,X: a] :
( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_r_coset_a_b @ r @ H @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% a_r_coset_subset_G
thf(fact_269_a__setmult__rcos__assoc,axiom,
! [H: set_a,K2: set_a,X: a] :
( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ H @ ( a_r_coset_a_b @ r @ K2 @ X ) )
= ( a_r_coset_a_b @ r @ ( set_add_a_b @ r @ H @ K2 ) @ X ) ) ) ) ) ).
% a_setmult_rcos_assoc
thf(fact_270_units__of__units,axiom,
! [G: partia1833973666xt_a_b] :
( ( units_a_Product_unit @ ( units_1411277569xt_a_b @ G ) )
= ( units_a_ring_ext_a_b @ G ) ) ).
% units_of_units
thf(fact_271_a__coset__add__zero,axiom,
! [M: set_a] :
( ( ord_less_eq_set_a @ M @ ( partia1066395285xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ M @ ( zero_a_b @ r ) )
= M ) ) ).
% a_coset_add_zero
thf(fact_272_units__of__carrier,axiom,
! [G: partia1833973666xt_a_b] :
( ( partia1955795460t_unit @ ( units_1411277569xt_a_b @ G ) )
= ( units_a_ring_ext_a_b @ G ) ) ).
% units_of_carrier
thf(fact_273_monoid_Ounits__group,axiom,
! [G: partia539144763t_unit] :
( ( monoid1242292471t_unit @ G )
=> ( group_52730354t_unit @ ( units_390215097t_unit @ G ) ) ) ).
% monoid.units_group
thf(fact_274_monoid_Ounits__group,axiom,
! [G: partia1833973666xt_a_b] :
( ( monoid1022105667xt_a_b @ G )
=> ( group_a_Product_unit @ ( units_1411277569xt_a_b @ G ) ) ) ).
% monoid.units_group
thf(fact_275_units__of__mult,axiom,
! [G: partia1833973666xt_a_b] :
( ( mult_a_Product_unit @ ( units_1411277569xt_a_b @ G ) )
= ( mult_a_ring_ext_a_b @ G ) ) ).
% units_of_mult
thf(fact_276_units__of__mult,axiom,
! [G: partia539144763t_unit] :
( ( mult_s977248631t_unit @ ( units_390215097t_unit @ G ) )
= ( mult_s977248631t_unit @ G ) ) ).
% units_of_mult
thf(fact_277_units__of__one,axiom,
! [G: partia1833973666xt_a_b] :
( ( one_a_Product_unit @ ( units_1411277569xt_a_b @ G ) )
= ( one_a_ring_ext_a_b @ G ) ) ).
% units_of_one
thf(fact_278_units__of__one,axiom,
! [G: partia539144763t_unit] :
( ( one_se1488596747t_unit @ ( units_390215097t_unit @ G ) )
= ( one_se1488596747t_unit @ G ) ) ).
% units_of_one
thf(fact_279_a__rcosetsI,axiom,
! [H: set_a,X: a] :
( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
=> ( member_set_a @ ( a_r_coset_a_b @ r @ H @ X ) @ ( a_RCOSETS_a_b @ r @ H ) ) ) ) ).
% a_rcosetsI
thf(fact_280_one__zeroI,axiom,
( ( ( partia1066395285xt_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_281_one__zeroD,axiom,
( ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) )
=> ( ( partia1066395285xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% one_zeroD
thf(fact_282_insert__absorb2,axiom,
! [X: a,A3: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A3 ) )
= ( insert_a @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_283_insert__iff,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
= ( ( A = B )
| ( member_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_284_insert__iff,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ A3 ) )
= ( ( A = B )
| ( member449909584od_a_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_285_insert__iff,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,A3: set_se1596668135od_a_a] :
( ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ B @ A3 ) )
= ( ( A = B )
| ( member1838126896od_a_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_286_insert__iff,axiom,
! [A: set_a,B: set_a,A3: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A3 ) )
= ( ( A = B )
| ( member_set_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_287_insertCI,axiom,
! [A: a,B4: set_a,B: a] :
( ( ~ ( member_a @ A @ B4 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_288_insertCI,axiom,
! [A: product_prod_a_a,B4: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ~ ( member449909584od_a_a @ A @ B4 )
=> ( A = B ) )
=> ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_289_insertCI,axiom,
! [A: set_Product_prod_a_a,B4: set_se1596668135od_a_a,B: set_Product_prod_a_a] :
( ( ~ ( member1838126896od_a_a @ A @ B4 )
=> ( A = B ) )
=> ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_290_insertCI,axiom,
! [A: set_a,B4: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B4 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_291_carrier__one__not__zero,axiom,
( ( ( partia1066395285xt_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_292_carrier__one__zero,axiom,
( ( ( partia1066395285xt_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_293_singletonI,axiom,
! [A: set_Product_prod_a_a] : ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ A @ bot_bo1424703315od_a_a ) ) ).
% singletonI
thf(fact_294_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_295_singletonI,axiom,
! [A: product_prod_a_a] : ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) ) ).
% singletonI
thf(fact_296_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_297_insert__subset,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ ( insert1116662519od_a_a @ X @ A3 ) @ B4 )
= ( ( member449909584od_a_a @ X @ B4 )
& ( ord_le1824328871od_a_a @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_298_insert__subset,axiom,
! [X: set_Product_prod_a_a,A3: set_se1596668135od_a_a,B4: set_se1596668135od_a_a] :
( ( ord_le391835783od_a_a @ ( insert1738756823od_a_a @ X @ A3 ) @ B4 )
= ( ( member1838126896od_a_a @ X @ B4 )
& ( ord_le391835783od_a_a @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_299_insert__subset,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A3 ) @ B4 )
= ( ( member_a @ X @ B4 )
& ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_300_insert__subset,axiom,
! [X: set_a,A3: set_set_a,B4: set_set_a] :
( ( ord_le318720350_set_a @ ( insert_set_a @ X @ A3 ) @ B4 )
= ( ( member_set_a @ X @ B4 )
& ( ord_le318720350_set_a @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_301_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a,B: product_prod_a_a] :
( ( ( insert1116662519od_a_a @ A @ A3 )
= ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) )
= ( ( A = B )
& ( ord_le1824328871od_a_a @ A3 @ ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_302_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_303_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A3: set_set_a,B: set_a] :
( ( ( insert_set_a @ A @ A3 )
= ( insert_set_a @ B @ bot_bot_set_set_a ) )
= ( ( A = B )
& ( ord_le318720350_set_a @ A3 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_304_singleton__insert__inj__eq,axiom,
! [B: product_prod_a_a,A: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a )
= ( insert1116662519od_a_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_le1824328871od_a_a @ A3 @ ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_305_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_306_singleton__insert__inj__eq,axiom,
! [B: set_a,A: set_a,A3: set_set_a] :
( ( ( insert_set_a @ B @ bot_bot_set_set_a )
= ( insert_set_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_le318720350_set_a @ A3 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_307_subset__singleton__iff,axiom,
! [X5: set_Product_prod_a_a,A: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ X5 @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) )
= ( ( X5 = bot_bo2131659635od_a_a )
| ( X5
= ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_308_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_309_subset__singleton__iff,axiom,
! [X5: set_set_a,A: set_a] :
( ( ord_le318720350_set_a @ X5 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( ( X5 = bot_bot_set_set_a )
| ( X5
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_310_subset__singletonD,axiom,
! [A3: set_Product_prod_a_a,X: product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A3 @ ( insert1116662519od_a_a @ X @ bot_bo2131659635od_a_a ) )
=> ( ( A3 = bot_bo2131659635od_a_a )
| ( A3
= ( insert1116662519od_a_a @ X @ bot_bo2131659635od_a_a ) ) ) ) ).
% subset_singletonD
thf(fact_311_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_312_subset__singletonD,axiom,
! [A3: set_set_a,X: set_a] :
( ( ord_le318720350_set_a @ A3 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
=> ( ( A3 = bot_bot_set_set_a )
| ( A3
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_313_mk__disjoint__insert,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ? [B6: set_a] :
( ( A3
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_314_mk__disjoint__insert,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ A3 )
=> ? [B6: set_Product_prod_a_a] :
( ( A3
= ( insert1116662519od_a_a @ A @ B6 ) )
& ~ ( member449909584od_a_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_315_mk__disjoint__insert,axiom,
! [A: set_Product_prod_a_a,A3: set_se1596668135od_a_a] :
( ( member1838126896od_a_a @ A @ A3 )
=> ? [B6: set_se1596668135od_a_a] :
( ( A3
= ( insert1738756823od_a_a @ A @ B6 ) )
& ~ ( member1838126896od_a_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_316_mk__disjoint__insert,axiom,
! [A: set_a,A3: set_set_a] :
( ( member_set_a @ A @ A3 )
=> ? [B6: set_set_a] :
( ( A3
= ( insert_set_a @ A @ B6 ) )
& ~ ( member_set_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_317_insert__commute,axiom,
! [X: a,Y: a,A3: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A3 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_318_insert__eq__iff,axiom,
! [A: a,A3: set_a,B: a,B4: set_a] :
( ~ ( member_a @ A @ A3 )
=> ( ~ ( member_a @ B @ B4 )
=> ( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C3: set_a] :
( ( A3
= ( insert_a @ B @ C3 ) )
& ~ ( member_a @ B @ C3 )
& ( B4
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_319_insert__eq__iff,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a,B: product_prod_a_a,B4: set_Product_prod_a_a] :
( ~ ( member449909584od_a_a @ A @ A3 )
=> ( ~ ( member449909584od_a_a @ B @ B4 )
=> ( ( ( insert1116662519od_a_a @ A @ A3 )
= ( insert1116662519od_a_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C3: set_Product_prod_a_a] :
( ( A3
= ( insert1116662519od_a_a @ B @ C3 ) )
& ~ ( member449909584od_a_a @ B @ C3 )
& ( B4
= ( insert1116662519od_a_a @ A @ C3 ) )
& ~ ( member449909584od_a_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_320_insert__eq__iff,axiom,
! [A: set_Product_prod_a_a,A3: set_se1596668135od_a_a,B: set_Product_prod_a_a,B4: set_se1596668135od_a_a] :
( ~ ( member1838126896od_a_a @ A @ A3 )
=> ( ~ ( member1838126896od_a_a @ B @ B4 )
=> ( ( ( insert1738756823od_a_a @ A @ A3 )
= ( insert1738756823od_a_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C3: set_se1596668135od_a_a] :
( ( A3
= ( insert1738756823od_a_a @ B @ C3 ) )
& ~ ( member1838126896od_a_a @ B @ C3 )
& ( B4
= ( insert1738756823od_a_a @ A @ C3 ) )
& ~ ( member1838126896od_a_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_321_insert__eq__iff,axiom,
! [A: set_a,A3: set_set_a,B: set_a,B4: set_set_a] :
( ~ ( member_set_a @ A @ A3 )
=> ( ~ ( member_set_a @ B @ B4 )
=> ( ( ( insert_set_a @ A @ A3 )
= ( insert_set_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C3: set_set_a] :
( ( A3
= ( insert_set_a @ B @ C3 ) )
& ~ ( member_set_a @ B @ C3 )
& ( B4
= ( insert_set_a @ A @ C3 ) )
& ~ ( member_set_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_322_insert__absorb,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_323_insert__absorb,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ A3 )
=> ( ( insert1116662519od_a_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_324_insert__absorb,axiom,
! [A: set_Product_prod_a_a,A3: set_se1596668135od_a_a] :
( ( member1838126896od_a_a @ A @ A3 )
=> ( ( insert1738756823od_a_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_325_insert__absorb,axiom,
! [A: set_a,A3: set_set_a] :
( ( member_set_a @ A @ A3 )
=> ( ( insert_set_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_326_insert__ident,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ~ ( member_a @ X @ B4 )
=> ( ( ( insert_a @ X @ A3 )
= ( insert_a @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_327_insert__ident,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ~ ( member449909584od_a_a @ X @ A3 )
=> ( ~ ( member449909584od_a_a @ X @ B4 )
=> ( ( ( insert1116662519od_a_a @ X @ A3 )
= ( insert1116662519od_a_a @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_328_insert__ident,axiom,
! [X: set_Product_prod_a_a,A3: set_se1596668135od_a_a,B4: set_se1596668135od_a_a] :
( ~ ( member1838126896od_a_a @ X @ A3 )
=> ( ~ ( member1838126896od_a_a @ X @ B4 )
=> ( ( ( insert1738756823od_a_a @ X @ A3 )
= ( insert1738756823od_a_a @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_329_insert__ident,axiom,
! [X: set_a,A3: set_set_a,B4: set_set_a] :
( ~ ( member_set_a @ X @ A3 )
=> ( ~ ( member_set_a @ X @ B4 )
=> ( ( ( insert_set_a @ X @ A3 )
= ( insert_set_a @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_330_Set_Oset__insert,axiom,
! [X: a,A3: set_a] :
( ( member_a @ X @ A3 )
=> ~ ! [B6: set_a] :
( ( A3
= ( insert_a @ X @ B6 ) )
=> ( member_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_331_Set_Oset__insert,axiom,
! [X: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member449909584od_a_a @ X @ A3 )
=> ~ ! [B6: set_Product_prod_a_a] :
( ( A3
= ( insert1116662519od_a_a @ X @ B6 ) )
=> ( member449909584od_a_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_332_Set_Oset__insert,axiom,
! [X: set_Product_prod_a_a,A3: set_se1596668135od_a_a] :
( ( member1838126896od_a_a @ X @ A3 )
=> ~ ! [B6: set_se1596668135od_a_a] :
( ( A3
= ( insert1738756823od_a_a @ X @ B6 ) )
=> ( member1838126896od_a_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_333_Set_Oset__insert,axiom,
! [X: set_a,A3: set_set_a] :
( ( member_set_a @ X @ A3 )
=> ~ ! [B6: set_set_a] :
( ( A3
= ( insert_set_a @ X @ B6 ) )
=> ( member_set_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_334_insertI2,axiom,
! [A: a,B4: set_a,B: a] :
( ( member_a @ A @ B4 )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_335_insertI2,axiom,
! [A: product_prod_a_a,B4: set_Product_prod_a_a,B: product_prod_a_a] :
( ( member449909584od_a_a @ A @ B4 )
=> ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_336_insertI2,axiom,
! [A: set_Product_prod_a_a,B4: set_se1596668135od_a_a,B: set_Product_prod_a_a] :
( ( member1838126896od_a_a @ A @ B4 )
=> ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_337_insertI2,axiom,
! [A: set_a,B4: set_set_a,B: set_a] :
( ( member_set_a @ A @ B4 )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_338_insertI1,axiom,
! [A: a,B4: set_a] : ( member_a @ A @ ( insert_a @ A @ B4 ) ) ).
% insertI1
thf(fact_339_insertI1,axiom,
! [A: product_prod_a_a,B4: set_Product_prod_a_a] : ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ A @ B4 ) ) ).
% insertI1
thf(fact_340_insertI1,axiom,
! [A: set_Product_prod_a_a,B4: set_se1596668135od_a_a] : ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ A @ B4 ) ) ).
% insertI1
thf(fact_341_insertI1,axiom,
! [A: set_a,B4: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B4 ) ) ).
% insertI1
thf(fact_342_insertE,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_343_insertE,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member449909584od_a_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_344_insertE,axiom,
! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,A3: set_se1596668135od_a_a] :
( ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member1838126896od_a_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_345_insertE,axiom,
! [A: set_a,B: set_a,A3: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_set_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_346_singleton__inject,axiom,
! [A: product_prod_a_a,B: product_prod_a_a] :
( ( ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a )
= ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_347_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_348_insert__not__empty,axiom,
! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
( ( insert1116662519od_a_a @ A @ A3 )
!= bot_bo2131659635od_a_a ) ).
% insert_not_empty
thf(fact_349_insert__not__empty,axiom,
! [A: a,A3: set_a] :
( ( insert_a @ A @ A3 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_350_doubleton__eq__iff,axiom,
! [A: product_prod_a_a,B: product_prod_a_a,C: product_prod_a_a,D: product_prod_a_a] :
( ( ( insert1116662519od_a_a @ A @ ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) )
= ( insert1116662519od_a_a @ C @ ( insert1116662519od_a_a @ D @ bot_bo2131659635od_a_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_351_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_352_field__intro2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( minus_minus_set_a @ ( partia1066395285xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) ) )
=> ( field_a_b @ r ) ) ) ).
% field_intro2
thf(fact_353_cring__fieldI,axiom,
( ( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia1066395285xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( field_a_b @ r ) ) ).
% cring_fieldI
thf(fact_354_rcosets__subset__PowG,axiom,
! [H: set_a] :
( ( additi2104487374up_a_b @ H @ r )
=> ( ord_le318720350_set_a @ ( a_RCOSETS_a_b @ r @ H ) @ ( pow_a @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% rcosets_subset_PowG
% Conjectures (1)
thf(conj_0,conjecture,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
= ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) ) ) ).
%------------------------------------------------------------------------------