TPTP Problem File: SLH0268^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Interpolation_Polynomials_HOL_Algebra/0001_Lagrange_Interpolation/prob_00302_011958__17406310_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1517 ( 254 unt; 237 typ; 0 def)
% Number of atoms : 4639 (1015 equ; 0 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 17471 ( 286 ~; 65 |; 253 &;14082 @)
% ( 0 <=>;2785 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 754 ( 754 >; 0 *; 0 +; 0 <<)
% Number of symbols : 220 ( 219 usr; 12 con; 0-4 aty)
% Number of variables : 3324 ( 102 ^;3093 !; 129 ?;3324 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:38:38.307
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
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thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
image_nat_a_a: ( ( nat > a ) > a ) > set_nat_a > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
image_6965494298868581957_nat_a: ( set_nat_a > set_nat_a ) > set_set_nat_a > set_set_nat_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_1042221919965026181_set_a: ( set_set_a > set_set_a ) > set_set_set_a > set_set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001_062_It__Nat__Onat_Mtf__a_J,type,
image_set_a_nat_a: ( set_a > nat > a ) > set_set_a > set_nat_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
image_set_a_a: ( set_a > a ) > set_set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
image_a_nat_a: ( a > nat > a ) > set_a > set_nat_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mtf__a_J,type,
insert_nat_a: ( nat > a ) > set_nat_a > set_nat_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Subrings_Osubcring_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
subcri4445174380595745425t_unit: set_set_a > partia6043505979758434576t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001tf__a_001tf__b,type,
subcring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubdomain_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
subdom4943114742163587044t_unit: set_set_a > partia6043505979758434576t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001tf__a_001tf__b,type,
subdomain_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubfield_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
subfie5224850075530046424t_unit: set_set_a > partia6043505979758434576t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
subrin1511138061850335568t_unit: set_set_a > partia6043505979758434576t_unit > $o ).
thf(sy_c_Subrings_Osubring_001tf__a_001tf__b,type,
subring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_UnivPoly_Obound_001t__Set__Oset_Itf__a_J,type,
bound_set_a: set_a > nat > ( nat > set_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_UnivPoly_Oup_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
up_set6103515193466504168t_unit: partia6043505979758434576t_unit > set_nat_set_a ).
thf(sy_c_UnivPoly_Oup_001tf__a_001tf__b,type,
up_a_b: partia2175431115845679010xt_a_b > set_nat_a ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Set__Oset_Itf__a_J_J,type,
member_nat_set_a: ( nat > set_a ) > set_nat_set_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
member_set_a_set_a: ( set_a > set_a ) > set_set_a_set_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mtf__a_J,type,
member_set_a_a: ( set_a > a ) > set_set_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Set__Oset_Itf__a_J_J,type,
member_a_set_a: ( a > set_a ) > set_a_set_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
member_set_nat_a: set_nat_a > set_set_nat_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_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: partia2175431115845679010xt_a_b ).
thf(sy_v_S,type,
s: set_a ).
thf(sy_v_f,type,
f: a > a ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1278)
thf(fact_0_factorial__domain__axioms,axiom,
ring_f5272581269873410839in_a_b @ r ).
% factorial_domain_axioms
thf(fact_1_local_Ofield__axioms,axiom,
field_a_b @ r ).
% local.field_axioms
thf(fact_2_assms_I1_J,axiom,
finite_finite_a @ s ).
% assms(1)
thf(fact_3_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_4_assms_I2_J,axiom,
ord_less_eq_set_a @ s @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(2)
thf(fact_5_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_6_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_7_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_8_assms_I3_J,axiom,
ord_less_eq_set_a @ ( image_a_a @ f @ s ) @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(3)
thf(fact_9_carrier__is__subcring,axiom,
subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subcring
thf(fact_10_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_11_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_12_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_13_add_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( add_a_b @ r @ C @ A )
= ( add_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.l_cancel
thf(fact_14_add_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( add_a_b @ r @ A @ C )
= ( add_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.r_cancel
thf(fact_15_a__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_16_a__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) ) ) ) ).
% a_comm
thf(fact_17_a__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
= ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_18_cgenideal__is__principalideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I ) @ r ) ) ).
% cgenideal_is_principalideal
thf(fact_19_a__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_20_local_Oadd_Oright__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X )
= ( add_a_b @ r @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_21_a__lcos__m__assoc,axiom,
! [M: set_a,G: a,H: a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H @ M ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H ) @ M ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_22_finite__imageI,axiom,
! [F: set_a,H: a > set_a] :
( ( finite_finite_a @ F )
=> ( finite_finite_set_a @ ( image_a_set_a @ H @ F ) ) ) ).
% finite_imageI
thf(fact_23_finite__imageI,axiom,
! [F: set_a,H: a > a] :
( ( finite_finite_a @ F )
=> ( finite_finite_a @ ( image_a_a @ H @ F ) ) ) ).
% finite_imageI
thf(fact_24_image__empty,axiom,
! [F2: a > set_a] :
( ( image_a_set_a @ F2 @ bot_bot_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_25_image__empty,axiom,
! [F2: a > a] :
( ( image_a_a @ F2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_26_empty__is__image,axiom,
! [F2: a > set_a,A2: set_a] :
( ( bot_bot_set_set_a
= ( image_a_set_a @ F2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_27_empty__is__image,axiom,
! [F2: a > a,A2: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_28_image__is__empty,axiom,
! [F2: a > set_a,A2: set_a] :
( ( ( image_a_set_a @ F2 @ A2 )
= bot_bot_set_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_29_image__is__empty,axiom,
! [F2: a > a,A2: set_a] :
( ( ( image_a_a @ F2 @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_30_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_31_subset__empty,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_32_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_33_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_34_subalgebra__in__carrier,axiom,
! [K: set_a,V: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_35_carrier__is__subalgebra,axiom,
! [K: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_36_a__l__coset__subset__G,axiom,
! [H2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_37_subset__Idl__subset,axiom,
! [I2: set_a,H2: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H2 @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H2 ) @ ( genideal_a_b @ r @ I2 ) ) ) ) ).
% subset_Idl_subset
thf(fact_38_genideal__self,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ r @ S ) ) ) ).
% genideal_self
thf(fact_39_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_40_subset__antisym,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_41_subsetI,axiom,
! [A2: set_nat_a,B2: set_nat_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_nat_a @ X2 @ B2 ) )
=> ( ord_le871467723717165285_nat_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_42_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_43_subsetI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( member_set_a @ X2 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_44_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_45_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_46_all__not__in__conv,axiom,
! [A2: set_nat_a] :
( ( ! [X3: nat > a] :
~ ( member_nat_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat_a ) ) ).
% all_not_in_conv
thf(fact_47_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_48_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_49_empty__iff,axiom,
! [C: nat > a] :
~ ( member_nat_a @ C @ bot_bot_set_nat_a ) ).
% empty_iff
thf(fact_50_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_51_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_52_image__eqI,axiom,
! [B: a,F2: a > a,X: a,A2: set_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ B @ ( image_a_a @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_53_image__eqI,axiom,
! [B: nat > a,F2: a > nat > a,X: a,A2: set_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_nat_a @ B @ ( image_a_nat_a @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_54_image__eqI,axiom,
! [B: set_a,F2: a > set_a,X: a,A2: set_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_set_a @ B @ ( image_a_set_a @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_55_image__eqI,axiom,
! [B: a,F2: ( nat > a ) > a,X: nat > a,A2: set_nat_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_nat_a @ X @ A2 )
=> ( member_a @ B @ ( image_nat_a_a @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_56_image__eqI,axiom,
! [B: nat > a,F2: ( nat > a ) > nat > a,X: nat > a,A2: set_nat_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_nat_a @ X @ A2 )
=> ( member_nat_a @ B @ ( image_nat_a_nat_a @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_57_image__eqI,axiom,
! [B: set_a,F2: ( nat > a ) > set_a,X: nat > a,A2: set_nat_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_nat_a @ X @ A2 )
=> ( member_set_a @ B @ ( image_nat_a_set_a @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_58_image__eqI,axiom,
! [B: a,F2: set_a > a,X: set_a,A2: set_set_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_set_a @ X @ A2 )
=> ( member_a @ B @ ( image_set_a_a @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_59_image__eqI,axiom,
! [B: nat > a,F2: set_a > nat > a,X: set_a,A2: set_set_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_set_a @ X @ A2 )
=> ( member_nat_a @ B @ ( image_set_a_nat_a @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_60_image__eqI,axiom,
! [B: set_a,F2: set_a > set_a,X: set_a,A2: set_set_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ B @ ( image_set_a_set_a @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_61_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_62_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_63_set__eq__subset,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_64_set__eq__subset,axiom,
( ( ^ [Y2: set_set_a,Z2: set_set_a] : ( Y2 = Z2 ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_65_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
! [A: nat > a,P: ( nat > a ) > $o] :
( ( member_nat_a @ A @ ( collect_nat_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_67_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_68_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A2: set_nat_a] :
( ( collect_nat_a
@ ^ [X3: nat > a] : ( member_nat_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_71_subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_72_subset__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_73_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_74_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_75_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_76_subset__refl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_77_subset__iff,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A3: set_nat_a,B3: set_nat_a] :
! [T: nat > a] :
( ( member_nat_a @ T @ A3 )
=> ( member_nat_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_78_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_79_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A3 )
=> ( member_set_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_80_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_81_equalityD2,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_82_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_83_equalityD1,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_84_subset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A3: set_nat_a,B3: set_nat_a] :
! [X3: nat > a] :
( ( member_nat_a @ X3 @ A3 )
=> ( member_nat_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_85_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_86_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( member_set_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_87_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_88_equalityE,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_89_subsetD,axiom,
! [A2: set_nat_a,B2: set_nat_a,C: nat > a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( ( member_nat_a @ C @ A2 )
=> ( member_nat_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_90_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_91_subsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_92_in__mono,axiom,
! [A2: set_nat_a,B2: set_nat_a,X: nat > a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( ( member_nat_a @ X @ A2 )
=> ( member_nat_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_93_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_94_in__mono,axiom,
! [A2: set_set_a,B2: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_95_ex__in__conv,axiom,
! [A2: set_nat_a] :
( ( ? [X3: nat > a] : ( member_nat_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat_a ) ) ).
% ex_in_conv
thf(fact_96_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_97_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_98_equals0I,axiom,
! [A2: set_nat_a] :
( ! [Y3: nat > a] :
~ ( member_nat_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat_a ) ) ).
% equals0I
thf(fact_99_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_100_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_101_equals0D,axiom,
! [A2: set_nat_a,A: nat > a] :
( ( A2 = bot_bot_set_nat_a )
=> ~ ( member_nat_a @ A @ A2 ) ) ).
% equals0D
thf(fact_102_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_103_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_104_emptyE,axiom,
! [A: nat > a] :
~ ( member_nat_a @ A @ bot_bot_set_nat_a ) ).
% emptyE
thf(fact_105_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_106_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_107_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: a,F2: a > a] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_a @ B @ ( image_a_a @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_108_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: nat > a,F2: a > nat > a] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_nat_a @ B @ ( image_a_nat_a @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_109_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: set_a,F2: a > set_a] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_set_a @ B @ ( image_a_set_a @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_110_rev__image__eqI,axiom,
! [X: nat > a,A2: set_nat_a,B: a,F2: ( nat > a ) > a] :
( ( member_nat_a @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_a @ B @ ( image_nat_a_a @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_111_rev__image__eqI,axiom,
! [X: nat > a,A2: set_nat_a,B: nat > a,F2: ( nat > a ) > nat > a] :
( ( member_nat_a @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_nat_a @ B @ ( image_nat_a_nat_a @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_112_rev__image__eqI,axiom,
! [X: nat > a,A2: set_nat_a,B: set_a,F2: ( nat > a ) > set_a] :
( ( member_nat_a @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_set_a @ B @ ( image_nat_a_set_a @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_113_rev__image__eqI,axiom,
! [X: set_a,A2: set_set_a,B: a,F2: set_a > a] :
( ( member_set_a @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_a @ B @ ( image_set_a_a @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_114_rev__image__eqI,axiom,
! [X: set_a,A2: set_set_a,B: nat > a,F2: set_a > nat > a] :
( ( member_set_a @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_nat_a @ B @ ( image_set_a_nat_a @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_115_rev__image__eqI,axiom,
! [X: set_a,A2: set_set_a,B: set_a,F2: set_a > set_a] :
( ( member_set_a @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_set_a @ B @ ( image_set_a_set_a @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_116_ball__imageD,axiom,
! [F2: a > a,A2: set_a,P: a > $o] :
( ! [X2: a] :
( ( member_a @ X2 @ ( image_a_a @ F2 @ A2 ) )
=> ( P @ X2 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_117_ball__imageD,axiom,
! [F2: a > set_a,A2: set_a,P: set_a > $o] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( image_a_set_a @ F2 @ A2 ) )
=> ( P @ X2 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_118_image__cong,axiom,
! [M: set_a,N: set_a,F2: a > a,G: a > a] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_a_a @ F2 @ M )
= ( image_a_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_119_image__cong,axiom,
! [M: set_a,N: set_a,F2: a > set_a,G: a > set_a] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_a_set_a @ F2 @ M )
= ( image_a_set_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_120_bex__imageD,axiom,
! [F2: a > a,A2: set_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_a_a @ F2 @ A2 ) )
& ( P @ X4 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_121_bex__imageD,axiom,
! [F2: a > set_a,A2: set_a,P: set_a > $o] :
( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( image_a_set_a @ F2 @ A2 ) )
& ( P @ X4 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_122_image__iff,axiom,
! [Z: a,F2: a > a,A2: set_a] :
( ( member_a @ Z @ ( image_a_a @ F2 @ A2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( Z
= ( F2 @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_123_image__iff,axiom,
! [Z: set_a,F2: a > set_a,A2: set_a] :
( ( member_set_a @ Z @ ( image_a_set_a @ F2 @ A2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( Z
= ( F2 @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_124_imageI,axiom,
! [X: a,A2: set_a,F2: a > a] :
( ( member_a @ X @ A2 )
=> ( member_a @ ( F2 @ X ) @ ( image_a_a @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_125_imageI,axiom,
! [X: a,A2: set_a,F2: a > nat > a] :
( ( member_a @ X @ A2 )
=> ( member_nat_a @ ( F2 @ X ) @ ( image_a_nat_a @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_126_imageI,axiom,
! [X: a,A2: set_a,F2: a > set_a] :
( ( member_a @ X @ A2 )
=> ( member_set_a @ ( F2 @ X ) @ ( image_a_set_a @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_127_imageI,axiom,
! [X: nat > a,A2: set_nat_a,F2: ( nat > a ) > a] :
( ( member_nat_a @ X @ A2 )
=> ( member_a @ ( F2 @ X ) @ ( image_nat_a_a @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_128_imageI,axiom,
! [X: nat > a,A2: set_nat_a,F2: ( nat > a ) > nat > a] :
( ( member_nat_a @ X @ A2 )
=> ( member_nat_a @ ( F2 @ X ) @ ( image_nat_a_nat_a @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_129_imageI,axiom,
! [X: nat > a,A2: set_nat_a,F2: ( nat > a ) > set_a] :
( ( member_nat_a @ X @ A2 )
=> ( member_set_a @ ( F2 @ X ) @ ( image_nat_a_set_a @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_130_imageI,axiom,
! [X: set_a,A2: set_set_a,F2: set_a > a] :
( ( member_set_a @ X @ A2 )
=> ( member_a @ ( F2 @ X ) @ ( image_set_a_a @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_131_imageI,axiom,
! [X: set_a,A2: set_set_a,F2: set_a > nat > a] :
( ( member_set_a @ X @ A2 )
=> ( member_nat_a @ ( F2 @ X ) @ ( image_set_a_nat_a @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_132_imageI,axiom,
! [X: set_a,A2: set_set_a,F2: set_a > set_a] :
( ( member_set_a @ X @ A2 )
=> ( member_set_a @ ( F2 @ X ) @ ( image_set_a_set_a @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_133_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_134_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_135_finite__has__minimal2,axiom,
! [A2: set_set_set_a,A: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( member_set_set_a @ A @ A2 )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
& ( ord_le3724670747650509150_set_a @ X2 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_136_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ A @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_137_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_138_finite__has__maximal2,axiom,
! [A2: set_set_set_a,A: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( member_set_set_a @ A @ A2 )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
& ( ord_le3724670747650509150_set_a @ A @ X2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_139_all__subset__image,axiom,
! [F2: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F2 @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( P @ ( image_a_a @ F2 @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_140_all__subset__image,axiom,
! [F2: set_a > a,A2: set_set_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_set_a_a @ F2 @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
=> ( P @ ( image_set_a_a @ F2 @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_141_all__subset__image,axiom,
! [F2: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ! [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_a_set_a @ F2 @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( P @ ( image_a_set_a @ F2 @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_142_all__subset__image,axiom,
! [F2: set_a > set_a,A2: set_set_a,P: set_set_a > $o] :
( ( ! [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_set_a_set_a @ F2 @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
=> ( P @ ( image_set_a_set_a @ F2 @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_143_subset__image__iff,axiom,
! [B2: set_a,F2: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F2 @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_144_subset__image__iff,axiom,
! [B2: set_a,F2: set_a > a,A2: set_set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_set_a_a @ F2 @ A2 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A2 )
& ( B2
= ( image_set_a_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_145_subset__image__iff,axiom,
! [B2: set_set_a,F2: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_a_set_a @ F2 @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_set_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_146_subset__image__iff,axiom,
! [B2: set_set_a,F2: set_a > set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_set_a_set_a @ F2 @ A2 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A2 )
& ( B2
= ( image_set_a_set_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_147_image__subset__iff,axiom,
! [F2: a > a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A2 ) @ B2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ ( F2 @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_148_image__subset__iff,axiom,
! [F2: a > set_a,A2: set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F2 @ A2 ) @ B2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_set_a @ ( F2 @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_149_subset__imageE,axiom,
! [B2: set_a,F2: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F2 @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B2
!= ( image_a_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_150_subset__imageE,axiom,
! [B2: set_a,F2: set_a > a,A2: set_set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_set_a_a @ F2 @ A2 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A2 )
=> ( B2
!= ( image_set_a_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_151_subset__imageE,axiom,
! [B2: set_set_a,F2: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_a_set_a @ F2 @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B2
!= ( image_a_set_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_152_subset__imageE,axiom,
! [B2: set_set_a,F2: set_a > set_a,A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_set_a_set_a @ F2 @ A2 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A2 )
=> ( B2
!= ( image_set_a_set_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_153_image__subsetI,axiom,
! [A2: set_a,F2: a > nat > a,B2: set_nat_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_nat_a @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_le871467723717165285_nat_a @ ( image_a_nat_a @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_154_image__subsetI,axiom,
! [A2: set_nat_a,F2: ( nat > a ) > nat > a,B2: set_nat_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_nat_a @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_le871467723717165285_nat_a @ ( image_nat_a_nat_a @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_155_image__subsetI,axiom,
! [A2: set_set_a,F2: set_a > nat > a,B2: set_nat_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( member_nat_a @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_le871467723717165285_nat_a @ ( image_set_a_nat_a @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_156_image__subsetI,axiom,
! [A2: set_a,F2: a > a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_157_image__subsetI,axiom,
! [A2: set_nat_a,F2: ( nat > a ) > a,B2: set_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_a @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_nat_a_a @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_158_image__subsetI,axiom,
! [A2: set_set_a,F2: set_a > a,B2: set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( member_a @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_159_image__subsetI,axiom,
! [A2: set_a,F2: a > set_a,B2: set_set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_set_a @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_160_image__subsetI,axiom,
! [A2: set_nat_a,F2: ( nat > a ) > set_a,B2: set_set_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_set_a @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_nat_a_set_a @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_161_image__subsetI,axiom,
! [A2: set_set_a,F2: set_a > set_a,B2: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( member_set_a @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_162_image__mono,axiom,
! [A2: set_a,B2: set_a,F2: a > a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A2 ) @ ( image_a_a @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_163_image__mono,axiom,
! [A2: set_a,B2: set_a,F2: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F2 @ A2 ) @ ( image_a_set_a @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_164_image__mono,axiom,
! [A2: set_set_a,B2: set_set_a,F2: set_a > a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F2 @ A2 ) @ ( image_set_a_a @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_165_image__mono,axiom,
! [A2: set_set_a,B2: set_set_a,F2: set_a > set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F2 @ A2 ) @ ( image_set_a_set_a @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_166_rev__finite__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_167_rev__finite__subset,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_168_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_169_infinite__super,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T2 )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_170_finite__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_171_finite__subset,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% finite_subset
thf(fact_172_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_173_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_174_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_175_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_176_finite__has__minimal,axiom,
! [A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_177_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_178_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_179_finite__has__maximal,axiom,
! [A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_180_all__finite__subset__image,axiom,
! [F2: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F2 @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A2 ) )
=> ( P @ ( image_a_a @ F2 @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_181_all__finite__subset__image,axiom,
! [F2: set_a > a,A2: set_set_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_set_a_a @ F2 @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_a] :
( ( ( finite_finite_set_a @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A2 ) )
=> ( P @ ( image_set_a_a @ F2 @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_182_all__finite__subset__image,axiom,
! [F2: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ! [B3: set_set_a] :
( ( ( finite_finite_set_a @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ ( image_a_set_a @ F2 @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A2 ) )
=> ( P @ ( image_a_set_a @ F2 @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_183_all__finite__subset__image,axiom,
! [F2: set_a > set_a,A2: set_set_a,P: set_set_a > $o] :
( ( ! [B3: set_set_a] :
( ( ( finite_finite_set_a @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ ( image_set_a_set_a @ F2 @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_a] :
( ( ( finite_finite_set_a @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A2 ) )
=> ( P @ ( image_set_a_set_a @ F2 @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_184_ex__finite__subset__image,axiom,
! [F2: a > a,A2: set_a,P: set_a > $o] :
( ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F2 @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A2 )
& ( P @ ( image_a_a @ F2 @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_185_ex__finite__subset__image,axiom,
! [F2: set_a > a,A2: set_set_a,P: set_a > $o] :
( ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_set_a_a @ F2 @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_set_a] :
( ( finite_finite_set_a @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A2 )
& ( P @ ( image_set_a_a @ F2 @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_186_ex__finite__subset__image,axiom,
! [F2: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ? [B3: set_set_a] :
( ( finite_finite_set_a @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ ( image_a_set_a @ F2 @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A2 )
& ( P @ ( image_a_set_a @ F2 @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_187_ex__finite__subset__image,axiom,
! [F2: set_a > set_a,A2: set_set_a,P: set_set_a > $o] :
( ( ? [B3: set_set_a] :
( ( finite_finite_set_a @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ ( image_set_a_set_a @ F2 @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_set_a] :
( ( finite_finite_set_a @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A2 )
& ( P @ ( image_set_a_set_a @ F2 @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_188_finite__subset__image,axiom,
! [B2: set_a,F2: a > a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F2 @ A2 ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
& ( finite_finite_a @ C3 )
& ( B2
= ( image_a_a @ F2 @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_189_finite__subset__image,axiom,
! [B2: set_a,F2: set_a > a,A2: set_set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_set_a_a @ F2 @ A2 ) )
=> ? [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A2 )
& ( finite_finite_set_a @ C3 )
& ( B2
= ( image_set_a_a @ F2 @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_190_finite__subset__image,axiom,
! [B2: set_set_a,F2: a > set_a,A2: set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ ( image_a_set_a @ F2 @ A2 ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
& ( finite_finite_a @ C3 )
& ( B2
= ( image_a_set_a @ F2 @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_191_finite__subset__image,axiom,
! [B2: set_set_a,F2: set_a > set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ ( image_set_a_set_a @ F2 @ A2 ) )
=> ? [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A2 )
& ( finite_finite_set_a @ C3 )
& ( B2
= ( image_set_a_set_a @ F2 @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_192_finite__surj,axiom,
! [A2: set_a,B2: set_a,F2: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F2 @ A2 ) )
=> ( finite_finite_a @ B2 ) ) ) ).
% finite_surj
thf(fact_193_finite__surj,axiom,
! [A2: set_a,B2: set_set_a,F2: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ ( image_a_set_a @ F2 @ A2 ) )
=> ( finite_finite_set_a @ B2 ) ) ) ).
% finite_surj
thf(fact_194_a__coset__hom_I1_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,I2: set_a,A: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( image_a_a @ H @ ( a_l_coset_a_b @ R @ A @ I2 ) )
= ( a_l_coset_a_b @ S @ ( H @ A ) @ ( image_a_a @ H @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_195_a__coset__hom_I1_J,axiom,
! [H: set_a > a,R: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,I2: set_set_a,A: set_a] :
( ( member_set_a_a @ H @ ( ring_h4811522740288071338it_a_b @ R @ S ) )
=> ( ( ord_le3724670747650509150_set_a @ I2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( image_set_a_a @ H @ ( a_l_co8760856729277977016t_unit @ R @ A @ I2 ) )
= ( a_l_coset_a_b @ S @ ( H @ A ) @ ( image_set_a_a @ H @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_196_noetherian__ringI,axiom,
( ! [I3: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ? [A4: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( finite_finite_a @ A4 )
& ( I3
= ( genideal_a_b @ r @ A4 ) ) ) )
=> ( ring_n3639167112692572309ng_a_b @ r ) ) ).
% noetherian_ringI
thf(fact_197_cgenideal__eq__genideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( cgenid547466209912283029xt_a_b @ r @ I )
= ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% cgenideal_eq_genideal
thf(fact_198_Idl__subset__ideal_H,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_199_a__lcos__mult__one,axiom,
! [M: set_a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M )
= M ) ) ).
% a_lcos_mult_one
thf(fact_200_genideal__self_H,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_201_line__extension__in__carrier,axiom,
! [K: set_a,A: a,E: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_202_finetely__gen,axiom,
! [I2: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ? [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( finite_finite_a @ A5 )
& ( I2
= ( genideal_a_b @ r @ A5 ) ) ) ) ).
% finetely_gen
thf(fact_203_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_204_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia6043505979758434576t_unit,X: set_a,Y: set_a] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ R ) )
=> ( member_set_a @ ( add_se3735415688806051380t_unit @ R @ X @ Y ) @ ( partia5907974310037520643t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_205_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_206_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia6043505979758434576t_unit,X: set_a,Y: set_a,Z: set_a] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Z @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( add_se3735415688806051380t_unit @ R @ ( add_se3735415688806051380t_unit @ R @ X @ Y ) @ Z )
= ( add_se3735415688806051380t_unit @ R @ X @ ( add_se3735415688806051380t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_207_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y )
= ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_208_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia6043505979758434576t_unit,X: set_a,Y: set_a] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( add_se3735415688806051380t_unit @ R @ X @ Y )
= ( add_se3735415688806051380t_unit @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_209_oneideal,axiom,
ideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% oneideal
thf(fact_210_local_Ominus__unique,axiom,
! [Y: a,X: a,Y4: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y4 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y4 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_211_add_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X2 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_212_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_213_add_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X2 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_214_add_Oinv__comm,axiom,
! [X: a,Y: a] :
( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_215_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_216_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_217_insert__iff,axiom,
! [A: nat > a,B: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ ( insert_nat_a @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_218_insert__iff,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A2 ) )
= ( ( A = B )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_219_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_220_insertCI,axiom,
! [A: nat > a,B2: set_nat_a,B: nat > a] :
( ( ~ ( member_nat_a @ A @ B2 )
=> ( A = B ) )
=> ( member_nat_a @ A @ ( insert_nat_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_221_insertCI,axiom,
! [A: set_a,B2: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B2 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_222_exists__gen,axiom,
! [I2: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( I2
= ( cgenid547466209912283029xt_a_b @ r @ X2 ) ) ) ) ).
% exists_gen
thf(fact_223_cgenideal__ideal,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ r ) ) ).
% cgenideal_ideal
thf(fact_224_cgenideal__minimal,axiom,
! [J: set_a,A: a] :
( ( ideal_a_b @ J @ r )
=> ( ( member_a @ A @ J )
=> ( ord_less_eq_set_a @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ J ) ) ) ).
% cgenideal_minimal
thf(fact_225_genideal__minimal,axiom,
! [I2: set_a,S: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ord_less_eq_set_a @ S @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ S ) @ I2 ) ) ) ).
% genideal_minimal
thf(fact_226_zeroideal,axiom,
ideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroideal
thf(fact_227_genideal__zero,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% genideal_zero
thf(fact_228_genideal__ideal,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ideal_a_b @ ( genideal_a_b @ r @ S ) @ r ) ) ).
% genideal_ideal
thf(fact_229_Idl__subset__ideal,axiom,
! [I2: set_a,H2: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H2 ) @ I2 )
= ( ord_less_eq_set_a @ H2 @ I2 ) ) ) ) ).
% Idl_subset_ideal
thf(fact_230_ideal__is__subalgebra,axiom,
! [K: set_a,I2: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ideal_a_b @ I2 @ r )
=> ( embedd9027525575939734154ra_a_b @ K @ I2 @ r ) ) ) ).
% ideal_is_subalgebra
thf(fact_231_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_232_insert__subset,axiom,
! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
= ( ( member_nat_a @ X @ B2 )
& ( ord_le871467723717165285_nat_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_233_insert__subset,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( ( member_a @ X @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_234_insert__subset,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( ( member_set_a @ X @ B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_235_singletonI,axiom,
! [A: nat > a] : ( member_nat_a @ A @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) ) ).
% singletonI
thf(fact_236_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_237_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_238_image__insert,axiom,
! [F2: a > set_a,A: a,B2: set_a] :
( ( image_a_set_a @ F2 @ ( insert_a @ A @ B2 ) )
= ( insert_set_a @ ( F2 @ A ) @ ( image_a_set_a @ F2 @ B2 ) ) ) ).
% image_insert
thf(fact_239_image__insert,axiom,
! [F2: a > a,A: a,B2: set_a] :
( ( image_a_a @ F2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ ( F2 @ A ) @ ( image_a_a @ F2 @ B2 ) ) ) ).
% image_insert
thf(fact_240_insert__image,axiom,
! [X: a,A2: set_a,F2: a > set_a] :
( ( member_a @ X @ A2 )
=> ( ( insert_set_a @ ( F2 @ X ) @ ( image_a_set_a @ F2 @ A2 ) )
= ( image_a_set_a @ F2 @ A2 ) ) ) ).
% insert_image
thf(fact_241_insert__image,axiom,
! [X: a,A2: set_a,F2: a > a] :
( ( member_a @ X @ A2 )
=> ( ( insert_a @ ( F2 @ X ) @ ( image_a_a @ F2 @ A2 ) )
= ( image_a_a @ F2 @ A2 ) ) ) ).
% insert_image
thf(fact_242_insert__image,axiom,
! [X: nat > a,A2: set_nat_a,F2: ( nat > a ) > a] :
( ( member_nat_a @ X @ A2 )
=> ( ( insert_a @ ( F2 @ X ) @ ( image_nat_a_a @ F2 @ A2 ) )
= ( image_nat_a_a @ F2 @ A2 ) ) ) ).
% insert_image
thf(fact_243_insert__image,axiom,
! [X: set_a,A2: set_set_a,F2: set_a > a] :
( ( member_set_a @ X @ A2 )
=> ( ( insert_a @ ( F2 @ X ) @ ( image_set_a_a @ F2 @ A2 ) )
= ( image_set_a_a @ F2 @ A2 ) ) ) ).
% insert_image
thf(fact_244_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_245_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_246_singleton__insert__inj__eq,axiom,
! [B: set_a,A: set_a,A2: set_set_a] :
( ( ( insert_set_a @ B @ bot_bot_set_set_a )
= ( insert_set_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_247_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_248_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A2: set_set_a,B: set_a] :
( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B @ bot_bot_set_set_a ) )
= ( ( A = B )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_249_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_250_r__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
= X ) ) ).
% r_zero
thf(fact_251_l__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
= X ) ) ).
% l_zero
thf(fact_252_add_Or__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ A @ X ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_253_add_Or__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_254_add_Ol__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ X @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_255_add_Ol__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ A )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_256_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B4: set_a] :
( ( A2
= ( insert_a @ A @ B4 ) )
& ~ ( member_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_257_mk__disjoint__insert,axiom,
! [A: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ A2 )
=> ? [B4: set_nat_a] :
( ( A2
= ( insert_nat_a @ A @ B4 ) )
& ~ ( member_nat_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_258_mk__disjoint__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ? [B4: set_set_a] :
( ( A2
= ( insert_set_a @ A @ B4 ) )
& ~ ( member_set_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_259_insert__commute,axiom,
! [X: a,Y: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A2 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_260_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_a] :
( ( A2
= ( insert_a @ B @ C4 ) )
& ~ ( member_a @ B @ C4 )
& ( B2
= ( insert_a @ A @ C4 ) )
& ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_261_insert__eq__iff,axiom,
! [A: nat > a,A2: set_nat_a,B: nat > a,B2: set_nat_a] :
( ~ ( member_nat_a @ A @ A2 )
=> ( ~ ( member_nat_a @ B @ B2 )
=> ( ( ( insert_nat_a @ A @ A2 )
= ( insert_nat_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_nat_a] :
( ( A2
= ( insert_nat_a @ B @ C4 ) )
& ~ ( member_nat_a @ B @ C4 )
& ( B2
= ( insert_nat_a @ A @ C4 ) )
& ~ ( member_nat_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_262_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B: set_a,B2: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B @ B2 )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_set_a] :
( ( A2
= ( insert_set_a @ B @ C4 ) )
& ~ ( member_set_a @ B @ C4 )
& ( B2
= ( insert_set_a @ A @ C4 ) )
& ~ ( member_set_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_263_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_264_insert__absorb,axiom,
! [A: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ A2 )
=> ( ( insert_nat_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_265_insert__absorb,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_266_insert__ident,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_267_insert__ident,axiom,
! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ~ ( member_nat_a @ X @ B2 )
=> ( ( ( insert_nat_a @ X @ A2 )
= ( insert_nat_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_268_insert__ident,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ~ ( member_set_a @ X @ B2 )
=> ( ( ( insert_set_a @ X @ A2 )
= ( insert_set_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_269_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B4: set_a] :
( ( A2
= ( insert_a @ X @ B4 ) )
=> ( member_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_270_Set_Oset__insert,axiom,
! [X: nat > a,A2: set_nat_a] :
( ( member_nat_a @ X @ A2 )
=> ~ ! [B4: set_nat_a] :
( ( A2
= ( insert_nat_a @ X @ B4 ) )
=> ( member_nat_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_271_Set_Oset__insert,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ~ ! [B4: set_set_a] :
( ( A2
= ( insert_set_a @ X @ B4 ) )
=> ( member_set_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_272_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a @ A @ B2 )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_273_insertI2,axiom,
! [A: nat > a,B2: set_nat_a,B: nat > a] :
( ( member_nat_a @ A @ B2 )
=> ( member_nat_a @ A @ ( insert_nat_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_274_insertI2,axiom,
! [A: set_a,B2: set_set_a,B: set_a] :
( ( member_set_a @ A @ B2 )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_275_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a @ A @ ( insert_a @ A @ B2 ) ) ).
% insertI1
thf(fact_276_insertI1,axiom,
! [A: nat > a,B2: set_nat_a] : ( member_nat_a @ A @ ( insert_nat_a @ A @ B2 ) ) ).
% insertI1
thf(fact_277_insertI1,axiom,
! [A: set_a,B2: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B2 ) ) ).
% insertI1
thf(fact_278_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_279_insertE,axiom,
! [A: nat > a,B: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ ( insert_nat_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_280_insertE,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_281_insert__mono,axiom,
! [C2: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_282_insert__mono,axiom,
! [C2: set_set_a,D: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ D )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A @ C2 ) @ ( insert_set_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_283_subset__insert,axiom,
! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ( ord_le871467723717165285_nat_a @ A2 @ ( insert_nat_a @ X @ B2 ) )
= ( ord_le871467723717165285_nat_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_284_subset__insert,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_285_subset__insert,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_286_subset__insertI,axiom,
! [B2: set_a,A: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_287_subset__insertI,axiom,
! [B2: set_set_a,A: set_a] : ( ord_le3724670747650509150_set_a @ B2 @ ( insert_set_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_288_subset__insertI2,axiom,
! [A2: set_a,B2: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_289_subset__insertI2,axiom,
! [A2: set_set_a,B2: set_set_a,B: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_290_singletonD,axiom,
! [B: nat > a,A: nat > a] :
( ( member_nat_a @ B @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_291_singletonD,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_292_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_293_singleton__iff,axiom,
! [B: nat > a,A: nat > a] :
( ( member_nat_a @ B @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_294_singleton__iff,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_295_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_296_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_297_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_298_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_299_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_300_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_301_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( semiri8432286899730879049t_unit @ R )
=> ( member_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ ( partia5907974310037520643t_unit @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_302_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_303_subset__singletonD,axiom,
! [A2: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
=> ( ( A2 = bot_bot_set_set_a )
| ( A2
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_304_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_305_subset__singleton__iff,axiom,
! [X5: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_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_306_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A5: set_a] :
( ? [A6: a] :
( A
= ( insert_a @ A6 @ A5 ) )
=> ~ ( finite_finite_a @ A5 ) ) ) ) ).
% finite.cases
thf(fact_307_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A7: set_a] :
( ( A7 = bot_bot_set_a )
| ? [A3: set_a,B5: a] :
( ( A7
= ( insert_a @ B5 @ A3 ) )
& ( finite_finite_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_308_finite__induct,axiom,
! [F: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ F )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [X2: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ~ ( member_nat_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_309_finite__induct,axiom,
! [F: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X2: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_310_finite__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_311_finite__ne__induct,axiom,
! [F: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ F )
=> ( ( F != bot_bot_set_nat_a )
=> ( ! [X2: nat > a] : ( P @ ( insert_nat_a @ X2 @ bot_bot_set_nat_a ) )
=> ( ! [X2: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( F3 != bot_bot_set_nat_a )
=> ( ~ ( member_nat_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_312_finite__ne__induct,axiom,
! [F: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( F != bot_bot_set_set_a )
=> ( ! [X2: set_a] : ( P @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
=> ( ! [X2: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( F3 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_313_finite__ne__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( F != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_314_infinite__finite__induct,axiom,
! [P: set_nat_a > $o,A2: set_nat_a] :
( ! [A5: set_nat_a] :
( ~ ( finite_finite_nat_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [X2: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ~ ( member_nat_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_315_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A2: set_set_a] :
( ! [A5: set_set_a] :
( ~ ( finite_finite_set_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X2: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_316_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A5: set_a] :
( ~ ( finite_finite_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_317_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_318_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia6043505979758434576t_unit,X: set_a] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( add_se3735415688806051380t_unit @ R @ X @ ( zero_s2174465271003423091t_unit @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_319_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_320_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia6043505979758434576t_unit,X: set_a] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( add_se3735415688806051380t_unit @ R @ ( zero_s2174465271003423091t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_321_finite__subset__induct,axiom,
! [F: set_nat_a,A2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ F )
=> ( ( ord_le871467723717165285_nat_a @ F @ A2 )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [A6: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( member_nat_a @ A6 @ A2 )
=> ( ~ ( member_nat_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_322_finite__subset__induct,axiom,
! [F: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A6 @ A2 )
=> ( ~ ( member_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_323_finite__subset__induct,axiom,
! [F: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A6: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A6 @ A2 )
=> ( ~ ( member_set_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_324_finite__subset__induct_H,axiom,
! [F: set_nat_a,A2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ F )
=> ( ( ord_le871467723717165285_nat_a @ F @ A2 )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [A6: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( member_nat_a @ A6 @ A2 )
=> ( ( ord_le871467723717165285_nat_a @ F3 @ A2 )
=> ( ~ ( member_nat_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_325_finite__subset__induct_H,axiom,
! [F: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A6 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_326_finite__subset__induct_H,axiom,
! [F: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A6: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A6 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A2 )
=> ( ~ ( member_set_a @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_327_ring__hom__closed,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_328_ring__hom__closed,axiom,
! [H: a > set_a,R: partia2175431115845679010xt_a_b,S: partia6043505979758434576t_unit,X: a] :
( ( member_a_set_a @ H @ ( ring_h7909251507493334186t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_set_a @ ( H @ X ) @ ( partia5907974310037520643t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_329_ring__hom__closed,axiom,
! [H: set_a > a,R: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,X: set_a] :
( ( member_set_a_a @ H @ ( ring_h4811522740288071338it_a_b @ R @ S ) )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_330_ring__hom__closed,axiom,
! [H: set_a > set_a,R: partia6043505979758434576t_unit,S: partia6043505979758434576t_unit,X: set_a] :
( ( member_set_a_set_a @ H @ ( ring_h4754363464466234712t_unit @ R @ S ) )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( member_set_a @ ( H @ X ) @ ( partia5907974310037520643t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_331_ring__iso__memE_I1_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_332_ring__iso__memE_I1_J,axiom,
! [H: a > set_a,R: partia2175431115845679010xt_a_b,S: partia6043505979758434576t_unit,X: a] :
( ( member_a_set_a @ H @ ( ring_i7849008455817099456t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_set_a @ ( H @ X ) @ ( partia5907974310037520643t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_333_ring__iso__memE_I1_J,axiom,
! [H: set_a > a,R: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,X: set_a] :
( ( member_set_a_a @ H @ ( ring_i4751279688611836608it_a_b @ R @ S ) )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_334_ring__iso__memE_I1_J,axiom,
! [H: set_a > set_a,R: partia6043505979758434576t_unit,S: partia6043505979758434576t_unit,X: set_a] :
( ( member_set_a_set_a @ H @ ( ring_i7821126338072249198t_unit @ R @ S ) )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( member_set_a @ ( H @ X ) @ ( partia5907974310037520643t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_335_ring__hom__add,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_336_ring__hom__add,axiom,
! [H: set_a > a,R: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,X: set_a,Y: set_a] :
( ( member_set_a_a @ H @ ( ring_h4811522740288071338it_a_b @ R @ S ) )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( H @ ( add_se3735415688806051380t_unit @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_337_ring__iso__memE_I3_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_338_ring__iso__memE_I3_J,axiom,
! [H: set_a > a,R: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,X: set_a,Y: set_a] :
( ( member_set_a_a @ H @ ( ring_i4751279688611836608it_a_b @ R @ S ) )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( H @ ( add_se3735415688806051380t_unit @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_339_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_340_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia6043505979758434576t_unit,X: set_a,Y: set_a,Z: set_a] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Z @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( add_se3735415688806051380t_unit @ R @ X @ ( add_se3735415688806051380t_unit @ R @ Y @ Z ) )
= ( add_se3735415688806051380t_unit @ R @ Y @ ( add_se3735415688806051380t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_341_zeromaximalideal__eq__field,axiom,
( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
= ( field_a_b @ r ) ) ).
% zeromaximalideal_eq_field
thf(fact_342_zeromaximalideal__fieldI,axiom,
( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
=> ( field_a_b @ r ) ) ).
% zeromaximalideal_fieldI
thf(fact_343_zeromaximalideal,axiom,
maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeromaximalideal
thf(fact_344_ring__primeE_I1_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( P2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_345_ring__irreducibleE_I1_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( R2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_346_noetherian__ring_Ofinetely__gen,axiom,
! [R: partia6043505979758434576t_unit,I2: set_set_a] :
( ( ring_n5014428767265248323t_unit @ R )
=> ( ( ideal_4463284918206690523t_unit @ I2 @ R )
=> ? [A5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ ( partia5907974310037520643t_unit @ R ) )
& ( finite_finite_set_a @ A5 )
& ( I2
= ( genide4542190045063241085t_unit @ R @ A5 ) ) ) ) ) ).
% noetherian_ring.finetely_gen
thf(fact_347_noetherian__ring_Ofinetely__gen,axiom,
! [R: partia2175431115845679010xt_a_b,I2: set_a] :
( ( ring_n3639167112692572309ng_a_b @ R )
=> ( ( ideal_a_b @ I2 @ R )
=> ? [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ ( partia707051561876973205xt_a_b @ R ) )
& ( finite_finite_a @ A5 )
& ( I2
= ( genideal_a_b @ R @ A5 ) ) ) ) ) ).
% noetherian_ring.finetely_gen
thf(fact_348_principalidealI,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( ideal_a_b @ I2 @ R )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
& ( I2
= ( genideal_a_b @ R @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( principalideal_a_b @ I2 @ R ) ) ) ).
% principalidealI
thf(fact_349_principalidealI,axiom,
! [I2: set_set_a,R: partia6043505979758434576t_unit] :
( ( ideal_4463284918206690523t_unit @ I2 @ R )
=> ( ? [X4: set_a] :
( ( member_set_a @ X4 @ ( partia5907974310037520643t_unit @ R ) )
& ( I2
= ( genide4542190045063241085t_unit @ R @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) )
=> ( princi3104115052557732031t_unit @ I2 @ R ) ) ) ).
% principalidealI
thf(fact_350_zeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroprimeideal
thf(fact_351_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_352_principalideal_Ogenerate,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
& ( I2
= ( genideal_a_b @ R @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_353_principalideal_Ogenerate,axiom,
! [I2: set_set_a,R: partia6043505979758434576t_unit] :
( ( princi3104115052557732031t_unit @ I2 @ R )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ ( partia5907974310037520643t_unit @ R ) )
& ( I2
= ( genide4542190045063241085t_unit @ R @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_354_maximalideal__prime,axiom,
! [I2: set_a] :
( ( maximalideal_a_b @ I2 @ r )
=> ( primeideal_a_b @ I2 @ r ) ) ).
% maximalideal_prime
thf(fact_355_primeness__condition,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P2 )
= ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% primeness_condition
thf(fact_356_ring__primeE_I3_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( prime_a_ring_ext_a_b @ r @ P2 ) ) ) ).
% ring_primeE(3)
thf(fact_357_ring__primeI,axiom,
! [P2: a] :
( ( P2
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P2 )
=> ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% ring_primeI
thf(fact_358_irreducible__imp__maximalideal,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P2 )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P2 ) @ r ) ) ) ).
% irreducible_imp_maximalideal
thf(fact_359_ring__prime__def,axiom,
( ring_ring_prime_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,A7: a] :
( ( A7
!= ( zero_a_b @ R3 ) )
& ( prime_a_ring_ext_a_b @ R3 @ A7 ) ) ) ) ).
% ring_prime_def
thf(fact_360_maximalideal_Ois__maximalideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( maximalideal_a_b @ I2 @ R )
=> ( maximalideal_a_b @ I2 @ R ) ) ).
% maximalideal.is_maximalideal
thf(fact_361_primeideal_Oprimeideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( primeideal_a_b @ I2 @ R )
=> ( primeideal_a_b @ I2 @ R ) ) ).
% primeideal.primeideal
thf(fact_362_primeideal_OI__notcarr,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( primeideal_a_b @ I2 @ R )
=> ( ( partia707051561876973205xt_a_b @ R )
!= I2 ) ) ).
% primeideal.I_notcarr
thf(fact_363_primeideal_OI__notcarr,axiom,
! [I2: set_set_a,R: partia6043505979758434576t_unit] :
( ( primei7645216761534224334t_unit @ I2 @ R )
=> ( ( partia5907974310037520643t_unit @ R )
!= I2 ) ) ).
% primeideal.I_notcarr
thf(fact_364_maximalideal_OI__notcarr,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( maximalideal_a_b @ I2 @ R )
=> ( ( partia707051561876973205xt_a_b @ R )
!= I2 ) ) ).
% maximalideal.I_notcarr
thf(fact_365_maximalideal_OI__notcarr,axiom,
! [I2: set_set_a,R: partia6043505979758434576t_unit] :
( ( maxima2253313296322093082t_unit @ I2 @ R )
=> ( ( partia5907974310037520643t_unit @ R )
!= I2 ) ) ).
% maximalideal.I_notcarr
thf(fact_366_primeideal_Oaxioms_I1_J,axiom,
! [I2: set_set_a,R: partia6043505979758434576t_unit] :
( ( primei7645216761534224334t_unit @ I2 @ R )
=> ( ideal_4463284918206690523t_unit @ I2 @ R ) ) ).
% primeideal.axioms(1)
thf(fact_367_primeideal_Oaxioms_I1_J,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( primeideal_a_b @ I2 @ R )
=> ( ideal_a_b @ I2 @ R ) ) ).
% primeideal.axioms(1)
thf(fact_368_maximalideal_Oaxioms_I1_J,axiom,
! [I2: set_set_a,R: partia6043505979758434576t_unit] :
( ( maxima2253313296322093082t_unit @ I2 @ R )
=> ( ideal_4463284918206690523t_unit @ I2 @ R ) ) ).
% maximalideal.axioms(1)
thf(fact_369_maximalideal_Oaxioms_I1_J,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( maximalideal_a_b @ I2 @ R )
=> ( ideal_a_b @ I2 @ R ) ) ).
% maximalideal.axioms(1)
thf(fact_370_principal__domain_Oprimeness__condition,axiom,
! [R: partia6043505979758434576t_unit,P2: set_a] :
( ( ring_p2862007038493914190t_unit @ R )
=> ( ( member_set_a @ P2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ P2 )
= ( ring_r6795642478576035723t_unit @ R @ P2 ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_371_principal__domain_Oprimeness__condition,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ P2 )
= ( ring_ring_prime_a_b @ R @ P2 ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_372_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia6043505979758434576t_unit,P2: set_a] :
( ( ring_p2862007038493914190t_unit @ R )
=> ( ( member_set_a @ P2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ P2 )
=> ( maxima2253313296322093082t_unit @ ( cgenid6682780793756002467t_unit @ R @ P2 ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_373_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ P2 )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ R @ P2 ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_374_maximalidealI,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( ideal_a_b @ I2 @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
!= I2 )
=> ( ! [J2: set_a] :
( ( ideal_a_b @ J2 @ R )
=> ( ( ord_less_eq_set_a @ I2 @ J2 )
=> ( ( ord_less_eq_set_a @ J2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( J2 = I2 )
| ( J2
= ( partia707051561876973205xt_a_b @ R ) ) ) ) ) )
=> ( maximalideal_a_b @ I2 @ R ) ) ) ) ).
% maximalidealI
thf(fact_375_maximalidealI,axiom,
! [I2: set_set_a,R: partia6043505979758434576t_unit] :
( ( ideal_4463284918206690523t_unit @ I2 @ R )
=> ( ( ( partia5907974310037520643t_unit @ R )
!= I2 )
=> ( ! [J2: set_set_a] :
( ( ideal_4463284918206690523t_unit @ J2 @ R )
=> ( ( ord_le3724670747650509150_set_a @ I2 @ J2 )
=> ( ( ord_le3724670747650509150_set_a @ J2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( J2 = I2 )
| ( J2
= ( partia5907974310037520643t_unit @ R ) ) ) ) ) )
=> ( maxima2253313296322093082t_unit @ I2 @ R ) ) ) ) ).
% maximalidealI
thf(fact_376_maximalideal_OI__maximal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b,J: set_a] :
( ( maximalideal_a_b @ I2 @ R )
=> ( ( ideal_a_b @ J @ R )
=> ( ( ord_less_eq_set_a @ I2 @ J )
=> ( ( ord_less_eq_set_a @ J @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( J = I2 )
| ( J
= ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ).
% maximalideal.I_maximal
thf(fact_377_maximalideal_OI__maximal,axiom,
! [I2: set_set_a,R: partia6043505979758434576t_unit,J: set_set_a] :
( ( maxima2253313296322093082t_unit @ I2 @ R )
=> ( ( ideal_4463284918206690523t_unit @ J @ R )
=> ( ( ord_le3724670747650509150_set_a @ I2 @ J )
=> ( ( ord_le3724670747650509150_set_a @ J @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( J = I2 )
| ( J
= ( partia5907974310037520643t_unit @ R ) ) ) ) ) ) ) ).
% maximalideal.I_maximal
thf(fact_378_ideal_Ois__ideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( ideal_a_b @ I2 @ R )
=> ( ideal_a_b @ I2 @ R ) ) ).
% ideal.is_ideal
thf(fact_379_ideal_Ois__ideal,axiom,
! [I2: set_set_a,R: partia6043505979758434576t_unit] :
( ( ideal_4463284918206690523t_unit @ I2 @ R )
=> ( ideal_4463284918206690523t_unit @ I2 @ R ) ) ).
% ideal.is_ideal
thf(fact_380_field_Ozeromaximalideal,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( maxima2253313296322093082t_unit @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) @ R ) ) ).
% field.zeromaximalideal
thf(fact_381_field_Ozeromaximalideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ).
% field.zeromaximalideal
thf(fact_382_principalideal_Ois__principalideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ( principalideal_a_b @ I2 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_383_ideal_OIcarr,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b,I: a] :
( ( ideal_a_b @ I2 @ R )
=> ( ( member_a @ I @ I2 )
=> ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ideal.Icarr
thf(fact_384_ideal_OIcarr,axiom,
! [I2: set_set_a,R: partia6043505979758434576t_unit,I: set_a] :
( ( ideal_4463284918206690523t_unit @ I2 @ R )
=> ( ( member_set_a @ I @ I2 )
=> ( member_set_a @ I @ ( partia5907974310037520643t_unit @ R ) ) ) ) ).
% ideal.Icarr
thf(fact_385_principalideal_Oaxioms_I1_J,axiom,
! [I2: set_set_a,R: partia6043505979758434576t_unit] :
( ( princi3104115052557732031t_unit @ I2 @ R )
=> ( ideal_4463284918206690523t_unit @ I2 @ R ) ) ).
% principalideal.axioms(1)
thf(fact_386_principalideal_Oaxioms_I1_J,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ( ideal_a_b @ I2 @ R ) ) ).
% principalideal.axioms(1)
thf(fact_387_noetherian__domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_n4045954140777738665in_a_b @ R )
=> ( ring_n3639167112692572309ng_a_b @ R ) ) ).
% noetherian_domain.axioms(1)
thf(fact_388_principal__domain_Oexists__gen,axiom,
! [R: partia6043505979758434576t_unit,I2: set_set_a] :
( ( ring_p2862007038493914190t_unit @ R )
=> ( ( ideal_4463284918206690523t_unit @ I2 @ R )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ ( partia5907974310037520643t_unit @ R ) )
& ( I2
= ( cgenid6682780793756002467t_unit @ R @ X2 ) ) ) ) ) ).
% principal_domain.exists_gen
thf(fact_389_principal__domain_Oexists__gen,axiom,
! [R: partia2175431115845679010xt_a_b,I2: set_a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( ideal_a_b @ I2 @ R )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
& ( I2
= ( cgenid547466209912283029xt_a_b @ R @ X2 ) ) ) ) ) ).
% principal_domain.exists_gen
thf(fact_390_primeideal__iff__prime,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P2 ) @ r )
= ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% primeideal_iff_prime
thf(fact_391_genideal__one,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) )
= ( partia707051561876973205xt_a_b @ r ) ) ).
% genideal_one
thf(fact_392_add_Oone__in__subset,axiom,
! [H2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H2 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H2 )
=> ( member_a @ ( a_inv_a_b @ r @ X2 ) @ H2 ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H2 )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ H2 )
=> ( member_a @ ( add_a_b @ r @ X2 @ Xa2 ) @ H2 ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H2 ) ) ) ) ) ).
% add.one_in_subset
thf(fact_393_zeroprimeideal__domainI,axiom,
( ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
=> ( domain_a_b @ r ) ) ).
% zeroprimeideal_domainI
thf(fact_394_domain__eq__zeroprimeideal,axiom,
( ( domain_a_b @ r )
= ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ) ) ).
% domain_eq_zeroprimeideal
thf(fact_395_carrier__one__not__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) ) ) ).
% carrier_one_not_zero
thf(fact_396_carrier__one__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% carrier_one_zero
thf(fact_397_one__zeroD,axiom,
( ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) )
=> ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% one_zeroD
thf(fact_398_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_399_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_400_r__neg2,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_401_r__neg1,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_402_local_Ominus__add,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_403_a__transpose__inv,axiom,
! [X: a,Y: a,Z: a] :
( ( ( add_a_b @ r @ X @ Y )
= Z )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_404_add_Oinv__solve__right_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
= A )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right'
thf(fact_405_add_Oinv__solve__right,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right
thf(fact_406_add_Oinv__solve__left_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
= A )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left'
thf(fact_407_add_Oinv__solve__left,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left
thf(fact_408_add_Oinv__mult__group,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_409_Diff__idemp,axiom,
! [A2: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_410_Diff__iff,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
= ( ( member_nat_a @ C @ A2 )
& ~ ( member_nat_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_411_Diff__iff,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( ( member_set_a @ C @ A2 )
& ~ ( member_set_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_412_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_413_DiffI,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ A2 )
=> ( ~ ( member_nat_a @ C @ B2 )
=> ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_414_DiffI,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ~ ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_415_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_416_sum__zero__eq__neg,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( X
= ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% sum_zero_eq_neg
thf(fact_417_r__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_418_minus__equality,axiom,
! [Y: a,X: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X )
= Y ) ) ) ) ).
% minus_equality
thf(fact_419_l__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_420_one__zeroI,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% one_zeroI
thf(fact_421_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_422_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_423_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_424_finite__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_425_finite__Diff2,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_426_insert__Diff1,axiom,
! [X: nat > a,B2: set_nat_a,A2: set_nat_a] :
( ( member_nat_a @ X @ B2 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
= ( minus_490503922182417452_nat_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_427_insert__Diff1,axiom,
! [X: set_a,B2: set_set_a,A2: set_set_a] :
( ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_428_insert__Diff1,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_429_Diff__insert0,axiom,
! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ X @ B2 ) )
= ( minus_490503922182417452_nat_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_430_Diff__insert0,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_431_Diff__insert0,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_432_Diff__eq__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_433_Diff__eq__empty__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_434_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_435_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_436_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_437_local_Ominus__minus,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
= X ) ) ).
% local.minus_minus
thf(fact_438_a__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_439_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_440_add_Oinv__eq__1__iff,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X )
= ( zero_a_b @ r ) )
= ( X
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_441_domain_Oone__not__zero,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( one_se211549098623999037t_unit @ R )
!= ( zero_s2174465271003423091t_unit @ R ) ) ) ).
% domain.one_not_zero
thf(fact_442_domain_Oone__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% domain.one_not_zero
thf(fact_443_domain_Ozero__not__one,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( zero_s2174465271003423091t_unit @ R )
!= ( one_se211549098623999037t_unit @ R ) ) ) ).
% domain.zero_not_one
thf(fact_444_domain_Ozero__not__one,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( zero_a_b @ R )
!= ( one_a_ring_ext_a_b @ R ) ) ) ).
% domain.zero_not_one
thf(fact_445_DiffD2,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
=> ~ ( member_nat_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_446_DiffD2,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ~ ( member_set_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_447_DiffD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_448_DiffD1,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
=> ( member_nat_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_449_DiffD1,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_450_DiffD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_451_DiffE,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
=> ~ ( ( member_nat_a @ C @ A2 )
=> ( member_nat_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_452_DiffE,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_453_DiffE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_454_Diff__mono,axiom,
! [A2: set_a,C2: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_455_Diff__mono,axiom,
! [A2: set_set_a,C2: set_set_a,D: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ D @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) @ ( minus_5736297505244876581_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_456_Diff__subset,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_457_Diff__subset,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_458_double__diff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_459_double__diff,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ( minus_5736297505244876581_set_a @ B2 @ ( minus_5736297505244876581_set_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_460_Diff__infinite__finite,axiom,
! [T2: set_a,S: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_461_insert__Diff__if,axiom,
! [X: nat > a,B2: set_nat_a,A2: set_nat_a] :
( ( ( member_nat_a @ X @ B2 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
= ( minus_490503922182417452_nat_a @ A2 @ B2 ) ) )
& ( ~ ( member_nat_a @ X @ B2 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
= ( insert_nat_a @ X @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_462_insert__Diff__if,axiom,
! [X: set_a,B2: set_set_a,A2: set_set_a] :
( ( ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( insert_set_a @ X @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_463_insert__Diff__if,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_464_field_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% field.axioms(1)
thf(fact_465_field_Oaxioms_I1_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( domain4236798911309298543t_unit @ R ) ) ).
% field.axioms(1)
thf(fact_466_ring__hom__one,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_hom_one
thf(fact_467_ring__iso__memE_I4_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_468_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_p2862007038493914190t_unit @ R )
=> ( domain4236798911309298543t_unit @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_469_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_470_noetherian__domain_Oaxioms_I2_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_n3212398840814694743t_unit @ R )
=> ( domain4236798911309298543t_unit @ R ) ) ).
% noetherian_domain.axioms(2)
thf(fact_471_noetherian__domain_Oaxioms_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_n4045954140777738665in_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% noetherian_domain.axioms(2)
thf(fact_472_factorial__domain_Oaxioms_I1_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_f6820247627256571077t_unit @ R )
=> ( domain4236798911309298543t_unit @ R ) ) ).
% factorial_domain.axioms(1)
thf(fact_473_factorial__domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_f5272581269873410839in_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% factorial_domain.axioms(1)
thf(fact_474_ideal_Oone__imp__carrier,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( ideal_a_b @ I2 @ R )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ R ) @ I2 )
=> ( I2
= ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ideal.one_imp_carrier
thf(fact_475_ideal_Oone__imp__carrier,axiom,
! [I2: set_set_a,R: partia6043505979758434576t_unit] :
( ( ideal_4463284918206690523t_unit @ I2 @ R )
=> ( ( member_set_a @ ( one_se211549098623999037t_unit @ R ) @ I2 )
=> ( I2
= ( partia5907974310037520643t_unit @ R ) ) ) ) ).
% ideal.one_imp_carrier
thf(fact_476_image__diff__subset,axiom,
! [F2: a > a,A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F2 @ A2 ) @ ( image_a_a @ F2 @ B2 ) ) @ ( image_a_a @ F2 @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_477_image__diff__subset,axiom,
! [F2: a > set_a,A2: set_a,B2: set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ ( image_a_set_a @ F2 @ A2 ) @ ( image_a_set_a @ F2 @ B2 ) ) @ ( image_a_set_a @ F2 @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_478_subset__Diff__insert,axiom,
! [A2: set_nat_a,B2: set_nat_a,X: nat > a,C2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A2 @ ( minus_490503922182417452_nat_a @ B2 @ ( insert_nat_a @ X @ C2 ) ) )
= ( ( ord_le871467723717165285_nat_a @ A2 @ ( minus_490503922182417452_nat_a @ B2 @ C2 ) )
& ~ ( member_nat_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_479_subset__Diff__insert,axiom,
! [A2: set_a,B2: set_a,X: a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ ( insert_a @ X @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C2 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_480_subset__Diff__insert,axiom,
! [A2: set_set_a,B2: set_set_a,X: set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B2 @ ( insert_set_a @ X @ C2 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B2 @ C2 ) )
& ~ ( member_set_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_481_in__image__insert__iff,axiom,
! [B2: set_set_nat_a,X: nat > a,A2: set_nat_a] :
( ! [C3: set_nat_a] :
( ( member_set_nat_a @ C3 @ B2 )
=> ~ ( member_nat_a @ X @ C3 ) )
=> ( ( member_set_nat_a @ A2 @ ( image_6965494298868581957_nat_a @ ( insert_nat_a @ X ) @ B2 ) )
= ( ( member_nat_a @ X @ A2 )
& ( member_set_nat_a @ ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ X @ bot_bot_set_nat_a ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_482_in__image__insert__iff,axiom,
! [B2: set_set_set_a,X: set_a,A2: set_set_a] :
( ! [C3: set_set_a] :
( ( member_set_set_a @ C3 @ B2 )
=> ~ ( member_set_a @ X @ C3 ) )
=> ( ( member_set_set_a @ A2 @ ( image_1042221919965026181_set_a @ ( insert_set_a @ X ) @ B2 ) )
= ( ( member_set_a @ X @ A2 )
& ( member_set_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_483_in__image__insert__iff,axiom,
! [B2: set_set_a,X: a,A2: set_a] :
( ! [C3: set_a] :
( ( member_set_a @ C3 @ B2 )
=> ~ ( member_a @ X @ C3 ) )
=> ( ( member_set_a @ A2 @ ( image_set_a_set_a @ ( insert_a @ X ) @ B2 ) )
= ( ( member_a @ X @ A2 )
& ( member_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_484_Diff__insert__absorb,axiom,
! [X: nat > a,A2: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ ( insert_nat_a @ X @ bot_bot_set_nat_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_485_Diff__insert__absorb,axiom,
! [X: set_a,A2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_486_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_487_Diff__insert2,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_488_insert__Diff,axiom,
! [A: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ A2 )
=> ( ( insert_nat_a @ A @ ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_489_insert__Diff,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_490_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_491_Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_492_Ring_Oone__not__zero,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( one_se211549098623999037t_unit @ R )
!= ( zero_s2174465271003423091t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_493_Ring_Oone__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_494_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_495_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( semiri8432286899730879049t_unit @ R )
=> ( member_set_a @ ( one_se211549098623999037t_unit @ R ) @ ( partia5907974310037520643t_unit @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_496_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( prime_4522187476880896870t_unit @ R @ ( zero_s2174465271003423091t_unit @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_497_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( prime_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_498_noetherian__domain_Ointro,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_n5014428767265248323t_unit @ R )
=> ( ( domain4236798911309298543t_unit @ R )
=> ( ring_n3212398840814694743t_unit @ R ) ) ) ).
% noetherian_domain.intro
thf(fact_499_noetherian__domain_Ointro,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_n3639167112692572309ng_a_b @ R )
=> ( ( domain_a_b @ R )
=> ( ring_n4045954140777738665in_a_b @ R ) ) ) ).
% noetherian_domain.intro
thf(fact_500_noetherian__domain__def,axiom,
( ring_n3212398840814694743t_unit
= ( ^ [R3: partia6043505979758434576t_unit] :
( ( ring_n5014428767265248323t_unit @ R3 )
& ( domain4236798911309298543t_unit @ R3 ) ) ) ) ).
% noetherian_domain_def
thf(fact_501_noetherian__domain__def,axiom,
( ring_n4045954140777738665in_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b] :
( ( ring_n3639167112692572309ng_a_b @ R3 )
& ( domain_a_b @ R3 ) ) ) ) ).
% noetherian_domain_def
thf(fact_502_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia6043505979758434576t_unit,R2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ R2 )
=> ( R2
!= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_503_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ( R2
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_504_domain_Oring__primeE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P2 )
=> ( P2
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_505_domain_Oring__primeE_I1_J,axiom,
! [R: partia6043505979758434576t_unit,P2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ P2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r6795642478576035723t_unit @ R @ P2 )
=> ( P2
!= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_506_subset__insert__iff,axiom,
! [A2: set_nat_a,X: nat > a,B2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A2 @ ( insert_nat_a @ X @ B2 ) )
= ( ( ( member_nat_a @ X @ A2 )
=> ( ord_le871467723717165285_nat_a @ ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ X @ bot_bot_set_nat_a ) ) @ B2 ) )
& ( ~ ( member_nat_a @ X @ A2 )
=> ( ord_le871467723717165285_nat_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_507_subset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_508_subset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( ( ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B2 ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_509_Diff__single__insert,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_510_Diff__single__insert,axiom,
! [A2: set_set_a,X: set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_511_infinite__remove,axiom,
! [S: set_a,A: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_512_infinite__coinduct,axiom,
! [X5: set_a > $o,A2: set_a] :
( ( X5 @ A2 )
=> ( ! [A5: set_a] :
( ( X5 @ A5 )
=> ? [X4: a] :
( ( member_a @ X4 @ A5 )
& ( ( X5 @ ( minus_minus_set_a @ A5 @ ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_513_finite__empty__induct,axiom,
! [A2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: nat > a,A5: set_nat_a] :
( ( finite_finite_nat_a @ A5 )
=> ( ( member_nat_a @ A6 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_490503922182417452_nat_a @ A5 @ ( insert_nat_a @ A6 @ bot_bot_set_nat_a ) ) ) ) ) )
=> ( P @ bot_bot_set_nat_a ) ) ) ) ).
% finite_empty_induct
thf(fact_514_finite__empty__induct,axiom,
! [A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: set_a,A5: set_set_a] :
( ( finite_finite_set_a @ A5 )
=> ( ( member_set_a @ A6 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ A6 @ bot_bot_set_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_515_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: a,A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( member_a @ A6 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ A6 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_516_domain_Oring__primeE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P2 )
=> ( prime_a_ring_ext_a_b @ R @ P2 ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_517_domain_Oring__primeE_I3_J,axiom,
! [R: partia6043505979758434576t_unit,P2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ P2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r6795642478576035723t_unit @ R @ P2 )
=> ( prime_4522187476880896870t_unit @ R @ P2 ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_518_domain_Oprimeideal__iff__prime,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ R @ P2 ) @ R )
= ( ring_ring_prime_a_b @ R @ P2 ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_519_domain_Oprimeideal__iff__prime,axiom,
! [R: partia6043505979758434576t_unit,P2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ P2 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ( primei7645216761534224334t_unit @ ( cgenid6682780793756002467t_unit @ R @ P2 ) @ R )
= ( ring_r6795642478576035723t_unit @ R @ P2 ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_520_domain_Ozeroprimeideal,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( primei7645216761534224334t_unit @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) @ R ) ) ).
% domain.zeroprimeideal
thf(fact_521_domain_Ozeroprimeideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ).
% domain.zeroprimeideal
thf(fact_522_remove__induct,axiom,
! [P: set_nat_a > $o,B2: set_nat_a] :
( ( P @ bot_bot_set_nat_a )
=> ( ( ~ ( finite_finite_nat_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A5: set_nat_a] :
( ( finite_finite_nat_a @ A5 )
=> ( ( A5 != bot_bot_set_nat_a )
=> ( ( ord_le871467723717165285_nat_a @ A5 @ B2 )
=> ( ! [X4: nat > a] :
( ( member_nat_a @ X4 @ A5 )
=> ( P @ ( minus_490503922182417452_nat_a @ A5 @ ( insert_nat_a @ X4 @ bot_bot_set_nat_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_523_remove__induct,axiom,
! [P: set_a > $o,B2: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A5 @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_524_remove__induct,axiom,
! [P: set_set_a > $o,B2: set_set_a] :
( ( P @ bot_bot_set_set_a )
=> ( ( ~ ( finite_finite_set_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A5: set_set_a] :
( ( finite_finite_set_a @ A5 )
=> ( ( A5 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ B2 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
=> ( P @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_525_finite__remove__induct,axiom,
! [B2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ B2 )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [A5: set_nat_a] :
( ( finite_finite_nat_a @ A5 )
=> ( ( A5 != bot_bot_set_nat_a )
=> ( ( ord_le871467723717165285_nat_a @ A5 @ B2 )
=> ( ! [X4: nat > a] :
( ( member_nat_a @ X4 @ A5 )
=> ( P @ ( minus_490503922182417452_nat_a @ A5 @ ( insert_nat_a @ X4 @ bot_bot_set_nat_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_526_finite__remove__induct,axiom,
! [B2: set_a,P: set_a > $o] :
( ( finite_finite_a @ B2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A5 @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_527_finite__remove__induct,axiom,
! [B2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ B2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A5: set_set_a] :
( ( finite_finite_set_a @ A5 )
=> ( ( A5 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ B2 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A5 )
=> ( P @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_528_semiring_Oone__zeroD,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) )
=> ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_529_semiring_Oone__zeroD,axiom,
! [R: partia6043505979758434576t_unit] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( ( one_se211549098623999037t_unit @ R )
= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( partia5907974310037520643t_unit @ R )
= ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_530_semiring_Oone__zeroI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_531_semiring_Oone__zeroI,axiom,
! [R: partia6043505979758434576t_unit] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( ( partia5907974310037520643t_unit @ R )
= ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) )
=> ( ( one_se211549098623999037t_unit @ R )
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_532_semiring_Ocarrier__one__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_533_semiring_Ocarrier__one__zero,axiom,
! [R: partia6043505979758434576t_unit] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( ( partia5907974310037520643t_unit @ R )
= ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) )
= ( ( one_se211549098623999037t_unit @ R )
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_534_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
!= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_535_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia6043505979758434576t_unit] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( ( partia5907974310037520643t_unit @ R )
!= ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) )
= ( ( one_se211549098623999037t_unit @ R )
!= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_536_field__iff__prime,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( field_6045675692312731021t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) ) )
= ( ring_ring_prime_a_b @ r @ A ) ) ) ).
% field_iff_prime
thf(fact_537_minus__eq,axiom,
! [X: a,Y: a] :
( ( a_minus_a_b @ r @ X @ Y )
= ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).
% minus_eq
thf(fact_538_field__intro2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_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_539_cring__fieldI,axiom,
( ( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( field_a_b @ r ) ) ).
% cring_fieldI
thf(fact_540_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A6: a] :
( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A6
!= ( zero_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A6 @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_541_local_Ofield__Units,axiom,
( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% local.field_Units
thf(fact_542_finite__ranking__induct,axiom,
! [S: set_nat_a,P: set_nat_a > $o,F2: ( nat > a ) > nat] :
( ( finite_finite_nat_a @ S )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [X2: nat > a,S2: set_nat_a] :
( ( finite_finite_nat_a @ S2 )
=> ( ! [Y5: nat > a] :
( ( member_nat_a @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F2 @ Y5 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_543_finite__ranking__induct,axiom,
! [S: set_set_a,P: set_set_a > $o,F2: set_a > nat] :
( ( finite_finite_set_a @ S )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X2: set_a,S2: set_set_a] :
( ( finite_finite_set_a @ S2 )
=> ( ! [Y5: set_a] :
( ( member_set_a @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F2 @ Y5 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_set_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_544_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F2: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F2 @ Y5 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_545_square__eq__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( X
= ( one_a_ring_ext_a_b @ r ) )
| ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% square_eq_one
thf(fact_546_m__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_547_m__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% m_comm
thf(fact_548_m__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_549_Units__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_550_local_Ointegral,axiom,
! [A: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_551_integral__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_552_m__lcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( mult_a_ring_ext_a_b @ r @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% m_lcancel
thf(fact_553_m__rcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B @ A )
= ( mult_a_ring_ext_a_b @ r @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% m_rcancel
thf(fact_554_l__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_555_r__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_556_inv__unique,axiom,
! [Y: a,X: a,Y4: 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 @ Y4 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y4 ) ) ) ) ) ) ).
% inv_unique
thf(fact_557_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_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_558_l__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% l_minus
thf(fact_559_r__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% r_minus
thf(fact_560_prod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_561_prod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_562_unit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_563_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_564_ideal__eq__carrier__iff,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( partia707051561876973205xt_a_b @ r )
= ( cgenid547466209912283029xt_a_b @ r @ A ) )
= ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ideal_eq_carrier_iff
thf(fact_565_line__extension__mem__iff,axiom,
! [U: a,K: set_a,A: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ K )
& ? [Y6: a] :
( ( member_a @ Y6 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ A ) @ Y6 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_566_ring__irreducibleE_I4_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_567_quot__domain__imp__primeideal,axiom,
! [P: set_a] :
( ( ideal_a_b @ P @ r )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ P ) )
=> ( primeideal_a_b @ P @ r ) ) ) ).
% quot_domain_imp_primeideal
thf(fact_568_quot__domain__iff__primeideal,axiom,
! [P: set_a] :
( ( ideal_a_b @ P @ r )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ P ) )
= ( primeideal_a_b @ P @ r ) ) ) ).
% quot_domain_iff_primeideal
thf(fact_569_Units__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_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_570_Units__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_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_571_ring__irreducibleE_I5_J,axiom,
! [R2: a,A: a,B: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_572_ring__irreducibleI,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ! [A6: a,B6: a] :
( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ r @ A6 @ B6 ) )
=> ( ( member_a @ A6 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B6 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ r @ R2 ) ) ) ) ).
% ring_irreducibleI
thf(fact_573_domain__iff__prime,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) ) )
= ( ring_ring_prime_a_b @ r @ A ) ) ) ).
% domain_iff_prime
thf(fact_574_m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_575_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_576_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_577_minus__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_578_l__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_579_r__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_580_l__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
= X ) ) ).
% l_one
thf(fact_581_r__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
= X ) ) ).
% r_one
thf(fact_582_Units__l__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% Units_l_cancel
thf(fact_583_finite__ring__finite__units,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( finite_finite_a @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% finite_ring_finite_units
thf(fact_584_Units__minus__one__closed,axiom,
member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_minus_one_closed
thf(fact_585_r__right__minus__eq,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( A = B ) ) ) ) ).
% r_right_minus_eq
thf(fact_586_FactRing__zeroideal_I2_J,axiom,
is_rin9099215527551818550t_unit @ r @ ( factRing_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% FactRing_zeroideal(2)
thf(fact_587_FactRing__zeroideal_I1_J,axiom,
is_rin6001486760346555702it_a_b @ ( factRing_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) @ r ).
% FactRing_zeroideal(1)
thf(fact_588_primeideal_Oquotient__is__domain,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( primeideal_a_b @ I2 @ R )
=> ( domain4236798911309298543t_unit @ ( factRing_a_b @ R @ I2 ) ) ) ).
% primeideal.quotient_is_domain
thf(fact_589_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia6043505979758434576t_unit,R2: set_a,A: set_a,B: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ R2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( R2
= ( mult_s7930653359683758801t_unit @ R @ A @ B ) )
=> ( ( member_set_a @ A @ ( units_2471184348132832486t_unit @ R ) )
| ( member_set_a @ B @ ( units_2471184348132832486t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_590_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ R @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ R ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_591_ideal_Ohelper__I__closed,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b,A: a,X: a,Y: a] :
( ( ideal_a_b @ I2 @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ R @ A @ X ) @ I2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ A @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) ) @ I2 ) ) ) ) ) ) ).
% ideal.helper_I_closed
thf(fact_592_ideal_Ohelper__I__closed,axiom,
! [I2: set_set_a,R: partia6043505979758434576t_unit,A: set_a,X: set_a,Y: set_a] :
( ( ideal_4463284918206690523t_unit @ I2 @ R )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ ( mult_s7930653359683758801t_unit @ R @ A @ X ) @ I2 )
=> ( member_set_a @ ( mult_s7930653359683758801t_unit @ R @ A @ ( mult_s7930653359683758801t_unit @ R @ X @ Y ) ) @ I2 ) ) ) ) ) ) ).
% ideal.helper_I_closed
thf(fact_593_ideal_OI__r__closed,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b,A: a,X: a] :
( ( ideal_a_b @ I2 @ R )
=> ( ( member_a @ A @ I2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ A @ X ) @ I2 ) ) ) ) ).
% ideal.I_r_closed
thf(fact_594_ideal_OI__r__closed,axiom,
! [I2: set_set_a,R: partia6043505979758434576t_unit,A: set_a,X: set_a] :
( ( ideal_4463284918206690523t_unit @ I2 @ R )
=> ( ( member_set_a @ A @ I2 )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( member_set_a @ ( mult_s7930653359683758801t_unit @ R @ A @ X ) @ I2 ) ) ) ) ).
% ideal.I_r_closed
thf(fact_595_ideal_OI__l__closed,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b,A: a,X: a] :
( ( ideal_a_b @ I2 @ R )
=> ( ( member_a @ A @ I2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ A ) @ I2 ) ) ) ) ).
% ideal.I_l_closed
thf(fact_596_ideal_OI__l__closed,axiom,
! [I2: set_set_a,R: partia6043505979758434576t_unit,A: set_a,X: set_a] :
( ( ideal_4463284918206690523t_unit @ I2 @ R )
=> ( ( member_set_a @ A @ I2 )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( member_set_a @ ( mult_s7930653359683758801t_unit @ R @ X @ A ) @ I2 ) ) ) ) ).
% ideal.I_l_closed
thf(fact_597_ring__hom__mult,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_598_ring__hom__mult,axiom,
! [H: set_a > a,R: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,X: set_a,Y: set_a] :
( ( member_set_a_a @ H @ ( ring_h4811522740288071338it_a_b @ R @ S ) )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( H @ ( mult_s7930653359683758801t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_599_primeideal_OI__prime,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( primeideal_a_b @ I2 @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ R @ A @ B ) @ I2 )
=> ( ( member_a @ A @ I2 )
| ( member_a @ B @ I2 ) ) ) ) ) ) ).
% primeideal.I_prime
thf(fact_600_primeideal_OI__prime,axiom,
! [I2: set_set_a,R: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( primei7645216761534224334t_unit @ I2 @ R )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ ( mult_s7930653359683758801t_unit @ R @ A @ B ) @ I2 )
=> ( ( member_set_a @ A @ I2 )
| ( member_set_a @ B @ I2 ) ) ) ) ) ) ).
% primeideal.I_prime
thf(fact_601_ring__iso__memE_I2_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_602_ring__iso__memE_I2_J,axiom,
! [H: set_a > a,R: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,X: set_a,Y: set_a] :
( ( member_set_a_a @ H @ ( ring_i4751279688611836608it_a_b @ R @ S ) )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( H @ ( mult_s7930653359683758801t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_603_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_604_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia6043505979758434576t_unit,X: set_a,Y: set_a] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ R ) )
=> ( member_set_a @ ( mult_s7930653359683758801t_unit @ R @ X @ Y ) @ ( partia5907974310037520643t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_605_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_606_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia6043505979758434576t_unit,X: set_a,Y: set_a,Z: set_a] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Z @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( mult_s7930653359683758801t_unit @ R @ ( mult_s7930653359683758801t_unit @ R @ X @ Y ) @ Z )
= ( mult_s7930653359683758801t_unit @ R @ X @ ( mult_s7930653359683758801t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_607_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia6043505979758434576t_unit,R2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ R2 )
=> ~ ( member_set_a @ R2 @ ( units_2471184348132832486t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_608_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_609_domain_Ointegral__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
= ( ( A
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_610_domain_Ointegral__iff,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ A @ B )
= ( zero_s2174465271003423091t_unit @ R ) )
= ( ( A
= ( zero_s2174465271003423091t_unit @ R ) )
| ( B
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_611_domain_Om__rcancel,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R )
=> ( ( A
!= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ B @ A )
= ( mult_a_ring_ext_a_b @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_612_domain_Om__rcancel,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a,C: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( A
!= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ B @ A )
= ( mult_s7930653359683758801t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_613_domain_Om__lcancel,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R )
=> ( ( A
!= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( mult_a_ring_ext_a_b @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_614_domain_Om__lcancel,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a,C: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( A
!= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ A @ B )
= ( mult_s7930653359683758801t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_615_domain_Ointegral,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_616_domain_Ointegral,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ A @ B )
= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( A
= ( zero_s2174465271003423091t_unit @ R ) )
| ( B
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_617_Ring_Ointegral,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( field_a_b @ R )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_618_Ring_Ointegral,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ A @ B )
= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( A
= ( zero_s2174465271003423091t_unit @ R ) )
| ( B
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_619_a__minus__def,axiom,
( a_minus_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,X3: a,Y6: a] : ( add_a_b @ R3 @ X3 @ ( a_inv_a_b @ R3 @ Y6 ) ) ) ) ).
% a_minus_def
thf(fact_620_semiring_Or__null,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.r_null
thf(fact_621_semiring_Or__null,axiom,
! [R: partia6043505979758434576t_unit,X: set_a] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( mult_s7930653359683758801t_unit @ R @ X @ ( zero_s2174465271003423091t_unit @ R ) )
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ).
% semiring.r_null
thf(fact_622_semiring_Ol__null,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.l_null
thf(fact_623_semiring_Ol__null,axiom,
! [R: partia6043505979758434576t_unit,X: set_a] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( mult_s7930653359683758801t_unit @ R @ ( zero_s2174465271003423091t_unit @ R ) @ X )
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ).
% semiring.l_null
thf(fact_624_semiring_Or__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_625_semiring_Or__distr,axiom,
! [R: partia6043505979758434576t_unit,X: set_a,Y: set_a,Z: set_a] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Z @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( mult_s7930653359683758801t_unit @ R @ Z @ ( add_se3735415688806051380t_unit @ R @ X @ Y ) )
= ( add_se3735415688806051380t_unit @ R @ ( mult_s7930653359683758801t_unit @ R @ Z @ X ) @ ( mult_s7930653359683758801t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_626_semiring_Ol__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_627_semiring_Ol__distr,axiom,
! [R: partia6043505979758434576t_unit,X: set_a,Y: set_a,Z: set_a] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Z @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( mult_s7930653359683758801t_unit @ R @ ( add_se3735415688806051380t_unit @ R @ X @ Y ) @ Z )
= ( add_se3735415688806051380t_unit @ R @ ( mult_s7930653359683758801t_unit @ R @ X @ Z ) @ ( mult_s7930653359683758801t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_628_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_629_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia6043505979758434576t_unit,X: set_a] :
( ( semiri8432286899730879049t_unit @ R )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( mult_s7930653359683758801t_unit @ R @ ( one_se211549098623999037t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_630_domain_Oring__irreducibleI,axiom,
! [R: partia6043505979758434576t_unit,R2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R2 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ~ ( member_set_a @ R2 @ ( units_2471184348132832486t_unit @ R ) )
=> ( ! [A6: set_a,B6: set_a] :
( ( member_set_a @ A6 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B6 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( R2
= ( mult_s7930653359683758801t_unit @ R @ A6 @ B6 ) )
=> ( ( member_set_a @ A6 @ ( units_2471184348132832486t_unit @ R ) )
| ( member_set_a @ B6 @ ( units_2471184348132832486t_unit @ R ) ) ) ) ) )
=> ( ring_r7790391342995787508t_unit @ R @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleI
thf(fact_631_domain_Oring__irreducibleI,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ R ) )
=> ( ! [A6: a,B6: a] :
( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ R @ A6 @ B6 ) )
=> ( ( member_a @ A6 @ ( units_a_ring_ext_a_b @ R ) )
| ( member_a @ B6 @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ R @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleI
thf(fact_632_domain_Osquare__eq__one,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ X @ X )
= ( one_a_ring_ext_a_b @ R ) )
=> ( ( X
= ( one_a_ring_ext_a_b @ R ) )
| ( X
= ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_633_domain_Osquare__eq__one,axiom,
! [R: partia6043505979758434576t_unit,X: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ X @ X )
= ( one_se211549098623999037t_unit @ R ) )
=> ( ( X
= ( one_se211549098623999037t_unit @ R ) )
| ( X
= ( a_inv_3226179052963059835t_unit @ R @ ( one_se211549098623999037t_unit @ R ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_634_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H: a > a,S: partia2175431115845679010xt_a_b] :
( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y3 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X2 ) @ ( H @ Y3 ) ) ) ) )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X2 @ Y3 ) )
= ( add_a_b @ S @ ( H @ X2 ) @ ( H @ Y3 ) ) ) ) )
=> ( ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_635_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H: a > set_a,S: partia6043505979758434576t_unit] :
( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_set_a @ ( H @ X2 ) @ ( partia5907974310037520643t_unit @ S ) ) )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y3 ) )
= ( mult_s7930653359683758801t_unit @ S @ ( H @ X2 ) @ ( H @ Y3 ) ) ) ) )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X2 @ Y3 ) )
= ( add_se3735415688806051380t_unit @ S @ ( H @ X2 ) @ ( H @ Y3 ) ) ) ) )
=> ( ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_se211549098623999037t_unit @ S ) )
=> ( member_a_set_a @ H @ ( ring_h7909251507493334186t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_636_ring__hom__memI,axiom,
! [R: partia6043505979758434576t_unit,H: set_a > a,S: partia2175431115845679010xt_a_b] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( member_a @ ( H @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X2: set_a,Y3: set_a] :
( ( member_set_a @ X2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y3 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( H @ ( mult_s7930653359683758801t_unit @ R @ X2 @ Y3 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X2 ) @ ( H @ Y3 ) ) ) ) )
=> ( ! [X2: set_a,Y3: set_a] :
( ( member_set_a @ X2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y3 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( H @ ( add_se3735415688806051380t_unit @ R @ X2 @ Y3 ) )
= ( add_a_b @ S @ ( H @ X2 ) @ ( H @ Y3 ) ) ) ) )
=> ( ( ( H @ ( one_se211549098623999037t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_set_a_a @ H @ ( ring_h4811522740288071338it_a_b @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_637_ring__hom__memI,axiom,
! [R: partia6043505979758434576t_unit,H: set_a > set_a,S: partia6043505979758434576t_unit] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( member_set_a @ ( H @ X2 ) @ ( partia5907974310037520643t_unit @ S ) ) )
=> ( ! [X2: set_a,Y3: set_a] :
( ( member_set_a @ X2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y3 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( H @ ( mult_s7930653359683758801t_unit @ R @ X2 @ Y3 ) )
= ( mult_s7930653359683758801t_unit @ S @ ( H @ X2 ) @ ( H @ Y3 ) ) ) ) )
=> ( ! [X2: set_a,Y3: set_a] :
( ( member_set_a @ X2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ Y3 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( H @ ( add_se3735415688806051380t_unit @ R @ X2 @ Y3 ) )
= ( add_se3735415688806051380t_unit @ S @ ( H @ X2 ) @ ( H @ Y3 ) ) ) ) )
=> ( ( ( H @ ( one_se211549098623999037t_unit @ R ) )
= ( one_se211549098623999037t_unit @ S ) )
=> ( member_set_a_set_a @ H @ ( ring_h4754363464466234712t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_638_principal__domain_Odomain__iff__prime,axiom,
! [R: partia6043505979758434576t_unit,A: set_a] :
( ( ring_p2862007038493914190t_unit @ R )
=> ( ( member_set_a @ A @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ( domain5937373053834922511t_unit @ ( factRi9077435707053964944t_unit @ R @ ( cgenid6682780793756002467t_unit @ R @ A ) ) )
= ( ring_r6795642478576035723t_unit @ R @ A ) ) ) ) ).
% principal_domain.domain_iff_prime
thf(fact_639_principal__domain_Odomain__iff__prime,axiom,
! [R: partia2175431115845679010xt_a_b,A: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ R @ ( cgenid547466209912283029xt_a_b @ R @ A ) ) )
= ( ring_ring_prime_a_b @ R @ A ) ) ) ) ).
% principal_domain.domain_iff_prime
thf(fact_640_Ring_Ofield__Units,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( units_a_ring_ext_a_b @ R )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ).
% Ring.field_Units
thf(fact_641_Ring_Ofield__Units,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( units_2471184348132832486t_unit @ R )
= ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) ) ) ).
% Ring.field_Units
thf(fact_642_principal__domain_Ofield__iff__prime,axiom,
! [R: partia6043505979758434576t_unit,A: set_a] :
( ( ring_p2862007038493914190t_unit @ R )
=> ( ( member_set_a @ A @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ( field_6196704264051520557t_unit @ ( factRi9077435707053964944t_unit @ R @ ( cgenid6682780793756002467t_unit @ R @ A ) ) )
= ( ring_r6795642478576035723t_unit @ R @ A ) ) ) ) ).
% principal_domain.field_iff_prime
thf(fact_643_principal__domain_Ofield__iff__prime,axiom,
! [R: partia2175431115845679010xt_a_b,A: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( field_6045675692312731021t_unit @ ( factRing_a_b @ R @ ( cgenid547466209912283029xt_a_b @ R @ A ) ) )
= ( ring_ring_prime_a_b @ R @ A ) ) ) ) ).
% principal_domain.field_iff_prime
thf(fact_644_subdomainI,axiom,
! [H2: set_a] :
( ( subcring_a_b @ H2 @ r )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( ( ( mult_a_ring_ext_a_b @ r @ H1 @ H22 )
= ( zero_a_b @ r ) )
=> ( ( H1
= ( zero_a_b @ r ) )
| ( H22
= ( zero_a_b @ r ) ) ) ) ) )
=> ( subdomain_a_b @ H2 @ r ) ) ) ) ).
% subdomainI
thf(fact_645_exists__irreducible__divisor,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ~ ! [B6: a] :
( ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ B6 )
=> ~ ( factor8216151070175719842xt_a_b @ r @ B6 @ A ) ) ) ) ) ).
% exists_irreducible_divisor
thf(fact_646_monoid__cancelI,axiom,
( ! [A6: a,B6: a,C5: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C5 @ A6 )
= ( mult_a_ring_ext_a_b @ r @ C5 @ B6 ) )
=> ( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A6 = B6 ) ) ) ) )
=> ( ! [A6: a,B6: a,C5: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A6 @ C5 )
= ( mult_a_ring_ext_a_b @ r @ B6 @ C5 ) )
=> ( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A6 = B6 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_647_subfield__m__inv__simprule,axiom,
! [K: set_a,K2: a,A: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ A ) @ K )
=> ( member_a @ A @ K ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_648_subringI,axiom,
! [H2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ H2 )
=> ( ! [H3: a] :
( ( member_a @ H3 @ H2 )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ H2 ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ H2 ) ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ H2 ) ) )
=> ( subring_a_b @ H2 @ r ) ) ) ) ) ) ).
% subringI
thf(fact_649_mult__divides,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ B ) ) ) ) ) ).
% mult_divides
thf(fact_650_carrier__is__subfield,axiom,
subfield_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subfield
thf(fact_651_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_652_subring__props_I2_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K ) ) ).
% subring_props(2)
thf(fact_653_subring__props_I7_J,axiom,
! [K: set_a,H12: a,H23: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H12 @ K )
=> ( ( member_a @ H23 @ K )
=> ( member_a @ ( add_a_b @ r @ H12 @ H23 ) @ K ) ) ) ) ).
% subring_props(7)
thf(fact_654_divides__trans,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( factor8216151070175719842xt_a_b @ r @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ C ) ) ) ) ).
% divides_trans
thf(fact_655_subring__props_I6_J,axiom,
! [K: set_a,H12: a,H23: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H12 @ K )
=> ( ( member_a @ H23 @ K )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H23 ) @ K ) ) ) ) ).
% subring_props(6)
thf(fact_656_subring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( K != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_657_zero__divides,axiom,
! [A: a] :
( ( factor8216151070175719842xt_a_b @ r @ ( zero_a_b @ r ) @ A )
= ( A
= ( zero_a_b @ r ) ) ) ).
% zero_divides
thf(fact_658_subring__props_I3_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K ) ) ).
% subring_props(3)
thf(fact_659_subring__props_I5_J,axiom,
! [K: set_a,H: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H @ K )
=> ( member_a @ ( a_inv_a_b @ r @ H ) @ K ) ) ) ).
% subring_props(5)
thf(fact_660_subcringI_H,axiom,
! [H2: set_a] :
( ( subring_a_b @ H2 @ r )
=> ( subcring_a_b @ H2 @ r ) ) ).
% subcringI'
thf(fact_661_subdomainI_H,axiom,
! [H2: set_a] :
( ( subring_a_b @ H2 @ r )
=> ( subdomain_a_b @ H2 @ r ) ) ).
% subdomainI'
thf(fact_662_subring__props_I1_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_663_divides__zero,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( zero_a_b @ r ) ) ) ).
% divides_zero
thf(fact_664_divides__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ).
% divides_prod_r
thf(fact_665_divides__prod__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ) ).
% divides_prod_l
thf(fact_666_local_Odivides__mult,axiom,
! [A: a,C: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% local.divides_mult
thf(fact_667_one__divides,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ A ) ) ).
% one_divides
thf(fact_668_unit__divides,axiom,
! [U: a,A: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ U @ A ) ) ) ).
% unit_divides
thf(fact_669_divides__unit,axiom,
! [A: a,U: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ U )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% divides_unit
thf(fact_670_subcringI,axiom,
! [H2: set_a] :
( ( subring_a_b @ H2 @ r )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( ( mult_a_ring_ext_a_b @ r @ H1 @ H22 )
= ( mult_a_ring_ext_a_b @ r @ H22 @ H1 ) ) ) )
=> ( subcring_a_b @ H2 @ r ) ) ) ).
% subcringI
thf(fact_671_divides__one,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ ( one_a_ring_ext_a_b @ r ) )
= ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% divides_one
thf(fact_672_Unit__eq__dividesone,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
= ( factor8216151070175719842xt_a_b @ r @ U @ ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Unit_eq_dividesone
thf(fact_673_to__contain__is__to__divide,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( cgenid547466209912283029xt_a_b @ r @ B ) @ ( cgenid547466209912283029xt_a_b @ r @ A ) )
= ( factor8216151070175719842xt_a_b @ r @ A @ B ) ) ) ) ).
% to_contain_is_to_divide
thf(fact_674_line__extension__smult__closed,axiom,
! [K: set_a,E: set_a,A: a,K2: a,U: a] :
( ( subfield_a_b @ K @ r )
=> ( ! [K3: a,V2: a] :
( ( member_a @ K3 @ K )
=> ( ( member_a @ V2 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ V2 ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ U ) @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_675_divides__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ A ) ) ).
% divides_refl
thf(fact_676_divides__mult__rI,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% divides_mult_rI
thf(fact_677_divides__mult__lI,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% divides_mult_lI
thf(fact_678_ring__iso__trans,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,Q: partia6043505979758434576t_unit] :
( ( is_ring_iso_a_b_a_b @ R @ S )
=> ( ( is_rin9099215527551818550t_unit @ S @ Q )
=> ( is_rin9099215527551818550t_unit @ R @ Q ) ) ) ).
% ring_iso_trans
thf(fact_679_ring__iso__trans,axiom,
! [R: partia6043505979758434576t_unit,S: partia6043505979758434576t_unit,Q: partia2175431115845679010xt_a_b] :
( ( is_rin3089442944380665828t_unit @ R @ S )
=> ( ( is_rin6001486760346555702it_a_b @ S @ Q )
=> ( is_rin6001486760346555702it_a_b @ R @ Q ) ) ) ).
% ring_iso_trans
thf(fact_680_ring__iso__trans,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia6043505979758434576t_unit,Q: partia6043505979758434576t_unit] :
( ( is_rin9099215527551818550t_unit @ R @ S )
=> ( ( is_rin3089442944380665828t_unit @ S @ Q )
=> ( is_rin9099215527551818550t_unit @ R @ Q ) ) ) ).
% ring_iso_trans
thf(fact_681_ring__iso__trans,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia6043505979758434576t_unit,Q: partia2175431115845679010xt_a_b] :
( ( is_rin9099215527551818550t_unit @ R @ S )
=> ( ( is_rin6001486760346555702it_a_b @ S @ Q )
=> ( is_ring_iso_a_b_a_b @ R @ Q ) ) ) ).
% ring_iso_trans
thf(fact_682_ring__iso__trans,axiom,
! [R: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,Q: partia2175431115845679010xt_a_b] :
( ( is_rin6001486760346555702it_a_b @ R @ S )
=> ( ( is_ring_iso_a_b_a_b @ S @ Q )
=> ( is_rin6001486760346555702it_a_b @ R @ Q ) ) ) ).
% ring_iso_trans
thf(fact_683_ring__iso__trans,axiom,
! [R: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,Q: partia6043505979758434576t_unit] :
( ( is_rin6001486760346555702it_a_b @ R @ S )
=> ( ( is_rin9099215527551818550t_unit @ S @ Q )
=> ( is_rin3089442944380665828t_unit @ R @ Q ) ) ) ).
% ring_iso_trans
thf(fact_684_is__ring__iso__def,axiom,
( is_rin9099215527551818550t_unit
= ( ^ [R3: partia2175431115845679010xt_a_b,S3: partia6043505979758434576t_unit] :
( ( ring_i7849008455817099456t_unit @ R3 @ S3 )
!= bot_bot_set_a_set_a ) ) ) ).
% is_ring_iso_def
thf(fact_685_is__ring__iso__def,axiom,
( is_rin6001486760346555702it_a_b
= ( ^ [R3: partia6043505979758434576t_unit,S3: partia2175431115845679010xt_a_b] :
( ( ring_i4751279688611836608it_a_b @ R3 @ S3 )
!= bot_bot_set_set_a_a ) ) ) ).
% is_ring_iso_def
thf(fact_686_domain_Omult__divides,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ C @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( factor8216151070175719842xt_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ C @ A ) @ ( mult_a_ring_ext_a_b @ R @ C @ B ) )
=> ( factor8216151070175719842xt_a_b @ R @ A @ B ) ) ) ) ) ) ).
% domain.mult_divides
thf(fact_687_domain_Omult__divides,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a,C: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ( factor5460682277579321776t_unit @ R @ ( mult_s7930653359683758801t_unit @ R @ C @ A ) @ ( mult_s7930653359683758801t_unit @ R @ C @ B ) )
=> ( factor5460682277579321776t_unit @ R @ A @ B ) ) ) ) ) ) ).
% domain.mult_divides
thf(fact_688_noetherian__domain_Oexists__irreducible__divisor,axiom,
! [R: partia6043505979758434576t_unit,A: set_a] :
( ( ring_n3212398840814694743t_unit @ R )
=> ( ( member_set_a @ A @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ~ ( member_set_a @ A @ ( units_2471184348132832486t_unit @ R ) )
=> ~ ! [B6: set_a] :
( ( member_set_a @ B6 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ B6 )
=> ~ ( factor5460682277579321776t_unit @ R @ B6 @ A ) ) ) ) ) ) ).
% noetherian_domain.exists_irreducible_divisor
thf(fact_689_noetherian__domain_Oexists__irreducible__divisor,axiom,
! [R: partia2175431115845679010xt_a_b,A: a] :
( ( ring_n4045954140777738665in_a_b @ R )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ R ) )
=> ~ ! [B6: a] :
( ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ B6 )
=> ~ ( factor8216151070175719842xt_a_b @ R @ B6 @ A ) ) ) ) ) ) ).
% noetherian_domain.exists_irreducible_divisor
thf(fact_690_isgcd__divides__r,axiom,
! [B: a,A: a] :
( ( factor8216151070175719842xt_a_b @ r @ B @ A )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ B @ A @ B ) ) ) ) ).
% isgcd_divides_r
thf(fact_691_isgcd__divides__l,axiom,
! [A: a,B: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ A @ A @ B ) ) ) ) ).
% isgcd_divides_l
thf(fact_692_dividesI_H,axiom,
! [B: a,G2: partia2175431115845679010xt_a_b,A: a,C: a] :
( ( B
= ( mult_a_ring_ext_a_b @ G2 @ A @ C ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( factor8216151070175719842xt_a_b @ G2 @ A @ B ) ) ) ).
% dividesI'
thf(fact_693_dividesI_H,axiom,
! [B: a,G2: partia8223610829204095565t_unit,A: a,C: a] :
( ( B
= ( mult_a_Product_unit @ G2 @ A @ C ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( factor3040189038382604065t_unit @ G2 @ A @ B ) ) ) ).
% dividesI'
thf(fact_694_dividesI_H,axiom,
! [B: set_a,G2: partia6043505979758434576t_unit,A: set_a,C: set_a] :
( ( B
= ( mult_s7930653359683758801t_unit @ G2 @ A @ C ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( factor5460682277579321776t_unit @ G2 @ A @ B ) ) ) ).
% dividesI'
thf(fact_695_subfield__m__inv_I3_J,axiom,
! [K: set_a,K2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ K2 ) @ K2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% subfield_m_inv(3)
thf(fact_696_subfield__m__inv_I2_J,axiom,
! [K: set_a,K2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ K2 @ ( m_inv_a_ring_ext_a_b @ r @ K2 ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% subfield_m_inv(2)
thf(fact_697_inv__eq__imp__eq,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( m_inv_a_ring_ext_a_b @ r @ Y ) )
=> ( X = Y ) ) ) ) ).
% inv_eq_imp_eq
thf(fact_698_inv__eq__one__eq,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( one_a_ring_ext_a_b @ r ) )
= ( X
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% inv_eq_one_eq
thf(fact_699_comm__inv__char,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ X )
= Y ) ) ) ) ).
% comm_inv_char
thf(fact_700_inv__char,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ X )
= Y ) ) ) ) ) ).
% inv_char
thf(fact_701_inv__unique_H,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( Y
= ( m_inv_a_ring_ext_a_b @ r @ X ) ) ) ) ) ) ).
% inv_unique'
thf(fact_702_inv__eq__self,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( X
= ( m_inv_a_ring_ext_a_b @ r @ X ) )
=> ( ( X
= ( one_a_ring_ext_a_b @ r ) )
| ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% inv_eq_self
thf(fact_703_inv__eq__neg__one__eq,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) )
= ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% inv_eq_neg_one_eq
thf(fact_704_subfield__m__inv_I1_J,axiom,
! [K: set_a,K2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ K2 ) @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ).
% subfield_m_inv(1)
thf(fact_705_subfieldI_H,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ! [K3: a] :
( ( member_a @ K3 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ K3 ) @ K ) )
=> ( subfield_a_b @ K @ r ) ) ) ).
% subfieldI'
thf(fact_706_inv__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) )
= ( one_a_ring_ext_a_b @ r ) ) ).
% inv_one
thf(fact_707_Units__inv__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ X ) )
= X ) ) ).
% Units_inv_inv
thf(fact_708_Units__inv__Units,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_inv_Units
thf(fact_709_Units__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_inv_closed
thf(fact_710_inv__neg__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) )
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ).
% inv_neg_one
thf(fact_711_Units__r__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( m_inv_a_ring_ext_a_b @ r @ X ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% Units_r_inv
thf(fact_712_Units__l__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% Units_l_inv
thf(fact_713_isgcd__def,axiom,
( isgcd_a_ring_ext_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,X3: a,A7: a,B5: a] :
( ( factor8216151070175719842xt_a_b @ G3 @ X3 @ A7 )
& ( factor8216151070175719842xt_a_b @ G3 @ X3 @ B5 )
& ! [Y6: a] :
( ( member_a @ Y6 @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( ( ( factor8216151070175719842xt_a_b @ G3 @ Y6 @ A7 )
& ( factor8216151070175719842xt_a_b @ G3 @ Y6 @ B5 ) )
=> ( factor8216151070175719842xt_a_b @ G3 @ Y6 @ X3 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_714_isgcd__def,axiom,
( isgcd_a_Product_unit
= ( ^ [G3: partia8223610829204095565t_unit,X3: a,A7: a,B5: a] :
( ( factor3040189038382604065t_unit @ G3 @ X3 @ A7 )
& ( factor3040189038382604065t_unit @ G3 @ X3 @ B5 )
& ! [Y6: a] :
( ( member_a @ Y6 @ ( partia6735698275553448452t_unit @ G3 ) )
=> ( ( ( factor3040189038382604065t_unit @ G3 @ Y6 @ A7 )
& ( factor3040189038382604065t_unit @ G3 @ Y6 @ B5 ) )
=> ( factor3040189038382604065t_unit @ G3 @ Y6 @ X3 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_715_isgcd__def,axiom,
( isgcd_8277756548069700071t_unit
= ( ^ [G3: partia6043505979758434576t_unit,X3: set_a,A7: set_a,B5: set_a] :
( ( factor5460682277579321776t_unit @ G3 @ X3 @ A7 )
& ( factor5460682277579321776t_unit @ G3 @ X3 @ B5 )
& ! [Y6: set_a] :
( ( member_set_a @ Y6 @ ( partia5907974310037520643t_unit @ G3 ) )
=> ( ( ( factor5460682277579321776t_unit @ G3 @ Y6 @ A7 )
& ( factor5460682277579321776t_unit @ G3 @ Y6 @ B5 ) )
=> ( factor5460682277579321776t_unit @ G3 @ Y6 @ X3 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_716_domain_Oinv__eq__self,axiom,
! [R: partia6043505979758434576t_unit,X: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ X @ ( units_2471184348132832486t_unit @ R ) )
=> ( ( X
= ( m_inv_7491079437187478987t_unit @ R @ X ) )
=> ( ( X
= ( one_se211549098623999037t_unit @ R ) )
| ( X
= ( a_inv_3226179052963059835t_unit @ R @ ( one_se211549098623999037t_unit @ R ) ) ) ) ) ) ) ).
% domain.inv_eq_self
thf(fact_717_domain_Oinv__eq__self,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ R ) )
=> ( ( X
= ( m_inv_a_ring_ext_a_b @ R @ X ) )
=> ( ( X
= ( one_a_ring_ext_a_b @ R ) )
| ( X
= ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) ) ) ) ) ) ).
% domain.inv_eq_self
thf(fact_718_dividesD,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( factor8216151070175719842xt_a_b @ G2 @ A @ B )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G2 ) )
& ( B
= ( mult_a_ring_ext_a_b @ G2 @ A @ X2 ) ) ) ) ).
% dividesD
thf(fact_719_dividesD,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a] :
( ( factor3040189038382604065t_unit @ G2 @ A @ B )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ G2 ) )
& ( B
= ( mult_a_Product_unit @ G2 @ A @ X2 ) ) ) ) ).
% dividesD
thf(fact_720_dividesD,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( factor5460682277579321776t_unit @ G2 @ A @ B )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ ( partia5907974310037520643t_unit @ G2 ) )
& ( B
= ( mult_s7930653359683758801t_unit @ G2 @ A @ X2 ) ) ) ) ).
% dividesD
thf(fact_721_dividesE,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( factor8216151070175719842xt_a_b @ G2 @ A @ B )
=> ~ ! [C5: a] :
( ( B
= ( mult_a_ring_ext_a_b @ G2 @ A @ C5 ) )
=> ~ ( member_a @ C5 @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ).
% dividesE
thf(fact_722_dividesE,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a] :
( ( factor3040189038382604065t_unit @ G2 @ A @ B )
=> ~ ! [C5: a] :
( ( B
= ( mult_a_Product_unit @ G2 @ A @ C5 ) )
=> ~ ( member_a @ C5 @ ( partia6735698275553448452t_unit @ G2 ) ) ) ) ).
% dividesE
thf(fact_723_dividesE,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( factor5460682277579321776t_unit @ G2 @ A @ B )
=> ~ ! [C5: set_a] :
( ( B
= ( mult_s7930653359683758801t_unit @ G2 @ A @ C5 ) )
=> ~ ( member_set_a @ C5 @ ( partia5907974310037520643t_unit @ G2 ) ) ) ) ).
% dividesE
thf(fact_724_dividesI,axiom,
! [C: a,G2: partia2175431115845679010xt_a_b,B: a,A: a] :
( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( B
= ( mult_a_ring_ext_a_b @ G2 @ A @ C ) )
=> ( factor8216151070175719842xt_a_b @ G2 @ A @ B ) ) ) ).
% dividesI
thf(fact_725_dividesI,axiom,
! [C: a,G2: partia8223610829204095565t_unit,B: a,A: a] :
( ( member_a @ C @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( B
= ( mult_a_Product_unit @ G2 @ A @ C ) )
=> ( factor3040189038382604065t_unit @ G2 @ A @ B ) ) ) ).
% dividesI
thf(fact_726_dividesI,axiom,
! [C: set_a,G2: partia6043505979758434576t_unit,B: set_a,A: set_a] :
( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( B
= ( mult_s7930653359683758801t_unit @ G2 @ A @ C ) )
=> ( factor5460682277579321776t_unit @ G2 @ A @ B ) ) ) ).
% dividesI
thf(fact_727_factor__def,axiom,
( factor8216151070175719842xt_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,A7: a,B5: a] :
? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G3 ) )
& ( B5
= ( mult_a_ring_ext_a_b @ G3 @ A7 @ X3 ) ) ) ) ) ).
% factor_def
thf(fact_728_factor__def,axiom,
( factor3040189038382604065t_unit
= ( ^ [G3: partia8223610829204095565t_unit,A7: a,B5: a] :
? [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G3 ) )
& ( B5
= ( mult_a_Product_unit @ G3 @ A7 @ X3 ) ) ) ) ) ).
% factor_def
thf(fact_729_factor__def,axiom,
( factor5460682277579321776t_unit
= ( ^ [G3: partia6043505979758434576t_unit,A7: set_a,B5: set_a] :
? [X3: set_a] :
( ( member_set_a @ X3 @ ( partia5907974310037520643t_unit @ G3 ) )
& ( B5
= ( mult_s7930653359683758801t_unit @ G3 @ A7 @ X3 ) ) ) ) ) ).
% factor_def
thf(fact_730_monoid__cancel_Ol__cancel,axiom,
! [G2: partia2175431115845679010xt_a_b,C: a,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( ( mult_a_ring_ext_a_b @ G2 @ C @ A )
= ( mult_a_ring_ext_a_b @ G2 @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_731_monoid__cancel_Ol__cancel,axiom,
! [G2: partia8223610829204095565t_unit,C: a,A: a,B: a] :
( ( monoid1999574367301118026t_unit @ G2 )
=> ( ( ( mult_a_Product_unit @ G2 @ C @ A )
= ( mult_a_Product_unit @ G2 @ C @ B ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_732_monoid__cancel_Ol__cancel,axiom,
! [G2: partia6043505979758434576t_unit,C: set_a,A: set_a,B: set_a] :
( ( monoid759071382269063239t_unit @ G2 )
=> ( ( ( mult_s7930653359683758801t_unit @ G2 @ C @ A )
= ( mult_s7930653359683758801t_unit @ G2 @ C @ B ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_733_monoid__cancel_Or__cancel,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,C: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( ( mult_a_ring_ext_a_b @ G2 @ A @ C )
= ( mult_a_ring_ext_a_b @ G2 @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_734_monoid__cancel_Or__cancel,axiom,
! [G2: partia8223610829204095565t_unit,A: a,C: a,B: a] :
( ( monoid1999574367301118026t_unit @ G2 )
=> ( ( ( mult_a_Product_unit @ G2 @ A @ C )
= ( mult_a_Product_unit @ G2 @ B @ C ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_735_monoid__cancel_Or__cancel,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,C: set_a,B: set_a] :
( ( monoid759071382269063239t_unit @ G2 )
=> ( ( ( mult_s7930653359683758801t_unit @ G2 @ A @ C )
= ( mult_s7930653359683758801t_unit @ G2 @ B @ C ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_736_monoid__cancel_Odivides__mult__l,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ ( mult_a_ring_ext_a_b @ G2 @ C @ A ) @ ( mult_a_ring_ext_a_b @ G2 @ C @ B ) )
= ( factor8216151070175719842xt_a_b @ G2 @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_737_monoid__cancel_Odivides__mult__l,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( monoid1999574367301118026t_unit @ G2 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( factor3040189038382604065t_unit @ G2 @ ( mult_a_Product_unit @ G2 @ C @ A ) @ ( mult_a_Product_unit @ G2 @ C @ B ) )
= ( factor3040189038382604065t_unit @ G2 @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_738_monoid__cancel_Odivides__mult__l,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a,C: set_a] :
( ( monoid759071382269063239t_unit @ G2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( factor5460682277579321776t_unit @ G2 @ ( mult_s7930653359683758801t_unit @ G2 @ C @ A ) @ ( mult_s7930653359683758801t_unit @ G2 @ C @ B ) )
= ( factor5460682277579321776t_unit @ G2 @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_739_primeE,axiom,
! [G2: partia2175431115845679010xt_a_b,P2: a] :
( ( prime_a_ring_ext_a_b @ G2 @ P2 )
=> ~ ( ~ ( member_a @ P2 @ ( units_a_ring_ext_a_b @ G2 ) )
=> ~ ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ P2 @ ( mult_a_ring_ext_a_b @ G2 @ X4 @ Xa ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ P2 @ X4 )
| ( factor8216151070175719842xt_a_b @ G2 @ P2 @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_740_primeE,axiom,
! [G2: partia8223610829204095565t_unit,P2: a] :
( ( prime_a_Product_unit @ G2 @ P2 )
=> ~ ( ~ ( member_a @ P2 @ ( units_a_Product_unit @ G2 ) )
=> ~ ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( factor3040189038382604065t_unit @ G2 @ P2 @ ( mult_a_Product_unit @ G2 @ X4 @ Xa ) )
=> ( ( factor3040189038382604065t_unit @ G2 @ P2 @ X4 )
| ( factor3040189038382604065t_unit @ G2 @ P2 @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_741_primeE,axiom,
! [G2: partia6043505979758434576t_unit,P2: set_a] :
( ( prime_4522187476880896870t_unit @ G2 @ P2 )
=> ~ ( ~ ( member_set_a @ P2 @ ( units_2471184348132832486t_unit @ G2 ) )
=> ~ ! [X4: set_a] :
( ( member_set_a @ X4 @ ( partia5907974310037520643t_unit @ G2 ) )
=> ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( factor5460682277579321776t_unit @ G2 @ P2 @ ( mult_s7930653359683758801t_unit @ G2 @ X4 @ Xa ) )
=> ( ( factor5460682277579321776t_unit @ G2 @ P2 @ X4 )
| ( factor5460682277579321776t_unit @ G2 @ P2 @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_742_primeI,axiom,
! [P2: a,G2: partia2175431115845679010xt_a_b] :
( ~ ( member_a @ P2 @ ( units_a_ring_ext_a_b @ G2 ) )
=> ( ! [A6: a,B6: a] :
( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ P2 @ ( mult_a_ring_ext_a_b @ G2 @ A6 @ B6 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ P2 @ A6 )
| ( factor8216151070175719842xt_a_b @ G2 @ P2 @ B6 ) ) ) ) )
=> ( prime_a_ring_ext_a_b @ G2 @ P2 ) ) ) ).
% primeI
thf(fact_743_primeI,axiom,
! [P2: a,G2: partia8223610829204095565t_unit] :
( ~ ( member_a @ P2 @ ( units_a_Product_unit @ G2 ) )
=> ( ! [A6: a,B6: a] :
( ( member_a @ A6 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B6 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( factor3040189038382604065t_unit @ G2 @ P2 @ ( mult_a_Product_unit @ G2 @ A6 @ B6 ) )
=> ( ( factor3040189038382604065t_unit @ G2 @ P2 @ A6 )
| ( factor3040189038382604065t_unit @ G2 @ P2 @ B6 ) ) ) ) )
=> ( prime_a_Product_unit @ G2 @ P2 ) ) ) ).
% primeI
thf(fact_744_primeI,axiom,
! [P2: set_a,G2: partia6043505979758434576t_unit] :
( ~ ( member_set_a @ P2 @ ( units_2471184348132832486t_unit @ G2 ) )
=> ( ! [A6: set_a,B6: set_a] :
( ( member_set_a @ A6 @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B6 @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( factor5460682277579321776t_unit @ G2 @ P2 @ ( mult_s7930653359683758801t_unit @ G2 @ A6 @ B6 ) )
=> ( ( factor5460682277579321776t_unit @ G2 @ P2 @ A6 )
| ( factor5460682277579321776t_unit @ G2 @ P2 @ B6 ) ) ) ) )
=> ( prime_4522187476880896870t_unit @ G2 @ P2 ) ) ) ).
% primeI
thf(fact_745_prime__def,axiom,
( prime_a_ring_ext_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,P3: a] :
( ~ ( member_a @ P3 @ ( units_a_ring_ext_a_b @ G3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ! [Y6: a] :
( ( member_a @ Y6 @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( ( factor8216151070175719842xt_a_b @ G3 @ P3 @ ( mult_a_ring_ext_a_b @ G3 @ X3 @ Y6 ) )
=> ( ( factor8216151070175719842xt_a_b @ G3 @ P3 @ X3 )
| ( factor8216151070175719842xt_a_b @ G3 @ P3 @ Y6 ) ) ) ) ) ) ) ) ).
% prime_def
thf(fact_746_prime__def,axiom,
( prime_a_Product_unit
= ( ^ [G3: partia8223610829204095565t_unit,P3: a] :
( ~ ( member_a @ P3 @ ( units_a_Product_unit @ G3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G3 ) )
=> ! [Y6: a] :
( ( member_a @ Y6 @ ( partia6735698275553448452t_unit @ G3 ) )
=> ( ( factor3040189038382604065t_unit @ G3 @ P3 @ ( mult_a_Product_unit @ G3 @ X3 @ Y6 ) )
=> ( ( factor3040189038382604065t_unit @ G3 @ P3 @ X3 )
| ( factor3040189038382604065t_unit @ G3 @ P3 @ Y6 ) ) ) ) ) ) ) ) ).
% prime_def
thf(fact_747_prime__def,axiom,
( prime_4522187476880896870t_unit
= ( ^ [G3: partia6043505979758434576t_unit,P3: set_a] :
( ~ ( member_set_a @ P3 @ ( units_2471184348132832486t_unit @ G3 ) )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ ( partia5907974310037520643t_unit @ G3 ) )
=> ! [Y6: set_a] :
( ( member_set_a @ Y6 @ ( partia5907974310037520643t_unit @ G3 ) )
=> ( ( factor5460682277579321776t_unit @ G3 @ P3 @ ( mult_s7930653359683758801t_unit @ G3 @ X3 @ Y6 ) )
=> ( ( factor5460682277579321776t_unit @ G3 @ P3 @ X3 )
| ( factor5460682277579321776t_unit @ G3 @ P3 @ Y6 ) ) ) ) ) ) ) ) ).
% prime_def
thf(fact_748_field_OsubfieldI_H,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ K @ R )
=> ( ! [K3: set_a] :
( ( member_set_a @ K3 @ ( minus_5736297505244876581_set_a @ K @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( member_set_a @ ( m_inv_7491079437187478987t_unit @ R @ K3 ) @ K ) )
=> ( subfie5224850075530046424t_unit @ K @ R ) ) ) ) ).
% field.subfieldI'
thf(fact_749_field_OsubfieldI_H,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( field_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ! [K3: a] :
( ( member_a @ K3 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ R @ K3 ) @ K ) )
=> ( subfield_a_b @ K @ R ) ) ) ) ).
% field.subfieldI'
thf(fact_750_divides__imp__divides__mult,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ).
% divides_imp_divides_mult
thf(fact_751_space__subgroup__props_I6_J,axiom,
! [K: set_a,N2: nat,E: set_a,K2: a,A: a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ A ) @ E )
=> ( member_a @ A @ E ) ) ) ) ) ) ).
% space_subgroup_props(6)
thf(fact_752_subalbegra__incl__imp__finite__dimension,axiom,
! [K: set_a,E: set_a,V: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ( ord_less_eq_set_a @ V @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K @ V ) ) ) ) ) ).
% subalbegra_incl_imp_finite_dimension
thf(fact_753_finite__mult__of,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( multip3210463924028840165of_a_b @ r ) ) ) ) ).
% finite_mult_of
thf(fact_754_mult__of_Omonoid__cancel__axioms,axiom,
monoid1999574367301118026t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.monoid_cancel_axioms
thf(fact_755_zero__is__prime_I2_J,axiom,
prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ ( zero_a_b @ r ) ).
% zero_is_prime(2)
thf(fact_756_mult__of_OUnits__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.Units_closed
thf(fact_757_mult__of_Ogcdof__exists,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [C5: a] :
( ( member_a @ C5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ C5 @ A @ B ) ) ) ) ).
% mult_of.gcdof_exists
thf(fact_758_dimension__is__inj,axiom,
! [K: set_a,N2: nat,E: set_a,M2: nat] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M2 @ K @ E )
=> ( N2 = M2 ) ) ) ) ).
% dimension_is_inj
thf(fact_759_telescopic__base__dim_I1_J,axiom,
! [K: set_a,F: set_a,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( subfield_a_b @ F @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ F )
=> ( ( embedd8708762675212832759on_a_b @ r @ F @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K @ E ) ) ) ) ) ).
% telescopic_base_dim(1)
thf(fact_760_finite__dimensionE_H,axiom,
! [K: set_a,E: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ~ ! [N3: nat] :
~ ( embedd2795209813406577254on_a_b @ r @ N3 @ K @ E ) ) ).
% finite_dimensionE'
thf(fact_761_finite__dimensionI,axiom,
! [N2: nat,K: set_a,E: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K @ E ) ) ).
% finite_dimensionI
thf(fact_762_finite__dimension__def,axiom,
! [K: set_a,E: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K @ E )
= ( ? [N4: nat] : ( embedd2795209813406577254on_a_b @ r @ N4 @ K @ E ) ) ) ).
% finite_dimension_def
thf(fact_763_mult__of_Oprod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.prod_unit_l
thf(fact_764_mult__of_Oprod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.prod_unit_r
thf(fact_765_mult__of_Ounit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.unit_factor
thf(fact_766_mult__of_Omonoid__cancelI,axiom,
( ! [A6: a,B6: a,C5: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C5 @ A6 )
= ( mult_a_ring_ext_a_b @ r @ C5 @ B6 ) )
=> ( ( member_a @ A6 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B6 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A6 = B6 ) ) ) ) )
=> ( ! [A6: a,B6: a,C5: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A6 @ C5 )
= ( mult_a_ring_ext_a_b @ r @ B6 @ C5 ) )
=> ( ( member_a @ A6 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B6 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A6 = B6 ) ) ) ) )
=> ( monoid1999574367301118026t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.monoid_cancelI
thf(fact_767_mult__of_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C @ A )
= ( mult_a_ring_ext_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A = B ) ) ) ) ) ).
% mult_of.l_cancel
thf(fact_768_mult__of_Om__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ ( ring_mult_of_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 ) ) ) ) ) ) ).
% mult_of.m_assoc
thf(fact_769_mult__of_Om__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% mult_of.m_comm
thf(fact_770_mult__of_Om__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ ( ring_mult_of_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 ) ) ) ) ) ) ).
% mult_of.m_lcomm
thf(fact_771_mult__of_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ C )
= ( mult_a_ring_ext_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A = B ) ) ) ) ) ).
% mult_of.r_cancel
thf(fact_772_mult__of_Ocarrier__not__empty,axiom,
( ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) )
!= bot_bot_set_a ) ).
% mult_of.carrier_not_empty
thf(fact_773_mult__of_OUnits__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_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.Units_inv_comm
thf(fact_774_mult__of_Odivides__unit,axiom,
! [A: a,U: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ U )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.divides_unit
thf(fact_775_mult__of_Ounit__divides,axiom,
! [U: a,A: a] :
( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ U @ A ) ) ) ).
% mult_of.unit_divides
thf(fact_776_mult__of_Oisgcd__divides__l,axiom,
! [A: a,B: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A @ A @ B ) ) ) ) ).
% mult_of.isgcd_divides_l
thf(fact_777_mult__of_Oisgcd__divides__r,axiom,
! [B: a,A: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ B @ A )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ B @ A @ B ) ) ) ) ).
% mult_of.isgcd_divides_r
thf(fact_778_mult__of_Odivides__trans,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ C ) ) ) ) ).
% mult_of.divides_trans
thf(fact_779_space__subgroup__props_I2_J,axiom,
! [K: set_a,N2: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E )
=> ( member_a @ ( zero_a_b @ r ) @ E ) ) ) ).
% space_subgroup_props(2)
thf(fact_780_space__subgroup__props_I3_J,axiom,
! [K: set_a,N2: nat,E: set_a,V1: a,V22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E )
=> ( ( member_a @ V1 @ E )
=> ( ( member_a @ V22 @ E )
=> ( member_a @ ( add_a_b @ r @ V1 @ V22 ) @ E ) ) ) ) ) ).
% space_subgroup_props(3)
thf(fact_781_space__subgroup__props_I5_J,axiom,
! [K: set_a,N2: nat,E: set_a,K2: a,V3: a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ V3 ) @ E ) ) ) ) ) ).
% space_subgroup_props(5)
thf(fact_782_space__subgroup__props_I4_J,axiom,
! [K: set_a,N2: nat,E: set_a,V3: a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( a_inv_a_b @ r @ V3 ) @ E ) ) ) ) ).
% space_subgroup_props(4)
thf(fact_783_ring__primeE_I2_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P2 ) ) ) ).
% ring_primeE(2)
thf(fact_784_unique__dimension,axiom,
! [K: set_a,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ? [X2: nat] :
( ( embedd2795209813406577254on_a_b @ r @ X2 @ K @ E )
& ! [Y5: nat] :
( ( embedd2795209813406577254on_a_b @ r @ Y5 @ K @ E )
=> ( Y5 = X2 ) ) ) ) ) ).
% unique_dimension
thf(fact_785_prime__eq__prime__mult,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P2 )
= ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P2 ) ) ) ).
% prime_eq_prime_mult
thf(fact_786_finite__dimension__imp__subalgebra,axiom,
! [K: set_a,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ( embedd9027525575939734154ra_a_b @ K @ E @ r ) ) ) ).
% finite_dimension_imp_subalgebra
thf(fact_787_divides__mult__zero,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( zero_a_b @ r ) )
=> ( A
= ( zero_a_b @ r ) ) ) ) ).
% divides_mult_zero
thf(fact_788_mult__of_OUnits__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X2 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.Units_l_inv_ex
thf(fact_789_mult__of_OUnits__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.Units_r_inv_ex
thf(fact_790_mult__of_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.one_unique
thf(fact_791_mult__of_Oinv__unique,axiom,
! [Y: a,X: a,Y4: 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 @ Y4 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( Y = Y4 ) ) ) ) ) ) ).
% mult_of.inv_unique
thf(fact_792_mult__of_Oprime__divides,axiom,
! [A: a,B: a,P2: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P2 )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P2 @ ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P2 @ A )
| ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P2 @ B ) ) ) ) ) ) ).
% mult_of.prime_divides
thf(fact_793_mult__of_Odivides__prod__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ) ).
% mult_of.divides_prod_l
thf(fact_794_mult__of_Odivides__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ).
% mult_of.divides_prod_r
thf(fact_795_space__subgroup__props_I1_J,axiom,
! [K: set_a,N2: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E )
=> ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% space_subgroup_props(1)
thf(fact_796_mult__of_OUnit__eq__dividesone,axiom,
! [U: a] :
( ( member_a @ U @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ U @ ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.Unit_eq_dividesone
thf(fact_797_Ring__Divisibility_Omult__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( mult_a_Product_unit @ ( ring_mult_of_a_b @ R ) )
= ( mult_a_ring_ext_a_b @ R ) ) ).
% Ring_Divisibility.mult_mult_of
thf(fact_798_ring__primeI_H,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P2 )
=> ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% ring_primeI'
thf(fact_799_mult__of_Olcmof__exists,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [C5: a] :
( ( member_a @ C5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( islcm_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ C5 @ A @ B ) ) ) ) ).
% mult_of.lcmof_exists
thf(fact_800_mult__of_OUnits__m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.Units_m_closed
thf(fact_801_mult__of_OUnits__one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ).
% mult_of.Units_one_closed
thf(fact_802_Units__mult__eq__Units,axiom,
( ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) )
= ( units_a_ring_ext_a_b @ r ) ) ).
% Units_mult_eq_Units
thf(fact_803_mult__of_OSomeGcd__ex,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( A2 != bot_bot_set_a )
=> ( member_a @ ( someGc8133249837406473920t_unit @ ( ring_mult_of_a_b @ r ) @ A2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.SomeGcd_ex
thf(fact_804_mult__of_OUnits__l__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% mult_of.Units_l_cancel
thf(fact_805_mult__of_Om__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.m_closed
thf(fact_806_mult__of_Oone__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ).
% mult_of.one_closed
thf(fact_807_mult__of_Odivides__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A ) ) ).
% mult_of.divides_refl
thf(fact_808_mult__of_Or__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
= X ) ) ).
% mult_of.r_one
thf(fact_809_mult__of_Ol__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
= X ) ) ).
% mult_of.l_one
thf(fact_810_mult__of_Odivides__mult__rI,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% mult_of.divides_mult_rI
thf(fact_811_mult__of_Odivides__mult__r,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ) ).
% mult_of.divides_mult_r
thf(fact_812_mult__of_Odivides__mult__lI,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% mult_of.divides_mult_lI
thf(fact_813_mult__of_Odivides__mult__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ) ).
% mult_of.divides_mult_l
thf(fact_814_Ring__Divisibility_Ocarrier__mult__of,axiom,
! [R: partia6043505979758434576t_unit] :
( ( partia8299590604543202116t_unit @ ( ring_m2800496791135293897t_unit @ R ) )
= ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) ) ).
% Ring_Divisibility.carrier_mult_of
thf(fact_815_Ring__Divisibility_Ocarrier__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ R ) )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ).
% Ring_Divisibility.carrier_mult_of
thf(fact_816_mult__of_Odivisor__chain__condition__monoid__axioms,axiom,
diviso6259607970152342594t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.divisor_chain_condition_monoid_axioms
thf(fact_817_mult__of_Oprimeness__condition__monoid__axioms,axiom,
primen965786292471834261t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.primeness_condition_monoid_axioms
thf(fact_818_domain_OUnits__mult__eq__Units,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( units_3682061850359937035t_unit @ ( ring_m2800496791135293897t_unit @ R ) )
= ( units_2471184348132832486t_unit @ R ) ) ) ).
% domain.Units_mult_eq_Units
thf(fact_819_domain_OUnits__mult__eq__Units,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( units_a_Product_unit @ ( ring_mult_of_a_b @ R ) )
= ( units_a_ring_ext_a_b @ R ) ) ) ).
% domain.Units_mult_eq_Units
thf(fact_820_domain_Ozero__is__prime_I2_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( prime_8576247383786985867t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ ( zero_s2174465271003423091t_unit @ R ) ) ) ).
% domain.zero_is_prime(2)
thf(fact_821_domain_Ozero__is__prime_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ R ) @ ( zero_a_b @ R ) ) ) ).
% domain.zero_is_prime(2)
thf(fact_822_divides__mult__imp__divides,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ R ) @ A @ B )
=> ( factor8216151070175719842xt_a_b @ R @ A @ B ) ) ).
% divides_mult_imp_divides
thf(fact_823_domain_Oring__primeE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P2 )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ R ) @ P2 ) ) ) ) ).
% domain.ring_primeE(2)
thf(fact_824_domain_Oring__primeE_I2_J,axiom,
! [R: partia6043505979758434576t_unit,P2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ P2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r6795642478576035723t_unit @ R @ P2 )
=> ( prime_8576247383786985867t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ P2 ) ) ) ) ).
% domain.ring_primeE(2)
thf(fact_825_domain_Oprime__eq__prime__mult,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( prime_a_ring_ext_a_b @ R @ P2 )
= ( prime_a_Product_unit @ ( ring_mult_of_a_b @ R ) @ P2 ) ) ) ) ).
% domain.prime_eq_prime_mult
thf(fact_826_domain_Oprime__eq__prime__mult,axiom,
! [R: partia6043505979758434576t_unit,P2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ P2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( prime_4522187476880896870t_unit @ R @ P2 )
= ( prime_8576247383786985867t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ P2 ) ) ) ) ).
% domain.prime_eq_prime_mult
thf(fact_827_domain_Odivides__mult__zero,axiom,
! [R: partia2175431115845679010xt_a_b,A: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ R ) @ A @ ( zero_a_b @ R ) )
=> ( A
= ( zero_a_b @ R ) ) ) ) ) ).
% domain.divides_mult_zero
thf(fact_828_domain_Odivides__mult__zero,axiom,
! [R: partia6043505979758434576t_unit,A: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( factor8582526991245238721t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ A @ ( zero_s2174465271003423091t_unit @ R ) )
=> ( A
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ).
% domain.divides_mult_zero
thf(fact_829_domain_Oring__primeI_H,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ R ) @ P2 )
=> ( ring_ring_prime_a_b @ R @ P2 ) ) ) ) ).
% domain.ring_primeI'
thf(fact_830_domain_Oring__primeI_H,axiom,
! [R: partia6043505979758434576t_unit,P2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ P2 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ( prime_8576247383786985867t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ P2 )
=> ( ring_r6795642478576035723t_unit @ R @ P2 ) ) ) ) ).
% domain.ring_primeI'
thf(fact_831_subringE_I2_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H2 @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H2 ) ) ).
% subringE(2)
thf(fact_832_subringE_I7_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subring_a_b @ H2 @ R )
=> ( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( add_a_b @ R @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subringE(7)
thf(fact_833_subringE_I4_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H2 @ R )
=> ( H2 != bot_bot_set_a ) ) ).
% subringE(4)
thf(fact_834_subringE_I3_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H2 @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H2 ) ) ).
% subringE(3)
thf(fact_835_subringE_I5_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H: a] :
( ( subring_a_b @ H2 @ R )
=> ( ( member_a @ H @ H2 )
=> ( member_a @ ( a_inv_a_b @ R @ H ) @ H2 ) ) ) ).
% subringE(5)
thf(fact_836_subcringE_I2_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H2 @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H2 ) ) ).
% subcringE(2)
thf(fact_837_subcringE_I7_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subcring_a_b @ H2 @ R )
=> ( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( add_a_b @ R @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subcringE(7)
thf(fact_838_subcringE_I4_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H2 @ R )
=> ( H2 != bot_bot_set_a ) ) ).
% subcringE(4)
thf(fact_839_subcring_Osub__m__comm,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subcring_a_b @ H2 @ R )
=> ( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( ( mult_a_ring_ext_a_b @ R @ H12 @ H23 )
= ( mult_a_ring_ext_a_b @ R @ H23 @ H12 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_840_subcringE_I6_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subcring_a_b @ H2 @ R )
=> ( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subcringE(6)
thf(fact_841_subfieldE_I2_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( subcring_a_b @ K @ R ) ) ).
% subfieldE(2)
thf(fact_842_subcringE_I3_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H2 @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H2 ) ) ).
% subcringE(3)
thf(fact_843_subcringE_I5_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H: a] :
( ( subcring_a_b @ H2 @ R )
=> ( ( member_a @ H @ H2 )
=> ( member_a @ ( a_inv_a_b @ R @ H ) @ H2 ) ) ) ).
% subcringE(5)
thf(fact_844_subcring_Oaxioms_I1_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H2 @ R )
=> ( subring_a_b @ H2 @ R ) ) ).
% subcring.axioms(1)
thf(fact_845_subdomainE_I2_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H2 @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H2 ) ) ).
% subdomainE(2)
thf(fact_846_subdomainE_I7_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subdomain_a_b @ H2 @ R )
=> ( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( add_a_b @ R @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subdomainE(7)
thf(fact_847_subdomainE_I4_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H2 @ R )
=> ( H2 != bot_bot_set_a ) ) ).
% subdomainE(4)
thf(fact_848_subdomainE_I3_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H2 @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H2 ) ) ).
% subdomainE(3)
thf(fact_849_subdomainE_I5_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H: a] :
( ( subdomain_a_b @ H2 @ R )
=> ( ( member_a @ H @ H2 )
=> ( member_a @ ( a_inv_a_b @ R @ H ) @ H2 ) ) ) ).
% subdomainE(5)
thf(fact_850_subdomain_Oaxioms_I1_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H2 @ R )
=> ( subcring_a_b @ H2 @ R ) ) ).
% subdomain.axioms(1)
thf(fact_851_domain_Odivides__imp__divides__mult,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( factor8216151070175719842xt_a_b @ R @ A @ B )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ R ) @ A @ B ) ) ) ) ) ).
% domain.divides_imp_divides_mult
thf(fact_852_domain_Odivides__imp__divides__mult,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ( factor5460682277579321776t_unit @ R @ A @ B )
=> ( factor8582526991245238721t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ A @ B ) ) ) ) ) ).
% domain.divides_imp_divides_mult
thf(fact_853_subfieldE_I3_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subfieldE(3)
thf(fact_854_subfieldE_I3_J,axiom,
! [K: set_set_a,R: partia6043505979758434576t_unit] :
( ( subfie5224850075530046424t_unit @ K @ R )
=> ( ord_le3724670747650509150_set_a @ K @ ( partia5907974310037520643t_unit @ R ) ) ) ).
% subfieldE(3)
thf(fact_855_subringE_I1_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H2 @ R )
=> ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subringE(1)
thf(fact_856_subringE_I1_J,axiom,
! [H2: set_set_a,R: partia6043505979758434576t_unit] :
( ( subrin1511138061850335568t_unit @ H2 @ R )
=> ( ord_le3724670747650509150_set_a @ H2 @ ( partia5907974310037520643t_unit @ R ) ) ) ).
% subringE(1)
thf(fact_857_subfieldE_I5_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K @ R )
=> ( ( member_a @ K1 @ K )
=> ( ( member_a @ K22 @ K )
=> ( ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
= ( zero_a_b @ R ) )
=> ( ( K1
= ( zero_a_b @ R ) )
| ( K22
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_858_subfieldE_I6_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% subfieldE(6)
thf(fact_859_subcringE_I1_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H2 @ R )
=> ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subcringE(1)
thf(fact_860_subcringE_I1_J,axiom,
! [H2: set_set_a,R: partia6043505979758434576t_unit] :
( ( subcri4445174380595745425t_unit @ H2 @ R )
=> ( ord_le3724670747650509150_set_a @ H2 @ ( partia5907974310037520643t_unit @ R ) ) ) ).
% subcringE(1)
thf(fact_861_field_Ocarrier__is__subfield,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( subfield_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% field.carrier_is_subfield
thf(fact_862_field_Ocarrier__is__subfield,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( subfie5224850075530046424t_unit @ ( partia5907974310037520643t_unit @ R ) @ R ) ) ).
% field.carrier_is_subfield
thf(fact_863_subdomainE_I1_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H2 @ R )
=> ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subdomainE(1)
thf(fact_864_subdomainE_I1_J,axiom,
! [H2: set_set_a,R: partia6043505979758434576t_unit] :
( ( subdom4943114742163587044t_unit @ H2 @ R )
=> ( ord_le3724670747650509150_set_a @ H2 @ ( partia5907974310037520643t_unit @ R ) ) ) ).
% subdomainE(1)
thf(fact_865_subdomain_Osubintegral,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subdomain_a_b @ H2 @ R )
=> ( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( ( ( mult_a_ring_ext_a_b @ R @ H12 @ H23 )
= ( zero_a_b @ R ) )
=> ( ( H12
= ( zero_a_b @ R ) )
| ( H23
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_866_subdomain_Osub__one__not__zero,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H2 @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_867_domain_OsubdomainI_H,axiom,
! [R: partia6043505979758434576t_unit,H2: set_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ H2 @ R )
=> ( subdom4943114742163587044t_unit @ H2 @ R ) ) ) ).
% domain.subdomainI'
thf(fact_868_domain_OsubdomainI_H,axiom,
! [R: partia2175431115845679010xt_a_b,H2: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ H2 @ R )
=> ( subdomain_a_b @ H2 @ R ) ) ) ).
% domain.subdomainI'
thf(fact_869_mult__of_Odivides__fcount,axiom,
! [A: a,B: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ord_less_eq_nat @ ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ A ) @ ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ B ) ) ) ) ) ).
% mult_of.divides_fcount
thf(fact_870_Multiplicative__Group_Ocarrier__mult__of,axiom,
! [R: partia6043505979758434576t_unit] :
( ( partia8299590604543202116t_unit @ ( multip3774352783277980819t_unit @ R ) )
= ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) ) ).
% Multiplicative_Group.carrier_mult_of
thf(fact_871_Multiplicative__Group_Ocarrier__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( partia6735698275553448452t_unit @ ( multip3210463924028840165of_a_b @ R ) )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ).
% Multiplicative_Group.carrier_mult_of
thf(fact_872_mult__of__is__Units,axiom,
( ( multip3210463924028840165of_a_b @ r )
= ( units_8174867845824275201xt_a_b @ r ) ) ).
% mult_of_is_Units
thf(fact_873_ring__irreducibleI_H,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ R2 )
=> ( ring_r999134135267193926le_a_b @ r @ R2 ) ) ) ).
% ring_irreducibleI'
thf(fact_874_dimension__zero,axiom,
! [K: set_a,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ zero_zero_nat @ K @ E )
=> ( E
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% dimension_zero
thf(fact_875_zero__is__irreducible__mult,axiom,
irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( zero_a_b @ r ) ).
% zero_is_irreducible_mult
thf(fact_876_mult__of_Oprime__irreducible,axiom,
! [P2: a] :
( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P2 )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ P2 ) ) ).
% mult_of.prime_irreducible
thf(fact_877_ring__irreducibleE_I3_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ R2 ) ) ) ).
% ring_irreducibleE(3)
thf(fact_878_mult__of_Oirreducible__prime,axiom,
! [P2: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ P2 )
=> ( ( member_a @ P2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P2 ) ) ) ).
% mult_of.irreducible_prime
thf(fact_879_zero__dim,axiom,
! [K: set_a] : ( embedd2795209813406577254on_a_b @ r @ zero_zero_nat @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% zero_dim
thf(fact_880_mult__of_Oirreducible__prod__rI,axiom,
! [A: a,B: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% mult_of.irreducible_prod_rI
thf(fact_881_mult__of_Oirreducible__prod__lI,axiom,
! [B: a,A: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ B )
=> ( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% mult_of.irreducible_prod_lI
thf(fact_882_mult__of_Oirreducible__prodE,axiom,
! [A: a,B: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A )
=> ~ ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) )
=> ~ ( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ~ ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ B ) ) ) ) ) ) ).
% mult_of.irreducible_prodE
thf(fact_883_primeness__condition__monoid_Oirreducible__prime,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a] :
( ( primen9005823089519874350xt_a_b @ G2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( irredu6211895646901577903xt_a_b @ G2 @ A )
=> ( prime_a_ring_ext_a_b @ G2 @ A ) ) ) ) ).
% primeness_condition_monoid.irreducible_prime
thf(fact_884_primeness__condition__monoid_Oirreducible__prime,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a] :
( ( primen4290209806047655228t_unit @ G2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( irredu5346329325703585725t_unit @ G2 @ A )
=> ( prime_4522187476880896870t_unit @ G2 @ A ) ) ) ) ).
% primeness_condition_monoid.irreducible_prime
thf(fact_885_primeness__condition__monoid_Oirreducible__prime,axiom,
! [G2: partia8223610829204095565t_unit,A: a] :
( ( primen965786292471834261t_unit @ G2 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( irredu4023057619401689684t_unit @ G2 @ A )
=> ( prime_a_Product_unit @ G2 @ A ) ) ) ) ).
% primeness_condition_monoid.irreducible_prime
thf(fact_886_field_Omult__of__is__Units,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( multip3774352783277980819t_unit @ R )
= ( units_1455294149231095823t_unit @ R ) ) ) ).
% field.mult_of_is_Units
thf(fact_887_field_Omult__of__is__Units,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( multip3210463924028840165of_a_b @ R )
= ( units_8174867845824275201xt_a_b @ R ) ) ) ).
% field.mult_of_is_Units
thf(fact_888_domain_Ozero__is__irreducible__mult,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( irredu8646402277169070324t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ ( zero_s2174465271003423091t_unit @ R ) ) ) ).
% domain.zero_is_irreducible_mult
thf(fact_889_domain_Ozero__is__irreducible__mult,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R ) @ ( zero_a_b @ R ) ) ) ).
% domain.zero_is_irreducible_mult
thf(fact_890_domain_Oring__irreducibleE_I3_J,axiom,
! [R: partia6043505979758434576t_unit,R2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ R2 )
=> ( irredu8646402277169070324t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ R2 ) ) ) ) ).
% domain.ring_irreducibleE(3)
thf(fact_891_domain_Oring__irreducibleE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R ) @ R2 ) ) ) ) ).
% domain.ring_irreducibleE(3)
thf(fact_892_islcm__def,axiom,
( islcm_a_ring_ext_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,X3: a,A7: a,B5: a] :
( ( factor8216151070175719842xt_a_b @ G3 @ A7 @ X3 )
& ( factor8216151070175719842xt_a_b @ G3 @ B5 @ X3 )
& ! [Y6: a] :
( ( member_a @ Y6 @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( ( ( factor8216151070175719842xt_a_b @ G3 @ A7 @ Y6 )
& ( factor8216151070175719842xt_a_b @ G3 @ B5 @ Y6 ) )
=> ( factor8216151070175719842xt_a_b @ G3 @ X3 @ Y6 ) ) ) ) ) ) ).
% islcm_def
thf(fact_893_islcm__def,axiom,
( islcm_a_Product_unit
= ( ^ [G3: partia8223610829204095565t_unit,X3: a,A7: a,B5: a] :
( ( factor3040189038382604065t_unit @ G3 @ A7 @ X3 )
& ( factor3040189038382604065t_unit @ G3 @ B5 @ X3 )
& ! [Y6: a] :
( ( member_a @ Y6 @ ( partia6735698275553448452t_unit @ G3 ) )
=> ( ( ( factor3040189038382604065t_unit @ G3 @ A7 @ Y6 )
& ( factor3040189038382604065t_unit @ G3 @ B5 @ Y6 ) )
=> ( factor3040189038382604065t_unit @ G3 @ X3 @ Y6 ) ) ) ) ) ) ).
% islcm_def
thf(fact_894_islcm__def,axiom,
( islcm_5941054137687769561t_unit
= ( ^ [G3: partia6043505979758434576t_unit,X3: set_a,A7: set_a,B5: set_a] :
( ( factor5460682277579321776t_unit @ G3 @ A7 @ X3 )
& ( factor5460682277579321776t_unit @ G3 @ B5 @ X3 )
& ! [Y6: set_a] :
( ( member_set_a @ Y6 @ ( partia5907974310037520643t_unit @ G3 ) )
=> ( ( ( factor5460682277579321776t_unit @ G3 @ A7 @ Y6 )
& ( factor5460682277579321776t_unit @ G3 @ B5 @ Y6 ) )
=> ( factor5460682277579321776t_unit @ G3 @ X3 @ Y6 ) ) ) ) ) ) ).
% islcm_def
thf(fact_895_domain_Oring__irreducibleI_H,axiom,
! [R: partia6043505979758434576t_unit,R2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R2 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ( irredu8646402277169070324t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ R2 )
=> ( ring_r7790391342995787508t_unit @ R @ R2 ) ) ) ) ).
% domain.ring_irreducibleI'
thf(fact_896_domain_Oring__irreducibleI_H,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R ) @ R2 )
=> ( ring_r999134135267193926le_a_b @ R @ R2 ) ) ) ) ).
% domain.ring_irreducibleI'
thf(fact_897_field_Ofinite__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ R ) )
=> ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( multip3210463924028840165of_a_b @ R ) ) ) ) ) ).
% field.finite_mult_of
thf(fact_898_field_Ofinite__mult__of,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( finite_finite_set_a @ ( partia5907974310037520643t_unit @ R ) )
=> ( finite_finite_set_a @ ( partia8299590604543202116t_unit @ ( multip3774352783277980819t_unit @ R ) ) ) ) ) ).
% field.finite_mult_of
thf(fact_899_euclidean__function,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ? [Q2: a,R4: a] :
( ( member_a @ Q2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ R4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( A
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ B @ Q2 ) @ R4 ) )
& ( ( R4
= ( zero_a_b @ r ) )
| ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ) ) ).
% euclidean_function
thf(fact_900_irreducible__mult__imp__irreducible,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A )
=> ( irredu6211895646901577903xt_a_b @ r @ A ) ) ) ).
% irreducible_mult_imp_irreducible
thf(fact_901_dimension_Osimps,axiom,
! [A1: nat,A22: set_a,A32: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ A1 @ A22 @ A32 )
= ( ? [K4: set_a] :
( ( A1 = zero_zero_nat )
& ( A22 = K4 )
& ( A32
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
| ? [V4: a,E2: set_a,N4: nat,K4: set_a] :
( ( A1
= ( suc @ N4 ) )
& ( A22 = K4 )
& ( A32
= ( embedd971793762689825387on_a_b @ r @ K4 @ V4 @ E2 ) )
& ( member_a @ V4 @ ( partia707051561876973205xt_a_b @ r ) )
& ~ ( member_a @ V4 @ E2 )
& ( embedd2795209813406577254on_a_b @ r @ N4 @ K4 @ E2 ) ) ) ) ).
% dimension.simps
thf(fact_902_ring__irreducibleE_I2_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( irredu6211895646901577903xt_a_b @ r @ R2 ) ) ) ).
% ring_irreducibleE(2)
thf(fact_903_zero__is__irreducible__iff__field,axiom,
( ( irredu6211895646901577903xt_a_b @ r @ ( zero_a_b @ r ) )
= ( field_a_b @ r ) ) ).
% zero_is_irreducible_iff_field
thf(fact_904_Suc__dim,axiom,
! [V3: a,E: set_a,N2: nat,K: set_a] :
( ( member_a @ V3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ V3 @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E )
=> ( embedd2795209813406577254on_a_b @ r @ ( suc @ N2 ) @ K @ ( embedd971793762689825387on_a_b @ r @ K @ V3 @ E ) ) ) ) ) ).
% Suc_dim
thf(fact_905_irreducible__prod__rI,axiom,
! [A: a,B: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_rI
thf(fact_906_irreducible__prod__lI,axiom,
! [B: a,A: a] :
( ( irredu6211895646901577903xt_a_b @ r @ B )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_lI
thf(fact_907_irreducible__imp__irreducible__mult,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A ) ) ) ).
% irreducible_imp_irreducible_mult
thf(fact_908_dimension__backwards,axiom,
! [K: set_a,N2: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ ( suc @ N2 ) @ K @ E )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ? [E3: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E3 )
& ~ ( member_a @ X2 @ E3 )
& ( E
= ( embedd971793762689825387on_a_b @ r @ K @ X2 @ E3 ) ) ) ) ) ) ).
% dimension_backwards
thf(fact_909_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_910_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_911_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_912_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_913_dimension_Ocases,axiom,
! [A1: nat,A22: set_a,A32: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ~ ! [V2: a,E4: set_a,N3: nat] :
( ( A1
= ( suc @ N3 ) )
=> ( ( A32
= ( embedd971793762689825387on_a_b @ r @ A22 @ V2 @ E4 ) )
=> ( ( member_a @ V2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ V2 @ E4 )
=> ~ ( embedd2795209813406577254on_a_b @ r @ N3 @ A22 @ E4 ) ) ) ) ) ) ) ).
% dimension.cases
thf(fact_914_euclidean__domainI,axiom,
! [Phi: a > nat] :
( ! [A6: a,B6: a] :
( ( member_a @ A6 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B6 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ? [Q3: a,R5: a] :
( ( member_a @ Q3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ R5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( A6
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ B6 @ Q3 ) @ R5 ) )
& ( ( R5
= ( zero_a_b @ r ) )
| ( ord_less_nat @ ( Phi @ R5 ) @ ( Phi @ B6 ) ) ) ) ) )
=> ( ring_e8745995371659049232in_a_b @ r @ Phi ) ) ).
% euclidean_domainI
thf(fact_915_infinite__growing,axiom,
! [X5: set_nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X5 )
& ( ord_less_nat @ X2 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X5 ) ) ) ).
% infinite_growing
thf(fact_916_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_917_ring__irreducible__def,axiom,
( ring_r999134135267193926le_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,A7: a] :
( ( A7
!= ( zero_a_b @ R3 ) )
& ( irredu6211895646901577903xt_a_b @ R3 @ A7 ) ) ) ) ).
% ring_irreducible_def
thf(fact_918_finite__linorder__max__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B6: nat,A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A5 )
=> ( ord_less_nat @ X4 @ B6 ) )
=> ( ( P @ A5 )
=> ( P @ ( insert_nat @ B6 @ A5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_919_finite__linorder__min__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B6: nat,A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A5 )
=> ( ord_less_nat @ B6 @ X4 ) )
=> ( ( P @ A5 )
=> ( P @ ( insert_nat @ B6 @ A5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_920_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_921_domain_Ozero__is__irreducible__iff__field,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( irredu5346329325703585725t_unit @ R @ ( zero_s2174465271003423091t_unit @ R ) )
= ( field_6045675692312731021t_unit @ R ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_922_domain_Ozero__is__irreducible__iff__field,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( irredu6211895646901577903xt_a_b @ R @ ( zero_a_b @ R ) )
= ( field_a_b @ R ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_923_domain_Oring__irreducibleE_I2_J,axiom,
! [R: partia6043505979758434576t_unit,R2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ R2 )
=> ( irredu5346329325703585725t_unit @ R @ R2 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_924_domain_Oring__irreducibleE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ( irredu6211895646901577903xt_a_b @ R @ R2 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_925_domain_Oirreducible__imp__irreducible__mult,axiom,
! [R: partia2175431115845679010xt_a_b,A: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( irredu6211895646901577903xt_a_b @ R @ A )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R ) @ A ) ) ) ) ).
% domain.irreducible_imp_irreducible_mult
thf(fact_926_domain_Oirreducible__imp__irreducible__mult,axiom,
! [R: partia6043505979758434576t_unit,A: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( irredu5346329325703585725t_unit @ R @ A )
=> ( irredu8646402277169070324t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ A ) ) ) ) ).
% domain.irreducible_imp_irreducible_mult
thf(fact_927_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_928_domain_Oirreducible__mult__imp__irreducible,axiom,
! [R: partia2175431115845679010xt_a_b,A: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R ) @ A )
=> ( irredu6211895646901577903xt_a_b @ R @ A ) ) ) ) ).
% domain.irreducible_mult_imp_irreducible
thf(fact_929_domain_Oirreducible__mult__imp__irreducible,axiom,
! [R: partia6043505979758434576t_unit,A: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ A @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ( irredu8646402277169070324t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ A )
=> ( irredu5346329325703585725t_unit @ R @ A ) ) ) ) ).
% domain.irreducible_mult_imp_irreducible
thf(fact_930_mult__of_Oorder__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
= ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.order_gt_0_iff_finite
thf(fact_931_order__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_a_ring_ext_a_b @ r ) )
= ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% order_gt_0_iff_finite
thf(fact_932_boundD__carrier,axiom,
! [N2: nat,F2: nat > a,M2: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N2 @ F2 )
=> ( ( ord_less_nat @ N2 @ M2 )
=> ( member_a @ ( F2 @ M2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_933_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_934_psubsetI,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_935_psubsetI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_set_a @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_936_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_937_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_938_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_939_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_940_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_941_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_942_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_943_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_944_psubsetE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_945_psubsetE,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_946_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_947_psubset__eq,axiom,
( ord_less_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_948_psubset__imp__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_949_psubset__imp__subset,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_950_psubset__subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_951_psubset__subset__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ord_less_set_set_a @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_952_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_953_subset__not__subset__eq,axiom,
( ord_less_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ~ ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_954_subset__psubset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_955_subset__psubset__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_set_set_a @ B2 @ C2 )
=> ( ord_less_set_set_a @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_956_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_957_subset__iff__psubset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_less_set_set_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_958_not__psubset__empty,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_959_finite__psubset__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ! [B7: set_a] :
( ( ord_less_set_a @ B7 @ A5 )
=> ( P @ B7 ) )
=> ( P @ A5 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_960_diff__commute,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J3 ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J3 ) ) ).
% diff_commute
thf(fact_961_psubset__imp__ex__mem,axiom,
! [A2: set_nat_a,B2: set_nat_a] :
( ( ord_less_set_nat_a @ A2 @ B2 )
=> ? [B6: nat > a] : ( member_nat_a @ B6 @ ( minus_490503922182417452_nat_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_962_psubset__imp__ex__mem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ? [B6: set_a] : ( member_set_a @ B6 @ ( minus_5736297505244876581_set_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_963_psubset__imp__ex__mem,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ? [B6: a] : ( member_a @ B6 @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_964_euclidean__domain_Oaxioms_I1_J,axiom,
! [R: partia6043505979758434576t_unit,Phi: set_a > nat] :
( ( ring_e187967263881214398t_unit @ R @ Phi )
=> ( domain4236798911309298543t_unit @ R ) ) ).
% euclidean_domain.axioms(1)
thf(fact_965_euclidean__domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,Phi: a > nat] :
( ( ring_e8745995371659049232in_a_b @ R @ Phi )
=> ( domain_a_b @ R ) ) ).
% euclidean_domain.axioms(1)
thf(fact_966_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_967_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_968_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I: nat] :
( ( P @ K2 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_969_less__imp__diff__less,axiom,
! [J3: nat,K2: nat,N2: nat] :
( ( ord_less_nat @ J3 @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J3 @ N2 ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_970_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_971_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_972_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_973_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_974_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_975_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_976_le__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_977_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K2 )
= ( minus_minus_nat @ N2 @ K2 ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_978_psubset__insert__iff,axiom,
! [A2: set_nat_a,X: nat > a,B2: set_nat_a] :
( ( ord_less_set_nat_a @ A2 @ ( insert_nat_a @ X @ B2 ) )
= ( ( ( member_nat_a @ X @ B2 )
=> ( ord_less_set_nat_a @ A2 @ B2 ) )
& ( ~ ( member_nat_a @ X @ B2 )
=> ( ( ( member_nat_a @ X @ A2 )
=> ( ord_less_set_nat_a @ ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ X @ bot_bot_set_nat_a ) ) @ B2 ) )
& ( ~ ( member_nat_a @ X @ A2 )
=> ( ord_le871467723717165285_nat_a @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_979_psubset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( ( member_a @ X @ A2 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_980_psubset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( ( ( member_set_a @ X @ B2 )
=> ( ord_less_set_set_a @ A2 @ B2 ) )
& ( ~ ( member_set_a @ X @ B2 )
=> ( ( ( member_set_a @ X @ A2 )
=> ( ord_less_set_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B2 ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_981_finite__induct__select,axiom,
! [S: set_a,P: set_a > $o] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [T3: set_a] :
( ( ord_less_set_a @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X4: a] :
( ( member_a @ X4 @ ( minus_minus_set_a @ S @ T3 ) )
& ( P @ ( insert_a @ X4 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_982_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_983_diff__less__Suc,axiom,
! [M2: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_984_Suc__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_985_Suc__diff__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_986_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_987_less__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_988_lift__Suc__antimono__le,axiom,
! [F2: nat > set_a,N2: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_set_a @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_set_a @ ( F2 @ N5 ) @ ( F2 @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_989_lift__Suc__antimono__le,axiom,
! [F2: nat > nat,N2: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_nat @ ( F2 @ N5 ) @ ( F2 @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_990_lift__Suc__antimono__le,axiom,
! [F2: nat > set_set_a,N2: nat,N5: nat] :
( ! [N3: nat] : ( ord_le3724670747650509150_set_a @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_le3724670747650509150_set_a @ ( F2 @ N5 ) @ ( F2 @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_991_lift__Suc__mono__le,axiom,
! [F2: nat > set_a,N2: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_set_a @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_set_a @ ( F2 @ N2 ) @ ( F2 @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_992_lift__Suc__mono__le,axiom,
! [F2: nat > nat,N2: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_less_eq_nat @ ( F2 @ N2 ) @ ( F2 @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_993_lift__Suc__mono__le,axiom,
! [F2: nat > set_set_a,N2: nat,N5: nat] :
( ! [N3: nat] : ( ord_le3724670747650509150_set_a @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ord_le3724670747650509150_set_a @ ( F2 @ N2 ) @ ( F2 @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_994_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_995_euclidean__domain_Oeuclidean__function,axiom,
! [R: partia6043505979758434576t_unit,Phi: set_a > nat,A: set_a,B: set_a] :
( ( ring_e187967263881214398t_unit @ R @ Phi )
=> ( ( member_set_a @ A @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ( member_set_a @ B @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ? [Q2: set_a,R4: set_a] :
( ( member_set_a @ Q2 @ ( partia5907974310037520643t_unit @ R ) )
& ( member_set_a @ R4 @ ( partia5907974310037520643t_unit @ R ) )
& ( A
= ( add_se3735415688806051380t_unit @ R @ ( mult_s7930653359683758801t_unit @ R @ B @ Q2 ) @ R4 ) )
& ( ( R4
= ( zero_s2174465271003423091t_unit @ R ) )
| ( ord_less_nat @ ( Phi @ R4 ) @ ( Phi @ B ) ) ) ) ) ) ) ).
% euclidean_domain.euclidean_function
thf(fact_996_euclidean__domain_Oeuclidean__function,axiom,
! [R: partia2175431115845679010xt_a_b,Phi: a > nat,A: a,B: a] :
( ( ring_e8745995371659049232in_a_b @ R @ Phi )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ? [Q2: a,R4: a] :
( ( member_a @ Q2 @ ( partia707051561876973205xt_a_b @ R ) )
& ( member_a @ R4 @ ( partia707051561876973205xt_a_b @ R ) )
& ( A
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ B @ Q2 ) @ R4 ) )
& ( ( R4
= ( zero_a_b @ R ) )
| ( ord_less_nat @ ( Phi @ R4 ) @ ( Phi @ B ) ) ) ) ) ) ) ).
% euclidean_domain.euclidean_function
thf(fact_997_domain_Oeuclidean__domainI,axiom,
! [R: partia6043505979758434576t_unit,Phi: set_a > nat] :
( ( domain4236798911309298543t_unit @ R )
=> ( ! [A6: set_a,B6: set_a] :
( ( member_set_a @ A6 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ( member_set_a @ B6 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ? [Q3: set_a,R5: set_a] :
( ( member_set_a @ Q3 @ ( partia5907974310037520643t_unit @ R ) )
& ( member_set_a @ R5 @ ( partia5907974310037520643t_unit @ R ) )
& ( A6
= ( add_se3735415688806051380t_unit @ R @ ( mult_s7930653359683758801t_unit @ R @ B6 @ Q3 ) @ R5 ) )
& ( ( R5
= ( zero_s2174465271003423091t_unit @ R ) )
| ( ord_less_nat @ ( Phi @ R5 ) @ ( Phi @ B6 ) ) ) ) ) )
=> ( ring_e187967263881214398t_unit @ R @ Phi ) ) ) ).
% domain.euclidean_domainI
thf(fact_998_domain_Oeuclidean__domainI,axiom,
! [R: partia2175431115845679010xt_a_b,Phi: a > nat] :
( ( domain_a_b @ R )
=> ( ! [A6: a,B6: a] :
( ( member_a @ A6 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B6 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ? [Q3: a,R5: a] :
( ( member_a @ Q3 @ ( partia707051561876973205xt_a_b @ R ) )
& ( member_a @ R5 @ ( partia707051561876973205xt_a_b @ R ) )
& ( A6
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ B6 @ Q3 ) @ R5 ) )
& ( ( R5
= ( zero_a_b @ R ) )
| ( ord_less_nat @ ( Phi @ R5 ) @ ( Phi @ B6 ) ) ) ) ) )
=> ( ring_e8745995371659049232in_a_b @ R @ Phi ) ) ) ).
% domain.euclidean_domainI
thf(fact_999_order__mult__of,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( order_a_Product_unit @ ( multip3210463924028840165of_a_b @ r ) )
= ( minus_minus_nat @ ( order_a_ring_ext_a_b @ r ) @ one_one_nat ) ) ) ).
% order_mult_of
thf(fact_1000_subgroup__mult__of,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( subgro3222307229058429633t_unit @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) @ ( multip3210463924028840165of_a_b @ r ) ) ) ).
% subgroup_mult_of
thf(fact_1001_euclidean__domain__axioms__def,axiom,
( ring_e1385719875880614195ms_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,Phi2: a > nat] :
! [A7: a,B5: a] :
( ( member_a @ A7 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R3 ) @ ( insert_a @ ( zero_a_b @ R3 ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B5 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R3 ) @ ( insert_a @ ( zero_a_b @ R3 ) @ bot_bot_set_a ) ) )
=> ? [Q4: a,R6: a] :
( ( member_a @ Q4 @ ( partia707051561876973205xt_a_b @ R3 ) )
& ( member_a @ R6 @ ( partia707051561876973205xt_a_b @ R3 ) )
& ( A7
= ( add_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ B5 @ Q4 ) @ R6 ) )
& ( ( R6
= ( zero_a_b @ R3 ) )
| ( ord_less_nat @ ( Phi2 @ R6 ) @ ( Phi2 @ B5 ) ) ) ) ) ) ) ) ).
% euclidean_domain_axioms_def
thf(fact_1002_euclidean__domain__axioms__def,axiom,
( ring_e1438410220843622113t_unit
= ( ^ [R3: partia6043505979758434576t_unit,Phi2: set_a > nat] :
! [A7: set_a,B5: set_a] :
( ( member_set_a @ A7 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R3 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R3 ) @ bot_bot_set_set_a ) ) )
=> ( ( member_set_a @ B5 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R3 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R3 ) @ bot_bot_set_set_a ) ) )
=> ? [Q4: set_a,R6: set_a] :
( ( member_set_a @ Q4 @ ( partia5907974310037520643t_unit @ R3 ) )
& ( member_set_a @ R6 @ ( partia5907974310037520643t_unit @ R3 ) )
& ( A7
= ( add_se3735415688806051380t_unit @ R3 @ ( mult_s7930653359683758801t_unit @ R3 @ B5 @ Q4 ) @ R6 ) )
& ( ( R6
= ( zero_s2174465271003423091t_unit @ R3 ) )
| ( ord_less_nat @ ( Phi2 @ R6 ) @ ( Phi2 @ B5 ) ) ) ) ) ) ) ) ).
% euclidean_domain_axioms_def
thf(fact_1003_telescopic__base__aux,axiom,
! [K: set_a,F: set_a,N2: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( subfield_a_b @ F @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ F )
=> ( ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ F @ E )
=> ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E ) ) ) ) ) ).
% telescopic_base_aux
thf(fact_1004_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_1005_dimension__one,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ K @ K ) ) ).
% dimension_one
thf(fact_1006_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_1007_psubsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1008_psubsetD,axiom,
! [A2: set_nat_a,B2: set_nat_a,C: nat > a] :
( ( ord_less_set_nat_a @ A2 @ B2 )
=> ( ( member_nat_a @ C @ A2 )
=> ( member_nat_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1009_psubsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1010_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1011_euclidean__domain_Oaxioms_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,Phi: a > nat] :
( ( ring_e8745995371659049232in_a_b @ R @ Phi )
=> ( ring_e1385719875880614195ms_a_b @ R @ Phi ) ) ).
% euclidean_domain.axioms(2)
thf(fact_1012_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1013_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1014_euclidean__domain_Ointro,axiom,
! [R: partia6043505979758434576t_unit,Phi: set_a > nat] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( ring_e1438410220843622113t_unit @ R @ Phi )
=> ( ring_e187967263881214398t_unit @ R @ Phi ) ) ) ).
% euclidean_domain.intro
thf(fact_1015_euclidean__domain_Ointro,axiom,
! [R: partia2175431115845679010xt_a_b,Phi: a > nat] :
( ( domain_a_b @ R )
=> ( ( ring_e1385719875880614195ms_a_b @ R @ Phi )
=> ( ring_e8745995371659049232in_a_b @ R @ Phi ) ) ) ).
% euclidean_domain.intro
thf(fact_1016_euclidean__domain__def,axiom,
( ring_e187967263881214398t_unit
= ( ^ [R3: partia6043505979758434576t_unit,Phi2: set_a > nat] :
( ( domain4236798911309298543t_unit @ R3 )
& ( ring_e1438410220843622113t_unit @ R3 @ Phi2 ) ) ) ) ).
% euclidean_domain_def
thf(fact_1017_euclidean__domain__def,axiom,
( ring_e8745995371659049232in_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,Phi2: a > nat] :
( ( domain_a_b @ R3 )
& ( ring_e1385719875880614195ms_a_b @ R3 @ Phi2 ) ) ) ) ).
% euclidean_domain_def
thf(fact_1018_field_Osubgroup__mult__of,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( subfie5224850075530046424t_unit @ K @ R )
=> ( subgro7904897551812261217t_unit @ ( minus_5736297505244876581_set_a @ K @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) @ ( multip3774352783277980819t_unit @ R ) ) ) ) ).
% field.subgroup_mult_of
thf(fact_1019_field_Osubgroup__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( field_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( subgro3222307229058429633t_unit @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) @ ( multip3210463924028840165of_a_b @ R ) ) ) ) ).
% field.subgroup_mult_of
thf(fact_1020_field_Oorder__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( order_a_Product_unit @ ( multip3210463924028840165of_a_b @ R ) )
= ( minus_minus_nat @ ( order_a_ring_ext_a_b @ R ) @ one_one_nat ) ) ) ) ).
% field.order_mult_of
thf(fact_1021_field_Oorder__mult__of,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( finite_finite_set_a @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( order_6731527964810494035t_unit @ ( multip3774352783277980819t_unit @ R ) )
= ( minus_minus_nat @ ( order_3761338894132136606t_unit @ R ) @ one_one_nat ) ) ) ) ).
% field.order_mult_of
thf(fact_1022_euclidean__domain__axioms_Ointro,axiom,
! [R: partia2175431115845679010xt_a_b,Phi: a > nat] :
( ! [A6: a,B6: a] :
( ( member_a @ A6 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B6 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ? [Q3: a,R5: a] :
( ( member_a @ Q3 @ ( partia707051561876973205xt_a_b @ R ) )
& ( member_a @ R5 @ ( partia707051561876973205xt_a_b @ R ) )
& ( A6
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ B6 @ Q3 ) @ R5 ) )
& ( ( R5
= ( zero_a_b @ R ) )
| ( ord_less_nat @ ( Phi @ R5 ) @ ( Phi @ B6 ) ) ) ) ) )
=> ( ring_e1385719875880614195ms_a_b @ R @ Phi ) ) ).
% euclidean_domain_axioms.intro
thf(fact_1023_euclidean__domain__axioms_Ointro,axiom,
! [R: partia6043505979758434576t_unit,Phi: set_a > nat] :
( ! [A6: set_a,B6: set_a] :
( ( member_set_a @ A6 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ( member_set_a @ B6 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ? [Q3: set_a,R5: set_a] :
( ( member_set_a @ Q3 @ ( partia5907974310037520643t_unit @ R ) )
& ( member_set_a @ R5 @ ( partia5907974310037520643t_unit @ R ) )
& ( A6
= ( add_se3735415688806051380t_unit @ R @ ( mult_s7930653359683758801t_unit @ R @ B6 @ Q3 ) @ R5 ) )
& ( ( R5
= ( zero_s2174465271003423091t_unit @ R ) )
| ( ord_less_nat @ ( Phi @ R5 ) @ ( Phi @ B6 ) ) ) ) ) )
=> ( ring_e1438410220843622113t_unit @ R @ Phi ) ) ).
% euclidean_domain_axioms.intro
thf(fact_1024_subgroup_Ointro,axiom,
! [H2: set_a,G2: partia2175431115845679010xt_a_b] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ H2 )
=> ( ( member_a @ Y3 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G2 @ X2 @ Y3 ) @ H2 ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G2 ) @ H2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H2 )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ G2 @ X2 ) @ H2 ) )
=> ( subgro1816942748394427906xt_a_b @ H2 @ G2 ) ) ) ) ) ).
% subgroup.intro
thf(fact_1025_subgroup_Ointro,axiom,
! [H2: set_a,G2: partia8223610829204095565t_unit] :
( ( ord_less_eq_set_a @ H2 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ H2 )
=> ( ( member_a @ Y3 @ H2 )
=> ( member_a @ ( mult_a_Product_unit @ G2 @ X2 @ Y3 ) @ H2 ) ) )
=> ( ( member_a @ ( one_a_Product_unit @ G2 ) @ H2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H2 )
=> ( member_a @ ( m_inv_a_Product_unit @ G2 @ X2 ) @ H2 ) )
=> ( subgro3222307229058429633t_unit @ H2 @ G2 ) ) ) ) ) ).
% subgroup.intro
thf(fact_1026_subgroup_Ointro,axiom,
! [H2: set_set_a,G2: partia6043505979758434576t_unit] :
( ( ord_le3724670747650509150_set_a @ H2 @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ! [X2: set_a,Y3: set_a] :
( ( member_set_a @ X2 @ H2 )
=> ( ( member_set_a @ Y3 @ H2 )
=> ( member_set_a @ ( mult_s7930653359683758801t_unit @ G2 @ X2 @ Y3 ) @ H2 ) ) )
=> ( ( member_set_a @ ( one_se211549098623999037t_unit @ G2 ) @ H2 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ H2 )
=> ( member_set_a @ ( m_inv_7491079437187478987t_unit @ G2 @ X2 ) @ H2 ) )
=> ( subgro1896602926325872144t_unit @ H2 @ G2 ) ) ) ) ) ).
% subgroup.intro
thf(fact_1027_subgroup__def,axiom,
( subgro1816942748394427906xt_a_b
= ( ^ [H4: set_a,G3: partia2175431115845679010xt_a_b] :
( ( ord_less_eq_set_a @ H4 @ ( partia707051561876973205xt_a_b @ G3 ) )
& ! [X3: a,Y6: a] :
( ( member_a @ X3 @ H4 )
=> ( ( member_a @ Y6 @ H4 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G3 @ X3 @ Y6 ) @ H4 ) ) )
& ( member_a @ ( one_a_ring_ext_a_b @ G3 ) @ H4 )
& ! [X3: a] :
( ( member_a @ X3 @ H4 )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ G3 @ X3 ) @ H4 ) ) ) ) ) ).
% subgroup_def
thf(fact_1028_subgroup__def,axiom,
( subgro3222307229058429633t_unit
= ( ^ [H4: set_a,G3: partia8223610829204095565t_unit] :
( ( ord_less_eq_set_a @ H4 @ ( partia6735698275553448452t_unit @ G3 ) )
& ! [X3: a,Y6: a] :
( ( member_a @ X3 @ H4 )
=> ( ( member_a @ Y6 @ H4 )
=> ( member_a @ ( mult_a_Product_unit @ G3 @ X3 @ Y6 ) @ H4 ) ) )
& ( member_a @ ( one_a_Product_unit @ G3 ) @ H4 )
& ! [X3: a] :
( ( member_a @ X3 @ H4 )
=> ( member_a @ ( m_inv_a_Product_unit @ G3 @ X3 ) @ H4 ) ) ) ) ) ).
% subgroup_def
thf(fact_1029_subgroup__def,axiom,
( subgro1896602926325872144t_unit
= ( ^ [H4: set_set_a,G3: partia6043505979758434576t_unit] :
( ( ord_le3724670747650509150_set_a @ H4 @ ( partia5907974310037520643t_unit @ G3 ) )
& ! [X3: set_a,Y6: set_a] :
( ( member_set_a @ X3 @ H4 )
=> ( ( member_set_a @ Y6 @ H4 )
=> ( member_set_a @ ( mult_s7930653359683758801t_unit @ G3 @ X3 @ Y6 ) @ H4 ) ) )
& ( member_set_a @ ( one_se211549098623999037t_unit @ G3 ) @ H4 )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ H4 )
=> ( member_set_a @ ( m_inv_7491079437187478987t_unit @ G3 @ X3 ) @ H4 ) ) ) ) ) ).
% subgroup_def
thf(fact_1030_mult__of_Oinv__eq__imp__eq,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X )
= ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Y ) )
=> ( X = Y ) ) ) ) ).
% mult_of.inv_eq_imp_eq
thf(fact_1031_mult__of_Oinv__eq__one__eq,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X )
= ( one_a_ring_ext_a_b @ r ) )
= ( X
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.inv_eq_one_eq
thf(fact_1032_m__inv__mult__of,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( multip3210463924028840165of_a_b @ r ) ) )
=> ( ( m_inv_a_Product_unit @ ( multip3210463924028840165of_a_b @ r ) @ X )
= ( m_inv_a_ring_ext_a_b @ r @ X ) ) ) ).
% m_inv_mult_of
thf(fact_1033_units__of__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_Product_unit @ ( units_8174867845824275201xt_a_b @ r ) @ X )
= ( m_inv_a_ring_ext_a_b @ r @ X ) ) ) ).
% units_of_inv
thf(fact_1034_mult__of_Oinv__unique_H,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( Y
= ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X ) ) ) ) ) ) ).
% mult_of.inv_unique'
thf(fact_1035_mult__of_Oinv__char,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X )
= Y ) ) ) ) ) ).
% mult_of.inv_char
thf(fact_1036_mult__of_Ocomm__inv__char,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X )
= Y ) ) ) ) ).
% mult_of.comm_inv_char
thf(fact_1037_Ring__Divisibility_Oone__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( one_a_Product_unit @ ( ring_mult_of_a_b @ R ) )
= ( one_a_ring_ext_a_b @ R ) ) ).
% Ring_Divisibility.one_mult_of
thf(fact_1038_Multiplicative__Group_Oone__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( one_a_Product_unit @ ( multip3210463924028840165of_a_b @ R ) )
= ( one_a_ring_ext_a_b @ R ) ) ).
% Multiplicative_Group.one_mult_of
thf(fact_1039_mult__of_Oinv__one,axiom,
( ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ ( one_a_ring_ext_a_b @ r ) )
= ( one_a_ring_ext_a_b @ r ) ) ).
% mult_of.inv_one
thf(fact_1040_mult__of_OUnits__inv__Units,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.Units_inv_Units
thf(fact_1041_mult__of_OUnits__inv__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X ) )
= X ) ) ).
% mult_of.Units_inv_inv
thf(fact_1042_mult__of_OUnits__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.Units_inv_closed
thf(fact_1043_mult__of_OUnits__l__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X ) @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% mult_of.Units_l_inv
thf(fact_1044_mult__of_OUnits__r__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% mult_of.Units_r_inv
thf(fact_1045_field_Om__inv__mult__of,axiom,
! [R: partia6043505979758434576t_unit,X: set_a] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( member_set_a @ X @ ( partia8299590604543202116t_unit @ ( multip3774352783277980819t_unit @ R ) ) )
=> ( ( m_inv_3738623195918084710t_unit @ ( multip3774352783277980819t_unit @ R ) @ X )
= ( m_inv_7491079437187478987t_unit @ R @ X ) ) ) ) ).
% field.m_inv_mult_of
thf(fact_1046_field_Om__inv__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( field_a_b @ R )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( multip3210463924028840165of_a_b @ R ) ) )
=> ( ( m_inv_a_Product_unit @ ( multip3210463924028840165of_a_b @ R ) @ X )
= ( m_inv_a_ring_ext_a_b @ R @ X ) ) ) ) ).
% field.m_inv_mult_of
thf(fact_1047_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1048_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1049_subgroup_Omem__carrier,axiom,
! [H2: set_a,G2: partia2175431115845679010xt_a_b,X: a] :
( ( subgro1816942748394427906xt_a_b @ H2 @ G2 )
=> ( ( member_a @ X @ H2 )
=> ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ).
% subgroup.mem_carrier
thf(fact_1050_subgroup_Omem__carrier,axiom,
! [H2: set_a,G2: partia8223610829204095565t_unit,X: a] :
( ( subgro3222307229058429633t_unit @ H2 @ G2 )
=> ( ( member_a @ X @ H2 )
=> ( member_a @ X @ ( partia6735698275553448452t_unit @ G2 ) ) ) ) ).
% subgroup.mem_carrier
thf(fact_1051_subgroup_Omem__carrier,axiom,
! [H2: set_set_a,G2: partia6043505979758434576t_unit,X: set_a] :
( ( subgro1896602926325872144t_unit @ H2 @ G2 )
=> ( ( member_set_a @ X @ H2 )
=> ( member_set_a @ X @ ( partia5907974310037520643t_unit @ G2 ) ) ) ) ).
% subgroup.mem_carrier
thf(fact_1052_subgroup__nonempty,axiom,
! [G2: partia8223610829204095565t_unit] :
~ ( subgro3222307229058429633t_unit @ bot_bot_set_a @ G2 ) ).
% subgroup_nonempty
thf(fact_1053_subgroup_Oone__closed,axiom,
! [H2: set_a,G2: partia8223610829204095565t_unit] :
( ( subgro3222307229058429633t_unit @ H2 @ G2 )
=> ( member_a @ ( one_a_Product_unit @ G2 ) @ H2 ) ) ).
% subgroup.one_closed
thf(fact_1054_subgroup_Oone__closed,axiom,
! [H2: set_a,G2: partia2175431115845679010xt_a_b] :
( ( subgro1816942748394427906xt_a_b @ H2 @ G2 )
=> ( member_a @ ( one_a_ring_ext_a_b @ G2 ) @ H2 ) ) ).
% subgroup.one_closed
thf(fact_1055_units__of__one,axiom,
! [G2: partia8223610829204095565t_unit] :
( ( one_a_Product_unit @ ( units_7501539392726747778t_unit @ G2 ) )
= ( one_a_Product_unit @ G2 ) ) ).
% units_of_one
thf(fact_1056_units__of__one,axiom,
! [G2: partia2175431115845679010xt_a_b] :
( ( one_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G2 ) )
= ( one_a_ring_ext_a_b @ G2 ) ) ).
% units_of_one
thf(fact_1057_subgroup_Osubset,axiom,
! [H2: set_a,G2: partia2175431115845679010xt_a_b] :
( ( subgro1816942748394427906xt_a_b @ H2 @ G2 )
=> ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ).
% subgroup.subset
thf(fact_1058_subgroup_Osubset,axiom,
! [H2: set_a,G2: partia8223610829204095565t_unit] :
( ( subgro3222307229058429633t_unit @ H2 @ G2 )
=> ( ord_less_eq_set_a @ H2 @ ( partia6735698275553448452t_unit @ G2 ) ) ) ).
% subgroup.subset
thf(fact_1059_subgroup_Osubset,axiom,
! [H2: set_set_a,G2: partia6043505979758434576t_unit] :
( ( subgro1896602926325872144t_unit @ H2 @ G2 )
=> ( ord_le3724670747650509150_set_a @ H2 @ ( partia5907974310037520643t_unit @ G2 ) ) ) ).
% subgroup.subset
thf(fact_1060_mult__of_Ounits__of__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( m_inv_a_Product_unit @ ( units_7501539392726747778t_unit @ ( ring_mult_of_a_b @ r ) ) @ X )
= ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X ) ) ) ).
% mult_of.units_of_inv
thf(fact_1061_mult__of_Ogcd__isgcd,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ A @ B ) ) ) ).
% mult_of.gcd_isgcd
thf(fact_1062_mult__of_Ogcd__divides,axiom,
! [Z: a,X: a,Y: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Z @ X )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Z @ Y )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Z @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X @ Y ) ) ) ) ) ) ) ).
% mult_of.gcd_divides
thf(fact_1063_mult__of_Ogcd__exists,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.gcd_exists
thf(fact_1064_mult__of_Ogcd__closed,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.gcd_closed
thf(fact_1065_mult__of_Ogcd__divides__r,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ B ) ) ) ).
% mult_of.gcd_divides_r
thf(fact_1066_mult__of_Ogcd__divides__l,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ A ) ) ) ).
% mult_of.gcd_divides_l
thf(fact_1067_mult__of_Orelprime__mult,axiom,
! [A: a,B: a,C: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ ( one_a_ring_ext_a_b @ r ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ C ) @ ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.relprime_mult
thf(fact_1068_mult__of_Ogcd__condition__monoid__axioms,axiom,
gcd_co701944698663231555t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.gcd_condition_monoid_axioms
thf(fact_1069_bound__upD,axiom,
! [F2: nat > a] :
( ( member_nat_a @ F2 @ ( up_a_b @ r ) )
=> ? [N3: nat] : ( bound_a @ ( zero_a_b @ r ) @ N3 @ F2 ) ) ).
% bound_upD
thf(fact_1070_mult__of_Oassociated__sym,axiom,
! [A: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ A ) ) ).
% mult_of.associated_sym
thf(fact_1071_mult__of_Oassoc__subst,axiom,
! [A: a,B: a,F2: a > a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ! [A6: a,B6: a] :
( ( ( member_a @ A6 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( member_a @ B6 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A6 @ B6 ) )
=> ( ( member_a @ ( F2 @ A6 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( member_a @ ( F2 @ B6 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( F2 @ A6 ) @ ( F2 @ B6 ) ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( F2 @ A ) @ ( F2 @ B ) ) ) ) ) ) ).
% mult_of.assoc_subst
thf(fact_1072_mult__of_Oassociated__trans,axiom,
! [A: a,B: a,C: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ C ) ) ) ) ) ).
% mult_of.associated_trans
thf(fact_1073_mult__of_OUnits__assoc,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ).
% mult_of.Units_assoc
thf(fact_1074_mult__of_Omult__cong__r,axiom,
! [B: a,B8: a,A: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ B8 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B8 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B8 ) ) ) ) ) ) ).
% mult_of.mult_cong_r
thf(fact_1075_mult__of_Omult__cong__l,axiom,
! [A: a,A8: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A8 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A8 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A8 @ B ) ) ) ) ) ) ).
% mult_of.mult_cong_l
thf(fact_1076_mult__of_Oassoc__r__cancel,axiom,
! [A: a,B: a,A8: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A8 @ B ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A8 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A8 ) ) ) ) ) ).
% mult_of.assoc_r_cancel
thf(fact_1077_mult__of_Oassoc__l__cancel,axiom,
! [A: a,B: a,B8: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B8 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B8 ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ B8 ) ) ) ) ) ).
% mult_of.assoc_l_cancel
thf(fact_1078_mult__of_Odivides__cong__l,axiom,
! [X: a,X6: a,Y: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ X @ X6 )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X6 @ Y )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X @ Y ) ) ) ) ).
% mult_of.divides_cong_l
thf(fact_1079_mult__of_Odivides__cong__r,axiom,
! [X: a,Y: a,Y4: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X @ Y )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ Y @ Y4 )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X @ Y4 ) ) ) ) ).
% mult_of.divides_cong_r
thf(fact_1080_mult__of_Oassoc__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.assoc_unit_r
thf(fact_1081_mult__of_Oassoc__unit__l,axiom,
! [A: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.assoc_unit_l
thf(fact_1082_mult__of_Oirreducible__cong,axiom,
! [A: a,A8: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A8 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A8 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A8 ) ) ) ) ) ).
% mult_of.irreducible_cong
thf(fact_1083_mult__of_Ogcd__cong__r,axiom,
! [Y: a,Y4: a,X: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ Y @ Y4 )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X @ Y ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X @ Y4 ) ) ) ) ) ) ).
% mult_of.gcd_cong_r
thf(fact_1084_mult__of_Ogcd__cong__l,axiom,
! [X: a,X6: a,Y: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ X @ X6 )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ X6 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X @ Y ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X6 @ Y ) ) ) ) ) ) ).
% mult_of.gcd_cong_l
thf(fact_1085_mult__of_Ogcd__assoc,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ C ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C ) ) ) ) ) ) ).
% mult_of.gcd_assoc
thf(fact_1086_mult__of_Oprime__cong,axiom,
! [P2: a,P4: a] :
( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P2 )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ P2 @ P4 )
=> ( ( member_a @ P2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ P4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P4 ) ) ) ) ) ).
% mult_of.prime_cong
thf(fact_1087_mult__of_Oassociated__fcount,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ A )
= ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ B ) ) ) ) ) ).
% mult_of.associated_fcount
thf(fact_1088_mult__of_Ogcdof__cong__l,axiom,
! [A8: a,A: a,B: a,C: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A8 @ A )
=> ( ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A @ B @ C )
=> ( ( member_a @ A8 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A8 @ B @ C ) ) ) ) ) ) ) ).
% mult_of.gcdof_cong_l
thf(fact_1089_mult__of_Oassociated__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( A
= ( mult_a_ring_ext_a_b @ r @ B @ X3 ) ) ) ) ) ) ) ).
% mult_of.associated_iff
thf(fact_1090_mult__of_OassociatedI2_H,axiom,
! [A: a,B: a,U: a] :
( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ).
% mult_of.associatedI2'
thf(fact_1091_mult__of_OassociatedI2,axiom,
! [U: a,A: a,B: a] :
( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ).
% mult_of.associatedI2
thf(fact_1092_mult__of_OassociatedE2,axiom,
! [A: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ! [U2: a] :
( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ~ ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.associatedE2
thf(fact_1093_mult__of_OassociatedD2,axiom,
! [A: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( A
= ( mult_a_ring_ext_a_b @ r @ B @ X2 ) ) ) ) ) ) ).
% mult_of.associatedD2
thf(fact_1094_mult__of_Ogcd__mult,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ) ).
% mult_of.gcd_mult
thf(fact_1095_mult__of_OgcdI,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ C )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Y3 @ B )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Y3 @ C )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Y3 @ A ) ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C ) ) ) ) ) ) ) ) ).
% mult_of.gcdI
thf(fact_1096_mult__of_OgcdI2,axiom,
! [A: a,B: a,C: a] :
( ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A @ B @ C )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C ) ) ) ) ) ) ).
% mult_of.gcdI2
thf(fact_1097_mult__of_Oassociated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A ) ) ).
% mult_of.associated_refl
thf(fact_1098_gcd__condition__monoid_Ogcd__mult,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( gcd_co8605360636473864704xt_a_b @ G2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( associ5860276527279195403xt_a_b @ G2 @ ( mult_a_ring_ext_a_b @ G2 @ C @ ( somegc1600592057159103747xt_a_b @ G2 @ A @ B ) ) @ ( somegc1600592057159103747xt_a_b @ G2 @ ( mult_a_ring_ext_a_b @ G2 @ C @ A ) @ ( mult_a_ring_ext_a_b @ G2 @ C @ B ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_mult
thf(fact_1099_gcd__condition__monoid_Ogcd__mult,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a,C: set_a] :
( ( gcd_co1135742393622391822t_unit @ G2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( associ6060417466526650393t_unit @ G2 @ ( mult_s7930653359683758801t_unit @ G2 @ C @ ( somegc5022794125158426641t_unit @ G2 @ A @ B ) ) @ ( somegc5022794125158426641t_unit @ G2 @ ( mult_s7930653359683758801t_unit @ G2 @ C @ A ) @ ( mult_s7930653359683758801t_unit @ G2 @ C @ B ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_mult
thf(fact_1100_gcd__condition__monoid_Ogcd__mult,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( gcd_co701944698663231555t_unit @ G2 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( associ6879500422977059064t_unit @ G2 @ ( mult_a_Product_unit @ G2 @ C @ ( somegc8962790057355718400t_unit @ G2 @ A @ B ) ) @ ( somegc8962790057355718400t_unit @ G2 @ ( mult_a_Product_unit @ G2 @ C @ A ) @ ( mult_a_Product_unit @ G2 @ C @ B ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_mult
thf(fact_1101_gcd__condition__monoid_OgcdI,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( gcd_co8605360636473864704xt_a_b @ G2 )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ A @ B )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ A @ C )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ Y3 @ B )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ Y3 @ C )
=> ( factor8216151070175719842xt_a_b @ G2 @ Y3 @ A ) ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( associ5860276527279195403xt_a_b @ G2 @ A @ ( somegc1600592057159103747xt_a_b @ G2 @ B @ C ) ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcdI
thf(fact_1102_gcd__condition__monoid_OgcdI,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a,C: set_a] :
( ( gcd_co1135742393622391822t_unit @ G2 )
=> ( ( factor5460682277579321776t_unit @ G2 @ A @ B )
=> ( ( factor5460682277579321776t_unit @ G2 @ A @ C )
=> ( ! [Y3: set_a] :
( ( member_set_a @ Y3 @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( factor5460682277579321776t_unit @ G2 @ Y3 @ B )
=> ( ( factor5460682277579321776t_unit @ G2 @ Y3 @ C )
=> ( factor5460682277579321776t_unit @ G2 @ Y3 @ A ) ) ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( associ6060417466526650393t_unit @ G2 @ A @ ( somegc5022794125158426641t_unit @ G2 @ B @ C ) ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcdI
thf(fact_1103_gcd__condition__monoid_OgcdI,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( gcd_co701944698663231555t_unit @ G2 )
=> ( ( factor3040189038382604065t_unit @ G2 @ A @ B )
=> ( ( factor3040189038382604065t_unit @ G2 @ A @ C )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( factor3040189038382604065t_unit @ G2 @ Y3 @ B )
=> ( ( factor3040189038382604065t_unit @ G2 @ Y3 @ C )
=> ( factor3040189038382604065t_unit @ G2 @ Y3 @ A ) ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( associ6879500422977059064t_unit @ G2 @ A @ ( somegc8962790057355718400t_unit @ G2 @ B @ C ) ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcdI
thf(fact_1104_gcd__condition__monoid_OgcdI2,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( gcd_co8605360636473864704xt_a_b @ G2 )
=> ( ( isgcd_a_ring_ext_a_b @ G2 @ A @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( associ5860276527279195403xt_a_b @ G2 @ A @ ( somegc1600592057159103747xt_a_b @ G2 @ B @ C ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcdI2
thf(fact_1105_gcd__condition__monoid_OgcdI2,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a,C: set_a] :
( ( gcd_co1135742393622391822t_unit @ G2 )
=> ( ( isgcd_8277756548069700071t_unit @ G2 @ A @ B @ C )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( associ6060417466526650393t_unit @ G2 @ A @ ( somegc5022794125158426641t_unit @ G2 @ B @ C ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcdI2
thf(fact_1106_gcd__condition__monoid_OgcdI2,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( gcd_co701944698663231555t_unit @ G2 )
=> ( ( isgcd_a_Product_unit @ G2 @ A @ B @ C )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( associ6879500422977059064t_unit @ G2 @ A @ ( somegc8962790057355718400t_unit @ G2 @ B @ C ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcdI2
thf(fact_1107_gcd__condition__monoid_Ogcd__assoc,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( gcd_co8605360636473864704xt_a_b @ G2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( associ5860276527279195403xt_a_b @ G2 @ ( somegc1600592057159103747xt_a_b @ G2 @ ( somegc1600592057159103747xt_a_b @ G2 @ A @ B ) @ C ) @ ( somegc1600592057159103747xt_a_b @ G2 @ A @ ( somegc1600592057159103747xt_a_b @ G2 @ B @ C ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_assoc
thf(fact_1108_gcd__condition__monoid_Ogcd__assoc,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a,C: set_a] :
( ( gcd_co1135742393622391822t_unit @ G2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( associ6060417466526650393t_unit @ G2 @ ( somegc5022794125158426641t_unit @ G2 @ ( somegc5022794125158426641t_unit @ G2 @ A @ B ) @ C ) @ ( somegc5022794125158426641t_unit @ G2 @ A @ ( somegc5022794125158426641t_unit @ G2 @ B @ C ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_assoc
thf(fact_1109_gcd__condition__monoid_Ogcd__assoc,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( gcd_co701944698663231555t_unit @ G2 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( associ6879500422977059064t_unit @ G2 @ ( somegc8962790057355718400t_unit @ G2 @ ( somegc8962790057355718400t_unit @ G2 @ A @ B ) @ C ) @ ( somegc8962790057355718400t_unit @ G2 @ A @ ( somegc8962790057355718400t_unit @ G2 @ B @ C ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_assoc
thf(fact_1110_gcd__condition__monoid_Ogcd__cong__l,axiom,
! [G2: partia2175431115845679010xt_a_b,X: a,X6: a,Y: a] :
( ( gcd_co8605360636473864704xt_a_b @ G2 )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ X @ X6 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ X6 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( associ5860276527279195403xt_a_b @ G2 @ ( somegc1600592057159103747xt_a_b @ G2 @ X @ Y ) @ ( somegc1600592057159103747xt_a_b @ G2 @ X6 @ Y ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_cong_l
thf(fact_1111_gcd__condition__monoid_Ogcd__cong__l,axiom,
! [G2: partia6043505979758434576t_unit,X: set_a,X6: set_a,Y: set_a] :
( ( gcd_co1135742393622391822t_unit @ G2 )
=> ( ( associ6060417466526650393t_unit @ G2 @ X @ X6 )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ X6 @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( associ6060417466526650393t_unit @ G2 @ ( somegc5022794125158426641t_unit @ G2 @ X @ Y ) @ ( somegc5022794125158426641t_unit @ G2 @ X6 @ Y ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_cong_l
thf(fact_1112_gcd__condition__monoid_Ogcd__cong__l,axiom,
! [G2: partia8223610829204095565t_unit,X: a,X6: a,Y: a] :
( ( gcd_co701944698663231555t_unit @ G2 )
=> ( ( associ6879500422977059064t_unit @ G2 @ X @ X6 )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ X6 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( associ6879500422977059064t_unit @ G2 @ ( somegc8962790057355718400t_unit @ G2 @ X @ Y ) @ ( somegc8962790057355718400t_unit @ G2 @ X6 @ Y ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_cong_l
thf(fact_1113_gcd__condition__monoid_Ogcd__cong__r,axiom,
! [G2: partia2175431115845679010xt_a_b,Y: a,Y4: a,X: a] :
( ( gcd_co8605360636473864704xt_a_b @ G2 )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ Y @ Y4 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( associ5860276527279195403xt_a_b @ G2 @ ( somegc1600592057159103747xt_a_b @ G2 @ X @ Y ) @ ( somegc1600592057159103747xt_a_b @ G2 @ X @ Y4 ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_cong_r
thf(fact_1114_gcd__condition__monoid_Ogcd__cong__r,axiom,
! [G2: partia6043505979758434576t_unit,Y: set_a,Y4: set_a,X: set_a] :
( ( gcd_co1135742393622391822t_unit @ G2 )
=> ( ( associ6060417466526650393t_unit @ G2 @ Y @ Y4 )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ Y4 @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( associ6060417466526650393t_unit @ G2 @ ( somegc5022794125158426641t_unit @ G2 @ X @ Y ) @ ( somegc5022794125158426641t_unit @ G2 @ X @ Y4 ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_cong_r
thf(fact_1115_gcd__condition__monoid_Ogcd__cong__r,axiom,
! [G2: partia8223610829204095565t_unit,Y: a,Y4: a,X: a] :
( ( gcd_co701944698663231555t_unit @ G2 )
=> ( ( associ6879500422977059064t_unit @ G2 @ Y @ Y4 )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ Y4 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( associ6879500422977059064t_unit @ G2 @ ( somegc8962790057355718400t_unit @ G2 @ X @ Y ) @ ( somegc8962790057355718400t_unit @ G2 @ X @ Y4 ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_cong_r
thf(fact_1116_gcd__condition__monoid_Ogcdof__cong__l,axiom,
! [G2: partia2175431115845679010xt_a_b,A8: a,A: a,B: a,C: a] :
( ( gcd_co8605360636473864704xt_a_b @ G2 )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ A8 @ A )
=> ( ( isgcd_a_ring_ext_a_b @ G2 @ A @ B @ C )
=> ( ( member_a @ A8 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( isgcd_a_ring_ext_a_b @ G2 @ A8 @ B @ C ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcdof_cong_l
thf(fact_1117_gcd__condition__monoid_Ogcdof__cong__l,axiom,
! [G2: partia6043505979758434576t_unit,A8: set_a,A: set_a,B: set_a,C: set_a] :
( ( gcd_co1135742393622391822t_unit @ G2 )
=> ( ( associ6060417466526650393t_unit @ G2 @ A8 @ A )
=> ( ( isgcd_8277756548069700071t_unit @ G2 @ A @ B @ C )
=> ( ( member_set_a @ A8 @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( isgcd_8277756548069700071t_unit @ G2 @ A8 @ B @ C ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcdof_cong_l
thf(fact_1118_gcd__condition__monoid_Ogcdof__cong__l,axiom,
! [G2: partia8223610829204095565t_unit,A8: a,A: a,B: a,C: a] :
( ( gcd_co701944698663231555t_unit @ G2 )
=> ( ( associ6879500422977059064t_unit @ G2 @ A8 @ A )
=> ( ( isgcd_a_Product_unit @ G2 @ A @ B @ C )
=> ( ( member_a @ A8 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( isgcd_a_Product_unit @ G2 @ A8 @ B @ C ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcdof_cong_l
thf(fact_1119_gcd__condition__monoid_Orelprime__mult,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( gcd_co8605360636473864704xt_a_b @ G2 )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ ( somegc1600592057159103747xt_a_b @ G2 @ A @ B ) @ ( one_a_ring_ext_a_b @ G2 ) )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ ( somegc1600592057159103747xt_a_b @ G2 @ A @ C ) @ ( one_a_ring_ext_a_b @ G2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( associ5860276527279195403xt_a_b @ G2 @ ( somegc1600592057159103747xt_a_b @ G2 @ A @ ( mult_a_ring_ext_a_b @ G2 @ B @ C ) ) @ ( one_a_ring_ext_a_b @ G2 ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.relprime_mult
thf(fact_1120_gcd__condition__monoid_Orelprime__mult,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a,C: set_a] :
( ( gcd_co1135742393622391822t_unit @ G2 )
=> ( ( associ6060417466526650393t_unit @ G2 @ ( somegc5022794125158426641t_unit @ G2 @ A @ B ) @ ( one_se211549098623999037t_unit @ G2 ) )
=> ( ( associ6060417466526650393t_unit @ G2 @ ( somegc5022794125158426641t_unit @ G2 @ A @ C ) @ ( one_se211549098623999037t_unit @ G2 ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( associ6060417466526650393t_unit @ G2 @ ( somegc5022794125158426641t_unit @ G2 @ A @ ( mult_s7930653359683758801t_unit @ G2 @ B @ C ) ) @ ( one_se211549098623999037t_unit @ G2 ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.relprime_mult
thf(fact_1121_gcd__condition__monoid_Orelprime__mult,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( gcd_co701944698663231555t_unit @ G2 )
=> ( ( associ6879500422977059064t_unit @ G2 @ ( somegc8962790057355718400t_unit @ G2 @ A @ B ) @ ( one_a_Product_unit @ G2 ) )
=> ( ( associ6879500422977059064t_unit @ G2 @ ( somegc8962790057355718400t_unit @ G2 @ A @ C ) @ ( one_a_Product_unit @ G2 ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( associ6879500422977059064t_unit @ G2 @ ( somegc8962790057355718400t_unit @ G2 @ A @ ( mult_a_Product_unit @ G2 @ B @ C ) ) @ ( one_a_Product_unit @ G2 ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.relprime_mult
thf(fact_1122_monoid__cancel_Oassoc__l__cancel,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a,B8: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B8 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ ( mult_a_ring_ext_a_b @ G2 @ A @ B ) @ ( mult_a_ring_ext_a_b @ G2 @ A @ B8 ) )
=> ( associ5860276527279195403xt_a_b @ G2 @ B @ B8 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_1123_monoid__cancel_Oassoc__l__cancel,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a,B8: a] :
( ( monoid1999574367301118026t_unit @ G2 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B8 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( associ6879500422977059064t_unit @ G2 @ ( mult_a_Product_unit @ G2 @ A @ B ) @ ( mult_a_Product_unit @ G2 @ A @ B8 ) )
=> ( associ6879500422977059064t_unit @ G2 @ B @ B8 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_1124_monoid__cancel_Oassoc__l__cancel,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a,B8: set_a] :
( ( monoid759071382269063239t_unit @ G2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B8 @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( associ6060417466526650393t_unit @ G2 @ ( mult_s7930653359683758801t_unit @ G2 @ A @ B ) @ ( mult_s7930653359683758801t_unit @ G2 @ A @ B8 ) )
=> ( associ6060417466526650393t_unit @ G2 @ B @ B8 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_1125_gcd__condition__monoid_Ogcd__exists,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( gcd_co8605360636473864704xt_a_b @ G2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( member_a @ ( somegc1600592057159103747xt_a_b @ G2 @ A @ B ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ) ).
% gcd_condition_monoid.gcd_exists
thf(fact_1126_gcd__condition__monoid_Ogcd__exists,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( gcd_co1135742393622391822t_unit @ G2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( member_set_a @ ( somegc5022794125158426641t_unit @ G2 @ A @ B ) @ ( partia5907974310037520643t_unit @ G2 ) ) ) ) ) ).
% gcd_condition_monoid.gcd_exists
thf(fact_1127_gcd__condition__monoid_Ogcd__exists,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a] :
( ( gcd_co701944698663231555t_unit @ G2 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( member_a @ ( somegc8962790057355718400t_unit @ G2 @ A @ B ) @ ( partia6735698275553448452t_unit @ G2 ) ) ) ) ) ).
% gcd_condition_monoid.gcd_exists
thf(fact_1128_gcd__condition__monoid_Ogcd__closed,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( gcd_co8605360636473864704xt_a_b @ G2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( member_a @ ( somegc1600592057159103747xt_a_b @ G2 @ A @ B ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ) ).
% gcd_condition_monoid.gcd_closed
thf(fact_1129_gcd__condition__monoid_Ogcd__closed,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( gcd_co1135742393622391822t_unit @ G2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( member_set_a @ ( somegc5022794125158426641t_unit @ G2 @ A @ B ) @ ( partia5907974310037520643t_unit @ G2 ) ) ) ) ) ).
% gcd_condition_monoid.gcd_closed
thf(fact_1130_gcd__condition__monoid_Ogcd__closed,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a] :
( ( gcd_co701944698663231555t_unit @ G2 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( member_a @ ( somegc8962790057355718400t_unit @ G2 @ A @ B ) @ ( partia6735698275553448452t_unit @ G2 ) ) ) ) ) ).
% gcd_condition_monoid.gcd_closed
thf(fact_1131_monoid__cancel_Oassoc__unit__r,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G2 ) )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ A @ B )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ G2 ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_1132_monoid__cancel_Oassoc__unit__r,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid1999574367301118026t_unit @ G2 )
=> ( ( member_a @ A @ ( units_a_Product_unit @ G2 ) )
=> ( ( associ6879500422977059064t_unit @ G2 @ A @ B )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( member_a @ B @ ( units_a_Product_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_1133_monoid__cancel_Oassoc__unit__r,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( monoid759071382269063239t_unit @ G2 )
=> ( ( member_set_a @ A @ ( units_2471184348132832486t_unit @ G2 ) )
=> ( ( associ6060417466526650393t_unit @ G2 @ A @ B )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( member_set_a @ B @ ( units_2471184348132832486t_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_1134_monoid__cancel_Oassoc__unit__l,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ A @ B )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ G2 ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_1135_monoid__cancel_Oassoc__unit__l,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid1999574367301118026t_unit @ G2 )
=> ( ( associ6879500422977059064t_unit @ G2 @ A @ B )
=> ( ( member_a @ B @ ( units_a_Product_unit @ G2 ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( member_a @ A @ ( units_a_Product_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_1136_monoid__cancel_Oassoc__unit__l,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( monoid759071382269063239t_unit @ G2 )
=> ( ( associ6060417466526650393t_unit @ G2 @ A @ B )
=> ( ( member_set_a @ B @ ( units_2471184348132832486t_unit @ G2 ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( member_set_a @ A @ ( units_2471184348132832486t_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_1137_gcd__condition__monoid_Ogcdof__exists,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( gcd_co8605360636473864704xt_a_b @ G2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ? [C5: a] :
( ( member_a @ C5 @ ( partia707051561876973205xt_a_b @ G2 ) )
& ( isgcd_a_ring_ext_a_b @ G2 @ C5 @ A @ B ) ) ) ) ) ).
% gcd_condition_monoid.gcdof_exists
thf(fact_1138_gcd__condition__monoid_Ogcdof__exists,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( gcd_co1135742393622391822t_unit @ G2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ? [C5: set_a] :
( ( member_set_a @ C5 @ ( partia5907974310037520643t_unit @ G2 ) )
& ( isgcd_8277756548069700071t_unit @ G2 @ C5 @ A @ B ) ) ) ) ) ).
% gcd_condition_monoid.gcdof_exists
thf(fact_1139_gcd__condition__monoid_Ogcdof__exists,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a] :
( ( gcd_co701944698663231555t_unit @ G2 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ? [C5: a] :
( ( member_a @ C5 @ ( partia6735698275553448452t_unit @ G2 ) )
& ( isgcd_a_Product_unit @ G2 @ C5 @ A @ B ) ) ) ) ) ).
% gcd_condition_monoid.gcdof_exists
thf(fact_1140_monoid__cancel_Oirreducible__cong,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,A8: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( irredu6211895646901577903xt_a_b @ G2 @ A )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ A @ A8 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ A8 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( irredu6211895646901577903xt_a_b @ G2 @ A8 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_1141_monoid__cancel_Oirreducible__cong,axiom,
! [G2: partia8223610829204095565t_unit,A: a,A8: a] :
( ( monoid1999574367301118026t_unit @ G2 )
=> ( ( irredu4023057619401689684t_unit @ G2 @ A )
=> ( ( associ6879500422977059064t_unit @ G2 @ A @ A8 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ A8 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( irredu4023057619401689684t_unit @ G2 @ A8 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_1142_monoid__cancel_Oirreducible__cong,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,A8: set_a] :
( ( monoid759071382269063239t_unit @ G2 )
=> ( ( irredu5346329325703585725t_unit @ G2 @ A )
=> ( ( associ6060417466526650393t_unit @ G2 @ A @ A8 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ A8 @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( irredu5346329325703585725t_unit @ G2 @ A8 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_1143_monoid__cancel_Oprime__cong,axiom,
! [G2: partia2175431115845679010xt_a_b,P2: a,P4: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( prime_a_ring_ext_a_b @ G2 @ P2 )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ P2 @ P4 )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ P4 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( prime_a_ring_ext_a_b @ G2 @ P4 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_1144_monoid__cancel_Oprime__cong,axiom,
! [G2: partia8223610829204095565t_unit,P2: a,P4: a] :
( ( monoid1999574367301118026t_unit @ G2 )
=> ( ( prime_a_Product_unit @ G2 @ P2 )
=> ( ( associ6879500422977059064t_unit @ G2 @ P2 @ P4 )
=> ( ( member_a @ P2 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ P4 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( prime_a_Product_unit @ G2 @ P4 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_1145_monoid__cancel_Oprime__cong,axiom,
! [G2: partia6043505979758434576t_unit,P2: set_a,P4: set_a] :
( ( monoid759071382269063239t_unit @ G2 )
=> ( ( prime_4522187476880896870t_unit @ G2 @ P2 )
=> ( ( associ6060417466526650393t_unit @ G2 @ P2 @ P4 )
=> ( ( member_set_a @ P2 @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ P4 @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( prime_4522187476880896870t_unit @ G2 @ P4 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_1146_divides__irreducible__condition,axiom,
! [G2: partia2175431115845679010xt_a_b,R2: a,A: a] :
( ( irredu6211895646901577903xt_a_b @ G2 @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ A @ R2 )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G2 ) )
| ( associ5860276527279195403xt_a_b @ G2 @ A @ R2 ) ) ) ) ) ).
% divides_irreducible_condition
thf(fact_1147_divides__irreducible__condition,axiom,
! [G2: partia8223610829204095565t_unit,R2: a,A: a] :
( ( irredu4023057619401689684t_unit @ G2 @ R2 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( factor3040189038382604065t_unit @ G2 @ A @ R2 )
=> ( ( member_a @ A @ ( units_a_Product_unit @ G2 ) )
| ( associ6879500422977059064t_unit @ G2 @ A @ R2 ) ) ) ) ) ).
% divides_irreducible_condition
thf(fact_1148_divides__irreducible__condition,axiom,
! [G2: partia6043505979758434576t_unit,R2: set_a,A: set_a] :
( ( irredu5346329325703585725t_unit @ G2 @ R2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( factor5460682277579321776t_unit @ G2 @ A @ R2 )
=> ( ( member_set_a @ A @ ( units_2471184348132832486t_unit @ G2 ) )
| ( associ6060417466526650393t_unit @ G2 @ A @ R2 ) ) ) ) ) ).
% divides_irreducible_condition
thf(fact_1149_monoid__cancel_OassociatedD2,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ G2 ) )
& ( A
= ( mult_a_ring_ext_a_b @ G2 @ B @ X2 ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_1150_monoid__cancel_OassociatedD2,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid1999574367301118026t_unit @ G2 )
=> ( ( associ6879500422977059064t_unit @ G2 @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( units_a_Product_unit @ G2 ) )
& ( A
= ( mult_a_Product_unit @ G2 @ B @ X2 ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_1151_monoid__cancel_OassociatedD2,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( monoid759071382269063239t_unit @ G2 )
=> ( ( associ6060417466526650393t_unit @ G2 @ A @ B )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ ( units_2471184348132832486t_unit @ G2 ) )
& ( A
= ( mult_s7930653359683758801t_unit @ G2 @ B @ X2 ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_1152_monoid__cancel_OassociatedE2,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ A @ B )
=> ( ! [U2: a] :
( ( A
= ( mult_a_ring_ext_a_b @ G2 @ B @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_ring_ext_a_b @ G2 ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ~ ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_1153_monoid__cancel_OassociatedE2,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid1999574367301118026t_unit @ G2 )
=> ( ( associ6879500422977059064t_unit @ G2 @ A @ B )
=> ( ! [U2: a] :
( ( A
= ( mult_a_Product_unit @ G2 @ B @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_Product_unit @ G2 ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ~ ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_1154_monoid__cancel_OassociatedE2,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( monoid759071382269063239t_unit @ G2 )
=> ( ( associ6060417466526650393t_unit @ G2 @ A @ B )
=> ( ! [U2: set_a] :
( ( A
= ( mult_s7930653359683758801t_unit @ G2 @ B @ U2 ) )
=> ~ ( member_set_a @ U2 @ ( units_2471184348132832486t_unit @ G2 ) ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ~ ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_1155_monoid__cancel_Oassociated__iff,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( associ5860276527279195403xt_a_b @ G2 @ A @ B )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ G2 ) )
& ( A
= ( mult_a_ring_ext_a_b @ G2 @ B @ X3 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_1156_monoid__cancel_Oassociated__iff,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a] :
( ( monoid1999574367301118026t_unit @ G2 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( associ6879500422977059064t_unit @ G2 @ A @ B )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( units_a_Product_unit @ G2 ) )
& ( A
= ( mult_a_Product_unit @ G2 @ B @ X3 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_1157_monoid__cancel_Oassociated__iff,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( monoid759071382269063239t_unit @ G2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( associ6060417466526650393t_unit @ G2 @ A @ B )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( units_2471184348132832486t_unit @ G2 ) )
& ( A
= ( mult_s7930653359683758801t_unit @ G2 @ B @ X3 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_1158_gcd__condition__monoid_Ogcd__divides,axiom,
! [G2: partia2175431115845679010xt_a_b,Z: a,X: a,Y: a] :
( ( gcd_co8605360636473864704xt_a_b @ G2 )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ Z @ X )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ Z @ Y )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( factor8216151070175719842xt_a_b @ G2 @ Z @ ( somegc1600592057159103747xt_a_b @ G2 @ X @ Y ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_divides
thf(fact_1159_gcd__condition__monoid_Ogcd__divides,axiom,
! [G2: partia6043505979758434576t_unit,Z: set_a,X: set_a,Y: set_a] :
( ( gcd_co1135742393622391822t_unit @ G2 )
=> ( ( factor5460682277579321776t_unit @ G2 @ Z @ X )
=> ( ( factor5460682277579321776t_unit @ G2 @ Z @ Y )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ Z @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( factor5460682277579321776t_unit @ G2 @ Z @ ( somegc5022794125158426641t_unit @ G2 @ X @ Y ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_divides
thf(fact_1160_gcd__condition__monoid_Ogcd__divides,axiom,
! [G2: partia8223610829204095565t_unit,Z: a,X: a,Y: a] :
( ( gcd_co701944698663231555t_unit @ G2 )
=> ( ( factor3040189038382604065t_unit @ G2 @ Z @ X )
=> ( ( factor3040189038382604065t_unit @ G2 @ Z @ Y )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( factor3040189038382604065t_unit @ G2 @ Z @ ( somegc8962790057355718400t_unit @ G2 @ X @ Y ) ) ) ) ) ) ) ) ).
% gcd_condition_monoid.gcd_divides
thf(fact_1161_gcd__condition__monoid_Ogcd__divides__l,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( gcd_co8605360636473864704xt_a_b @ G2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( factor8216151070175719842xt_a_b @ G2 @ ( somegc1600592057159103747xt_a_b @ G2 @ A @ B ) @ A ) ) ) ) ).
% gcd_condition_monoid.gcd_divides_l
thf(fact_1162_gcd__condition__monoid_Ogcd__divides__l,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( gcd_co1135742393622391822t_unit @ G2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( factor5460682277579321776t_unit @ G2 @ ( somegc5022794125158426641t_unit @ G2 @ A @ B ) @ A ) ) ) ) ).
% gcd_condition_monoid.gcd_divides_l
thf(fact_1163_gcd__condition__monoid_Ogcd__divides__l,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a] :
( ( gcd_co701944698663231555t_unit @ G2 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( factor3040189038382604065t_unit @ G2 @ ( somegc8962790057355718400t_unit @ G2 @ A @ B ) @ A ) ) ) ) ).
% gcd_condition_monoid.gcd_divides_l
thf(fact_1164_gcd__condition__monoid_Ogcd__divides__r,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( gcd_co8605360636473864704xt_a_b @ G2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( factor8216151070175719842xt_a_b @ G2 @ ( somegc1600592057159103747xt_a_b @ G2 @ A @ B ) @ B ) ) ) ) ).
% gcd_condition_monoid.gcd_divides_r
thf(fact_1165_gcd__condition__monoid_Ogcd__divides__r,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( gcd_co1135742393622391822t_unit @ G2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( factor5460682277579321776t_unit @ G2 @ ( somegc5022794125158426641t_unit @ G2 @ A @ B ) @ B ) ) ) ) ).
% gcd_condition_monoid.gcd_divides_r
thf(fact_1166_gcd__condition__monoid_Ogcd__divides__r,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a] :
( ( gcd_co701944698663231555t_unit @ G2 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( factor3040189038382604065t_unit @ G2 @ ( somegc8962790057355718400t_unit @ G2 @ A @ B ) @ B ) ) ) ) ).
% gcd_condition_monoid.gcd_divides_r
thf(fact_1167_gcd__condition__monoid_Ogcd__isgcd,axiom,
! [G2: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( gcd_co8605360636473864704xt_a_b @ G2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( isgcd_a_ring_ext_a_b @ G2 @ ( somegc1600592057159103747xt_a_b @ G2 @ A @ B ) @ A @ B ) ) ) ) ).
% gcd_condition_monoid.gcd_isgcd
thf(fact_1168_gcd__condition__monoid_Ogcd__isgcd,axiom,
! [G2: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( gcd_co1135742393622391822t_unit @ G2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( isgcd_8277756548069700071t_unit @ G2 @ ( somegc5022794125158426641t_unit @ G2 @ A @ B ) @ A @ B ) ) ) ) ).
% gcd_condition_monoid.gcd_isgcd
thf(fact_1169_gcd__condition__monoid_Ogcd__isgcd,axiom,
! [G2: partia8223610829204095565t_unit,A: a,B: a] :
( ( gcd_co701944698663231555t_unit @ G2 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( isgcd_a_Product_unit @ G2 @ ( somegc8962790057355718400t_unit @ G2 @ A @ B ) @ A @ B ) ) ) ) ).
% gcd_condition_monoid.gcd_isgcd
thf(fact_1170_gcd__condition__monoid_OSomeGcd__ex,axiom,
! [G2: partia2175431115845679010xt_a_b,A2: set_a] :
( ( gcd_co8605360636473864704xt_a_b @ G2 )
=> ( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( A2 != bot_bot_set_a )
=> ( member_a @ ( someGc1035247079102080323xt_a_b @ G2 @ A2 ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ) ) ).
% gcd_condition_monoid.SomeGcd_ex
thf(fact_1171_gcd__condition__monoid_OSomeGcd__ex,axiom,
! [G2: partia6043505979758434576t_unit,A2: set_set_a] :
( ( gcd_co1135742393622391822t_unit @ G2 )
=> ( ( finite_finite_set_a @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( partia5907974310037520643t_unit @ G2 ) )
=> ( ( A2 != bot_bot_set_set_a )
=> ( member_set_a @ ( someGc8699850889451378769t_unit @ G2 @ A2 ) @ ( partia5907974310037520643t_unit @ G2 ) ) ) ) ) ) ).
% gcd_condition_monoid.SomeGcd_ex
thf(fact_1172_gcd__condition__monoid_OSomeGcd__ex,axiom,
! [G2: partia8223610829204095565t_unit,A2: set_a] :
( ( gcd_co701944698663231555t_unit @ G2 )
=> ( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( A2 != bot_bot_set_a )
=> ( member_a @ ( someGc8133249837406473920t_unit @ G2 @ A2 ) @ ( partia6735698275553448452t_unit @ G2 ) ) ) ) ) ) ).
% gcd_condition_monoid.SomeGcd_ex
thf(fact_1173_mem__upI,axiom,
! [F2: nat > a,R: partia2175431115845679010xt_a_b] :
( ! [N3: nat] : ( member_a @ ( F2 @ N3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ? [N6: nat] : ( bound_a @ ( zero_a_b @ R ) @ N6 @ F2 )
=> ( member_nat_a @ F2 @ ( up_a_b @ R ) ) ) ) ).
% mem_upI
thf(fact_1174_mem__upI,axiom,
! [F2: nat > set_a,R: partia6043505979758434576t_unit] :
( ! [N3: nat] : ( member_set_a @ ( F2 @ N3 ) @ ( partia5907974310037520643t_unit @ R ) )
=> ( ? [N6: nat] : ( bound_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ N6 @ F2 )
=> ( member_nat_set_a @ F2 @ ( up_set6103515193466504168t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_1175_canonical__proj__vimage__in__carrier,axiom,
! [I2: set_a,J: set_set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ord_le3724670747650509150_set_a @ J @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ r @ I2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ J ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% canonical_proj_vimage_in_carrier
thf(fact_1176_ideal__sum__iff__gcd,axiom,
! [A: a,B: a,D2: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ D2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( cgenid547466209912283029xt_a_b @ r @ D2 )
= ( set_add_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ ( cgenid547466209912283029xt_a_b @ r @ B ) ) )
= ( isgcd_a_ring_ext_a_b @ r @ D2 @ A @ B ) ) ) ) ) ).
% ideal_sum_iff_gcd
thf(fact_1177_associated__sym,axiom,
! [A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( associ5860276527279195403xt_a_b @ r @ B @ A ) ) ).
% associated_sym
thf(fact_1178_associated__trans,axiom,
! [A: a,B: a,C: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ( associ5860276527279195403xt_a_b @ r @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ C ) ) ) ) ) ).
% associated_trans
thf(fact_1179_assoc__subst,axiom,
! [A: a,B: a,F2: a > a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ! [A6: a,B6: a] :
( ( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ A6 @ B6 ) )
=> ( ( member_a @ ( F2 @ A6 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ ( F2 @ B6 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ ( F2 @ A6 ) @ ( F2 @ B6 ) ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( F2 @ A ) @ ( F2 @ B ) ) ) ) ) ) ).
% assoc_subst
thf(fact_1180_Units__assoc,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ).
% Units_assoc
thf(fact_1181_mult__cong__r,axiom,
! [B: a,B8: a,A: a] :
( ( associ5860276527279195403xt_a_b @ r @ B @ B8 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B8 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B8 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_1182_mult__cong__l,axiom,
! [A: a,A8: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ A8 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A8 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A8 @ B ) ) ) ) ) ) ).
% mult_cong_l
thf(fact_1183_Units__cong,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_cong
thf(fact_1184_divides__cong__r,axiom,
! [X: a,Y: a,Y4: a] :
( ( factor8216151070175719842xt_a_b @ r @ X @ Y )
=> ( ( associ5860276527279195403xt_a_b @ r @ Y @ Y4 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ X @ Y4 ) ) ) ) ).
% divides_cong_r
thf(fact_1185_divides__cong__l,axiom,
! [X: a,X6: a,Y: a] :
( ( associ5860276527279195403xt_a_b @ r @ X @ X6 )
=> ( ( factor8216151070175719842xt_a_b @ r @ X6 @ Y )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ X @ Y ) ) ) ) ).
% divides_cong_l
thf(fact_1186_associated__iff__same__ideal,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
= ( ( cgenid547466209912283029xt_a_b @ r @ A )
= ( cgenid547466209912283029xt_a_b @ r @ B ) ) ) ) ) ).
% associated_iff_same_ideal
thf(fact_1187_add__ideals,axiom,
! [I2: set_a,J: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ideal_a_b @ J @ r )
=> ( ideal_a_b @ ( set_add_a_b @ r @ I2 @ J ) @ r ) ) ) ).
% add_ideals
thf(fact_1188_assoc__iff__assoc__mult,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
= ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ).
% assoc_iff_assoc_mult
thf(fact_1189_ring__associated__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
& ( A
= ( mult_a_ring_ext_a_b @ r @ X3 @ B ) ) ) ) ) ) ) ).
% ring_associated_iff
thf(fact_1190_associatedI2_H,axiom,
! [A: a,B: a,U: a] :
( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2'
thf(fact_1191_associatedI2,axiom,
! [U: a,A: a,B: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2
thf(fact_1192_setadd__subset__G,axiom,
! [H2: set_a,K: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H2 @ K ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% setadd_subset_G
thf(fact_1193_set__add__comm,axiom,
! [I2: set_a,J: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ J @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ I2 @ J )
= ( set_add_a_b @ r @ J @ I2 ) ) ) ) ).
% set_add_comm
thf(fact_1194_set__add__closed,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A2 @ B2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_add_closed
thf(fact_1195_bezout__identity,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ ( cgenid547466209912283029xt_a_b @ r @ B ) )
= ( cgenid547466209912283029xt_a_b @ r @ ( somegc1600592057159103747xt_a_b @ r @ A @ B ) ) ) ) ) ).
% bezout_identity
thf(fact_1196_sum__space__dim_I1_J,axiom,
! [K: set_a,E: set_a,F: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ F )
=> ( embedd8708762675212832759on_a_b @ r @ K @ ( set_add_a_b @ r @ E @ F ) ) ) ) ) ).
% sum_space_dim(1)
thf(fact_1197_quot__ideal__imp__ring__ideal,axiom,
! [I2: set_a,J: set_set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ideal_4463284918206690523t_unit @ J @ ( factRing_a_b @ r @ I2 ) )
=> ( ideal_a_b @ ( comple2307003609928055243_set_a @ J ) @ r ) ) ) ).
% quot_ideal_imp_ring_ideal
thf(fact_1198_finite__Union,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ! [M3: set_a] :
( ( member_set_a @ M3 @ A2 )
=> ( finite_finite_a @ M3 ) )
=> ( finite_finite_a @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% finite_Union
thf(fact_1199_associated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ A ) ) ).
% associated_refl
thf(fact_1200_finite__UN,axiom,
! [A2: set_a,B2: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A2 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( finite_finite_a @ ( B2 @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1201_finite__UnionD,axiom,
! [A2: set_set_a] :
( ( finite_finite_a @ ( comple2307003609928055243_set_a @ A2 ) )
=> ( finite_finite_set_a @ A2 ) ) ).
% finite_UnionD
thf(fact_1202_principal__domain_Obezout__identity,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( ring_p2862007038493914190t_unit @ R )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( set_ad3113708954360070384t_unit @ R @ ( cgenid6682780793756002467t_unit @ R @ A ) @ ( cgenid6682780793756002467t_unit @ R @ B ) )
= ( cgenid6682780793756002467t_unit @ R @ ( somegc5022794125158426641t_unit @ R @ A @ B ) ) ) ) ) ) ).
% principal_domain.bezout_identity
thf(fact_1203_principal__domain_Obezout__identity,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( set_add_a_b @ R @ ( cgenid547466209912283029xt_a_b @ R @ A ) @ ( cgenid547466209912283029xt_a_b @ R @ B ) )
= ( cgenid547466209912283029xt_a_b @ R @ ( somegc1600592057159103747xt_a_b @ R @ A @ B ) ) ) ) ) ) ).
% principal_domain.bezout_identity
thf(fact_1204_domain_Oring__associated__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( associ5860276527279195403xt_a_b @ R @ A @ B )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ R ) )
& ( A
= ( mult_a_ring_ext_a_b @ R @ X3 @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_1205_domain_Oring__associated__iff,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( associ6060417466526650393t_unit @ R @ A @ B )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( units_2471184348132832486t_unit @ R ) )
& ( A
= ( mult_s7930653359683758801t_unit @ R @ X3 @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_1206_domain_Oassoc__iff__assoc__mult,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( associ5860276527279195403xt_a_b @ R @ A @ B )
= ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ R ) @ A @ B ) ) ) ) ) ).
% domain.assoc_iff_assoc_mult
thf(fact_1207_domain_Oassoc__iff__assoc__mult,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( associ6060417466526650393t_unit @ R @ A @ B )
= ( associ2347642451913777048t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ A @ B ) ) ) ) ) ).
% domain.assoc_iff_assoc_mult
thf(fact_1208_set__add__hom,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,I2: set_a,J: set_a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ J @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( image_a_a @ H @ ( set_add_a_b @ R @ I2 @ J ) )
= ( set_add_a_b @ S @ ( image_a_a @ H @ I2 ) @ ( image_a_a @ H @ J ) ) ) ) ) ) ).
% set_add_hom
thf(fact_1209_set__add__hom,axiom,
! [H: set_a > a,R: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,I2: set_set_a,J: set_set_a] :
( ( member_set_a_a @ H @ ( ring_h4811522740288071338it_a_b @ R @ S ) )
=> ( ( ord_le3724670747650509150_set_a @ I2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ord_le3724670747650509150_set_a @ J @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( image_set_a_a @ H @ ( set_ad3113708954360070384t_unit @ R @ I2 @ J ) )
= ( set_add_a_b @ S @ ( image_set_a_a @ H @ I2 ) @ ( image_set_a_a @ H @ J ) ) ) ) ) ) ).
% set_add_hom
thf(fact_1210_principal__domain_Oideal__sum__iff__gcd,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a,D2: set_a] :
( ( ring_p2862007038493914190t_unit @ R )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ D2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ( cgenid6682780793756002467t_unit @ R @ D2 )
= ( set_ad3113708954360070384t_unit @ R @ ( cgenid6682780793756002467t_unit @ R @ A ) @ ( cgenid6682780793756002467t_unit @ R @ B ) ) )
= ( isgcd_8277756548069700071t_unit @ R @ D2 @ A @ B ) ) ) ) ) ) ).
% principal_domain.ideal_sum_iff_gcd
thf(fact_1211_principal__domain_Oideal__sum__iff__gcd,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,D2: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ D2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( cgenid547466209912283029xt_a_b @ R @ D2 )
= ( set_add_a_b @ R @ ( cgenid547466209912283029xt_a_b @ R @ A ) @ ( cgenid547466209912283029xt_a_b @ R @ B ) ) )
= ( isgcd_a_ring_ext_a_b @ R @ D2 @ A @ B ) ) ) ) ) ) ).
% principal_domain.ideal_sum_iff_gcd
thf(fact_1212_mem__upD,axiom,
! [F2: nat > a,R: partia2175431115845679010xt_a_b,N2: nat] :
( ( member_nat_a @ F2 @ ( up_a_b @ R ) )
=> ( member_a @ ( F2 @ N2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% mem_upD
thf(fact_1213_mem__upD,axiom,
! [F2: nat > set_a,R: partia6043505979758434576t_unit,N2: nat] :
( ( member_nat_set_a @ F2 @ ( up_set6103515193466504168t_unit @ R ) )
=> ( member_set_a @ ( F2 @ N2 ) @ ( partia5907974310037520643t_unit @ R ) ) ) ).
% mem_upD
thf(fact_1214_cSup__singleton,axiom,
! [X: set_a] :
( ( comple2307003609928055243_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= X ) ).
% cSup_singleton
thf(fact_1215_ccpo__Sup__singleton,axiom,
! [X: set_a] :
( ( comple2307003609928055243_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_1216_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C5: nat] :
( ( ord_less_eq_nat @ A @ C5 )
& ( ord_less_eq_nat @ C5 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C5 ) )
=> ( P @ X4 ) )
& ! [D3: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D3 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1217_cSup__eq__maximum,axiom,
! [Z: set_set_a,X5: set_set_set_a] :
( ( member_set_set_a @ Z @ X5 )
=> ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ X5 )
=> ( ord_le3724670747650509150_set_a @ X2 @ Z ) )
=> ( ( comple3958522678809307947_set_a @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1218_cSup__eq__maximum,axiom,
! [Z: set_a,X5: set_set_a] :
( ( member_set_a @ Z @ X5 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X5 )
=> ( ord_less_eq_set_a @ X2 @ Z ) )
=> ( ( comple2307003609928055243_set_a @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1219_canonical__proj__vimage__mem__iff,axiom,
! [I2: set_a,J: set_set_a,A: a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ord_le3724670747650509150_set_a @ J @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ r @ I2 ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( comple2307003609928055243_set_a @ J ) )
= ( member_set_a @ ( a_r_coset_a_b @ r @ I2 @ A ) @ J ) ) ) ) ) ).
% canonical_proj_vimage_mem_iff
thf(fact_1220_a__rcos__zero,axiom,
! [I2: set_a,I: a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( member_a @ I @ I2 )
=> ( ( a_r_coset_a_b @ r @ I2 @ I )
= I2 ) ) ) ).
% a_rcos_zero
thf(fact_1221_a__r__coset__subset__G,axiom,
! [H2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_r_coset_a_b @ r @ H2 @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_r_coset_subset_G
thf(fact_1222_a__rcosI,axiom,
! [H: a,H2: set_a,X: a] :
( ( member_a @ H @ H2 )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ H @ X ) @ ( a_r_coset_a_b @ r @ H2 @ X ) ) ) ) ) ).
% a_rcosI
thf(fact_1223_a__coset__add__assoc,axiom,
! [M: set_a,G: a,H: a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ ( a_r_coset_a_b @ r @ M @ G ) @ H )
= ( a_r_coset_a_b @ r @ M @ ( add_a_b @ r @ G @ H ) ) ) ) ) ) ).
% a_coset_add_assoc
thf(fact_1224_a__setmult__rcos__assoc,axiom,
! [H2: set_a,K: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ H2 @ ( a_r_coset_a_b @ r @ K @ X ) )
= ( a_r_coset_a_b @ r @ ( set_add_a_b @ r @ H2 @ K ) @ X ) ) ) ) ) ).
% a_setmult_rcos_assoc
thf(fact_1225_quotient__eq__iff__same__a__r__cos,axiom,
! [I2: set_a,A: a,B: a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( a_minus_a_b @ r @ A @ B ) @ I2 )
= ( ( a_r_coset_a_b @ r @ I2 @ A )
= ( a_r_coset_a_b @ r @ I2 @ B ) ) ) ) ) ) ).
% quotient_eq_iff_same_a_r_cos
thf(fact_1226_a__coset__add__inv2,axiom,
! [M: set_a,X: a,Y: a] :
( ( ( a_r_coset_a_b @ r @ M @ X )
= ( a_r_coset_a_b @ r @ M @ Y ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ M @ ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) )
= M ) ) ) ) ) ).
% a_coset_add_inv2
thf(fact_1227_a__coset__add__inv1,axiom,
! [M: set_a,X: a,Y: a] :
( ( ( a_r_coset_a_b @ r @ M @ ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) )
= M )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ M @ X )
= ( a_r_coset_a_b @ r @ M @ Y ) ) ) ) ) ) ).
% a_coset_add_inv1
thf(fact_1228_ring__ideal__imp__quot__ideal,axiom,
! [I2: set_a,J: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ideal_a_b @ J @ r )
=> ( ideal_4463284918206690523t_unit @ ( image_a_set_a @ ( a_r_coset_a_b @ r @ I2 ) @ J ) @ ( factRing_a_b @ r @ I2 ) ) ) ) ).
% ring_ideal_imp_quot_ideal
thf(fact_1229_a__rcos__assoc__lcos,axiom,
! [H2: set_a,K: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ ( a_r_coset_a_b @ r @ H2 @ X ) @ K )
= ( set_add_a_b @ r @ H2 @ ( a_l_coset_a_b @ r @ X @ K ) ) ) ) ) ) ).
% a_rcos_assoc_lcos
thf(fact_1230_a__coset__add__zero,axiom,
! [M: set_a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ M @ ( zero_a_b @ r ) )
= M ) ) ).
% a_coset_add_zero
thf(fact_1231_a__rcosetsI,axiom,
! [H2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_set_a @ ( a_r_coset_a_b @ r @ H2 @ X ) @ ( a_RCOSETS_a_b @ r @ H2 ) ) ) ) ).
% a_rcosetsI
thf(fact_1232_a__card__cosets__equal,axiom,
! [C: set_a,H2: set_a] :
( ( member_set_a @ C @ ( a_RCOSETS_a_b @ r @ H2 ) )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_card_a @ C )
= ( finite_card_a @ H2 ) ) ) ) ) ).
% a_card_cosets_equal
thf(fact_1233_mult__of_Oprime__pow__divides__iff,axiom,
! [P2: a,A: a,B: a,N2: nat] :
( ( member_a @ P2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P2 )
=> ( ~ ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P2 @ A )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ P2 @ N2 ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ P2 @ N2 ) @ B ) ) ) ) ) ) ) ).
% mult_of.prime_pow_divides_iff
thf(fact_1234_add_Opower__order__eq__one,axiom,
! [A: a] :
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ ( finite_card_a @ ( partia707051561876973205xt_a_b @ r ) ) @ A )
= ( zero_a_b @ r ) ) ) ) ).
% add.power_order_eq_one
thf(fact_1235_add_Ogroup__commutes__pow,axiom,
! [X: a,Y: a,N2: nat] :
( ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N2 @ X ) @ Y )
= ( add_a_b @ r @ Y @ ( add_pow_a_b_nat @ r @ N2 @ X ) ) ) ) ) ) ).
% add.group_commutes_pow
thf(fact_1236_add_Onat__pow__comm,axiom,
! [X: a,N2: nat,M2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N2 @ X ) @ ( add_pow_a_b_nat @ r @ M2 @ X ) )
= ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ M2 @ X ) @ ( add_pow_a_b_nat @ r @ N2 @ X ) ) ) ) ).
% add.nat_pow_comm
thf(fact_1237_add_Onat__pow__distrib,axiom,
! [X: a,Y: a,N2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ N2 @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N2 @ X ) @ ( add_pow_a_b_nat @ r @ N2 @ Y ) ) ) ) ) ).
% add.nat_pow_distrib
thf(fact_1238_add_Opow__mult__distrib,axiom,
! [X: a,Y: a,N2: nat] :
( ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ N2 @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N2 @ X ) @ ( add_pow_a_b_nat @ r @ N2 @ Y ) ) ) ) ) ) ).
% add.pow_mult_distrib
thf(fact_1239_add__pow__ldistr,axiom,
! [A: a,B: a,K2: nat] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_nat @ r @ K2 @ A ) @ B )
= ( add_pow_a_b_nat @ r @ K2 @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_ldistr
thf(fact_1240_add__pow__rdistr,axiom,
! [A: a,B: a,K2: nat] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_nat @ r @ K2 @ B ) )
= ( add_pow_a_b_nat @ r @ K2 @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_rdistr
thf(fact_1241_add_Onat__pow__inv,axiom,
! [X: a,I: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ I @ ( a_inv_a_b @ r @ X ) )
= ( a_inv_a_b @ r @ ( add_pow_a_b_nat @ r @ I @ X ) ) ) ) ).
% add.nat_pow_inv
thf(fact_1242_mult__of_OUnits__pow__closed,axiom,
! [X: a,D2: nat] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ D2 ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.Units_pow_closed
thf(fact_1243_add_Onat__pow__Suc2,axiom,
! [X: a,N2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ ( suc @ N2 ) @ X )
= ( add_a_b @ r @ X @ ( add_pow_a_b_nat @ r @ N2 @ X ) ) ) ) ).
% add.nat_pow_Suc2
thf(fact_1244_add_Opow__eq__div2,axiom,
! [X: a,M2: nat,N2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_pow_a_b_nat @ r @ M2 @ X )
= ( add_pow_a_b_nat @ r @ N2 @ X ) )
=> ( ( add_pow_a_b_nat @ r @ ( minus_minus_nat @ M2 @ N2 ) @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.pow_eq_div2
thf(fact_1245_mult__of_Opow__mult__distrib,axiom,
! [X: a,Y: a,N2: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N2 )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N2 ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ Y @ N2 ) ) ) ) ) ) ).
% mult_of.pow_mult_distrib
thf(fact_1246_mult__of_Onat__pow__distrib,axiom,
! [X: a,Y: a,N2: nat] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N2 )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N2 ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ Y @ N2 ) ) ) ) ) ).
% mult_of.nat_pow_distrib
thf(fact_1247_mult__of_Onat__pow__comm,axiom,
! [X: a,N2: nat,M2: nat] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N2 ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ M2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ M2 ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N2 ) ) ) ) ).
% mult_of.nat_pow_comm
thf(fact_1248_mult__of_Ogroup__commutes__pow,axiom,
! [X: a,Y: a,N2: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N2 ) @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N2 ) ) ) ) ) ) ).
% mult_of.group_commutes_pow
thf(fact_1249_mult__of_Ounits__of__pow,axiom,
! [X: a,N2: nat] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( units_7501539392726747778t_unit @ ( ring_mult_of_a_b @ r ) ) @ X @ N2 )
= ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N2 ) ) ) ).
% mult_of.units_of_pow
thf(fact_1250_mult__of_Onat__pow__Suc2,axiom,
! [X: a,N2: nat] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ ( suc @ N2 ) )
= ( mult_a_ring_ext_a_b @ r @ X @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N2 ) ) ) ) ).
% mult_of.nat_pow_Suc2
thf(fact_1251_add_Onat__pow__closed,axiom,
! [X: a,N2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_pow_a_b_nat @ r @ N2 @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.nat_pow_closed
thf(fact_1252_add_Onat__pow__one,axiom,
! [N2: nat] :
( ( add_pow_a_b_nat @ r @ N2 @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% add.nat_pow_one
thf(fact_1253_add_Onat__pow__eone,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ one_one_nat @ X )
= X ) ) ).
% add.nat_pow_eone
thf(fact_1254_add_Onat__pow__0,axiom,
! [X: a] :
( ( add_pow_a_b_nat @ r @ zero_zero_nat @ X )
= ( zero_a_b @ r ) ) ).
% add.nat_pow_0
thf(fact_1255_add_Onat__pow__Suc,axiom,
! [N2: nat,X: a] :
( ( add_pow_a_b_nat @ r @ ( suc @ N2 ) @ X )
= ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N2 @ X ) @ X ) ) ).
% add.nat_pow_Suc
thf(fact_1256_mult__of_Onat__pow__closed,axiom,
! [X: a,N2: nat] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.nat_pow_closed
thf(fact_1257_mult__of_Onat__pow__one,axiom,
! [N2: nat] :
( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ ( one_a_ring_ext_a_b @ r ) @ N2 )
= ( one_a_ring_ext_a_b @ r ) ) ).
% mult_of.nat_pow_one
thf(fact_1258_mult__of_Onat__pow__Suc,axiom,
! [X: a,N2: nat] :
( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ ( suc @ N2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N2 ) @ X ) ) ).
% mult_of.nat_pow_Suc
thf(fact_1259_mult__of_Onat__pow__eone,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ one_one_nat )
= X ) ) ).
% mult_of.nat_pow_eone
thf(fact_1260_mult__of_Onat__pow__0,axiom,
! [X: a] :
( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ r ) ) ).
% mult_of.nat_pow_0
thf(fact_1261_mult__of_Onat__pow__mult,axiom,
! [X: a,N2: nat,M2: nat] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N2 ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ M2 ) )
= ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ ( plus_plus_nat @ N2 @ M2 ) ) ) ) ).
% mult_of.nat_pow_mult
thf(fact_1262_mult__of_Oproperfactor__fcount,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ord_less_nat @ ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ A ) @ ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ B ) ) ) ) ) ).
% mult_of.properfactor_fcount
thf(fact_1263_Units__pow__closed,axiom,
! [X: a,D2: nat] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X @ D2 ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_pow_closed
thf(fact_1264_nat__pow__zero,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( zero_a_b @ r ) @ N2 )
= ( zero_a_b @ r ) ) ) ).
% nat_pow_zero
thf(fact_1265_pow__mult__distrib,axiom,
! [X: a,Y: a,N2: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N2 )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N2 ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N2 ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_1266_nat__pow__distrib,axiom,
! [X: a,Y: a,N2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N2 )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N2 ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N2 ) ) ) ) ) ).
% nat_pow_distrib
thf(fact_1267_nat__pow__comm,axiom,
! [X: a,N2: nat,M2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N2 ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M2 ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N2 ) ) ) ) ).
% nat_pow_comm
thf(fact_1268_group__commutes__pow,axiom,
! [X: a,Y: a,N2: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N2 ) @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N2 ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_1269_nat__pow__Suc2,axiom,
! [X: a,N2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ ( suc @ N2 ) )
= ( mult_a_ring_ext_a_b @ r @ X @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N2 ) ) ) ) ).
% nat_pow_Suc2
thf(fact_1270_nat__pow__mult,axiom,
! [X: a,N2: nat,M2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N2 ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M2 ) )
= ( pow_a_1026414303147256608_b_nat @ r @ X @ ( plus_plus_nat @ N2 @ M2 ) ) ) ) ).
% nat_pow_mult
thf(fact_1271_mult__of_Oproperfactor__divides,axiom,
! [A: a,B: a] :
( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ).
% mult_of.properfactor_divides
thf(fact_1272_units__of__pow,axiom,
! [X: a,N2: nat] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( units_8174867845824275201xt_a_b @ r ) @ X @ N2 )
= ( pow_a_1026414303147256608_b_nat @ r @ X @ N2 ) ) ) ).
% units_of_pow
thf(fact_1273_add_Onat__pow__mult,axiom,
! [X: a,N2: nat,M2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N2 @ X ) @ ( add_pow_a_b_nat @ r @ M2 @ X ) )
= ( add_pow_a_b_nat @ r @ ( plus_plus_nat @ N2 @ M2 ) @ X ) ) ) ).
% add.nat_pow_mult
thf(fact_1274_properfactor__of__zero_I1_J,axiom,
! [B: a] :
( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ~ ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ B @ ( zero_a_b @ r ) ) ) ).
% properfactor_of_zero(1)
thf(fact_1275_mult__of_Oproperfactor__prod__l,axiom,
! [A: a,B: a,C: a] :
( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ) ).
% mult_of.properfactor_prod_l
thf(fact_1276_mult__of_Oproperfactor__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% mult_of.properfactor_prod_r
thf(fact_1277_mult__of_Oproperfactor__cong__l,axiom,
! [X6: a,X: a,Y: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ X6 @ X )
=> ( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ X @ Y )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ X6 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ X6 @ Y ) ) ) ) ) ) ).
% mult_of.properfactor_cong_l
% Conjectures (2)
thf(conj_0,hypothesis,
member_a @ x @ s ).
thf(conj_1,conjecture,
member_a @ ( f @ x ) @ ( partia707051561876973205xt_a_b @ r ) ).
%------------------------------------------------------------------------------