TPTP Problem File: SLH0709^1.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 : Finite_Fields/0007_Monic_Polynomial_Factorization/prob_00153_005445__18285672_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1458 ( 338 unt; 179 typ; 0 def)
% Number of atoms : 4127 (1126 equ; 0 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 15396 ( 300 ~; 52 |; 214 &;12645 @)
% ( 0 <=>;2185 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 356 ( 356 >; 0 *; 0 +; 0 <<)
% Number of symbols : 170 ( 169 usr; 9 con; 0-4 aty)
% Number of variables : 2969 ( 82 ^;2783 !; 104 ?;2969 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:22:19.093
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Ideal_Ocgenideal_001t__Set__Oset_Itf__a_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_Itf__a_J_Mt__Product____Type__Ounit_J,type,
cgenid6682780793756002467t_unit: partia6043505979758434576t_unit > set_a > set_set_a ).
thf(sy_c_Ideal_Ocgenideal_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
cgenid547466209912283029xt_a_b: partia2175431115845679010xt_a_b > a > set_a ).
thf(sy_c_Ideal_Ogenideal_001tf__a_001tf__b,type,
genideal_a_b: partia2175431115845679010xt_a_b > set_a > set_a ).
thf(sy_c_Ideal_Omaximalideal_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
maxima2253313296322093082t_unit: set_set_a > partia6043505979758434576t_unit > $o ).
thf(sy_c_Ideal_Omaximalideal_001tf__a_001tf__b,type,
maximalideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ideal_Oprimeideal_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
primei7645216761534224334t_unit: set_set_a > partia6043505979758434576t_unit > $o ).
thf(sy_c_Ideal_Oprimeideal_001tf__a_001tf__b,type,
primeideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ideal_Oprincipalideal_001tf__a_001tf__b,type,
principalideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Multiplicative__Group_Ogroup_Oord_001tf__a_001t__Product____Type__Ounit,type,
multip1500854282228996350t_unit: partia8223610829204095565t_unit > a > nat ).
thf(sy_c_Multiplicative__Group_Omult__of_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
multip3774352783277980819t_unit: partia6043505979758434576t_unit > partia6838931692028023693t_unit ).
thf(sy_c_Multiplicative__Group_Omult__of_001tf__a_001tf__b,type,
multip3210463924028840165of_a_b: partia2175431115845679010xt_a_b > partia8223610829204095565t_unit ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_QuotRing_OFactRing_001tf__a_001tf__b,type,
factRing_a_b: partia2175431115845679010xt_a_b > set_a > partia6043505979758434576t_unit ).
thf(sy_c_QuotRing_Ois__ring__iso_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
is_rin6001486760346555702it_a_b: partia6043505979758434576t_unit > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_QuotRing_Ois__ring__iso_001tf__a_001tf__b_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
is_rin9099215527551818550t_unit: partia2175431115845679010xt_a_b > partia6043505979758434576t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
domain4236798911309298543t_unit: partia6043505979758434576t_unit > $o ).
thf(sy_c_Ring_Odomain_001tf__a_001tf__b,type,
domain_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Ofield_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
field_6045675692312731021t_unit: partia6043505979758434576t_unit > $o ).
thf(sy_c_Ring_Ofield_001tf__a_001tf__b,type,
field_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_Oadd_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
add_se3735415688806051380t_unit: partia6043505979758434576t_unit > set_a > set_a > set_a ).
thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
add_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
zero_s2174465271003423091t_unit: partia6043505979758434576t_unit > set_a ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring__Characteristic_Ochar_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
ring_c1190685578609594780t_unit: partia6043505979758434576t_unit > nat ).
thf(sy_c_Ring__Characteristic_Ochar_001tf__a_001tf__b,type,
ring_char_a_b: partia2175431115845679010xt_a_b > nat ).
thf(sy_c_Ring__Divisibility_Oeuclidean__domain_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
ring_e187967263881214398t_unit: partia6043505979758434576t_unit > ( set_a > nat ) > $o ).
thf(sy_c_Ring__Divisibility_Oeuclidean__domain_001tf__a_001tf__b,type,
ring_e8745995371659049232in_a_b: partia2175431115845679010xt_a_b > ( a > nat ) > $o ).
thf(sy_c_Ring__Divisibility_Ofactorial__domain_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
ring_f6820247627256571077t_unit: partia6043505979758434576t_unit > $o ).
thf(sy_c_Ring__Divisibility_Ofactorial__domain_001tf__a_001tf__b,type,
ring_f5272581269873410839in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Omult__of_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
ring_m2800496791135293897t_unit: partia6043505979758434576t_unit > partia6838931692028023693t_unit ).
thf(sy_c_Ring__Divisibility_Omult__of_001tf__a_001tf__b,type,
ring_mult_of_a_b: partia2175431115845679010xt_a_b > partia8223610829204095565t_unit ).
thf(sy_c_Ring__Divisibility_Onoetherian__domain_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
ring_n3212398840814694743t_unit: partia6043505979758434576t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__domain_001tf__a_001tf__b,type,
ring_n4045954140777738665in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
ring_n5014428767265248323t_unit: partia6043505979758434576t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001tf__a_001tf__b,type,
ring_n3639167112692572309ng_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
ring_p2862007038493914190t_unit: partia6043505979758434576t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001tf__a_001tf__b,type,
ring_p8803135361686045600in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r7790391342995787508t_unit: partia6043505979758434576t_unit > set_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001tf__a_001tf__b,type,
ring_r999134135267193926le_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r6795642478576035723t_unit: partia6043505979758434576t_unit > set_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001tf__a_001tf__b,type,
ring_ring_prime_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Subrings_Osubfield_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_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_a____,type,
a2: a ).
% Relevant facts (1278)
thf(fact_0_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_1_factorial__domain__axioms,axiom,
ring_f5272581269873410839in_a_b @ r ).
% factorial_domain_axioms
thf(fact_2_local_Ofield__axioms,axiom,
field_a_b @ r ).
% local.field_axioms
thf(fact_3_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_4_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_5_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_6_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_7_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_8_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_9_zeromaximalideal,axiom,
maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeromaximalideal
thf(fact_10_ring__primeE_I1_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( P
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_11_ring__irreducibleE_I1_J,axiom,
! [R: a] :
( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R )
=> ( R
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_12_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_13_zeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroprimeideal
thf(fact_14_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_15_Diff__insert0,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ B ) )
= ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_16_Diff__insert0,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_17_insert__Diff1,axiom,
! [X: set_a,B: set_set_a,A2: set_set_a] :
( ( member_set_a @ X @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B )
= ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_18_insert__Diff1,axiom,
! [X: a,B: set_a,A2: set_a] :
( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_19_maximalideal__prime,axiom,
! [I: set_a] :
( ( maximalideal_a_b @ I @ r )
=> ( primeideal_a_b @ I @ r ) ) ).
% maximalideal_prime
thf(fact_20_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X2: a] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_21_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_22_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_23_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_24_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_25_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_26_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_27_insert__iff,axiom,
! [A: a,B2: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_28_insert__iff,axiom,
! [A: set_a,B2: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_29_insertCI,axiom,
! [A: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = B2 ) )
=> ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_30_insertCI,axiom,
! [A: set_a,B: set_set_a,B2: set_a] :
( ( ~ ( member_set_a @ A @ B )
=> ( A = B2 ) )
=> ( member_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_31_Diff__idemp,axiom,
! [A2: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_32_Diff__iff,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= ( ( member_set_a @ C @ A2 )
& ~ ( member_set_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_33_Diff__iff,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_34_DiffI,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_35_DiffI,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_36_primeness__condition,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P )
= ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% primeness_condition
thf(fact_37_ring__primeE_I3_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( prime_a_ring_ext_a_b @ r @ P ) ) ) ).
% ring_primeE(3)
thf(fact_38_ring__primeI,axiom,
! [P: a] :
( ( P
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P )
=> ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% ring_primeI
thf(fact_39_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_40_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_41_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_42_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_43_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_44_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_45_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_46_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_47_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_48_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y: set_a] :
~ ( member_set_a @ Y @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_49_equals0I,axiom,
! [A2: set_a] :
( ! [Y: a] :
~ ( member_a @ Y @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_50_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_51_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_52_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_53_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_54_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B3: set_a] :
( ( A2
= ( insert_a @ A @ B3 ) )
& ~ ( member_a @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_55_mk__disjoint__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ? [B3: set_set_a] :
( ( A2
= ( insert_set_a @ A @ B3 ) )
& ~ ( member_set_a @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_56_insert__commute,axiom,
! [X: a,Y2: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y2 @ A2 ) )
= ( insert_a @ Y2 @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_57_insert__eq__iff,axiom,
! [A: a,A2: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_a] :
( ( A2
= ( insert_a @ B2 @ C2 ) )
& ~ ( member_a @ B2 @ C2 )
& ( B
= ( insert_a @ A @ C2 ) )
& ~ ( member_a @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_58_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B2: set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B2 @ B )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_set_a] :
( ( A2
= ( insert_set_a @ B2 @ C2 ) )
& ~ ( member_set_a @ B2 @ C2 )
& ( B
= ( insert_set_a @ A @ C2 ) )
& ~ ( member_set_a @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_59_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_60_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_61_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A: set_a,P2: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_63_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_65_insert__ident,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_66_insert__ident,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ~ ( member_set_a @ X @ B )
=> ( ( ( insert_set_a @ X @ A2 )
= ( insert_set_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_67_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B3: set_a] :
( ( A2
= ( insert_a @ X @ B3 ) )
=> ( member_a @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_68_Set_Oset__insert,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ~ ! [B3: set_set_a] :
( ( A2
= ( insert_set_a @ X @ B3 ) )
=> ( member_set_a @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_69_insertI2,axiom,
! [A: a,B: set_a,B2: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_70_insertI2,axiom,
! [A: set_a,B: set_set_a,B2: set_a] :
( ( member_set_a @ A @ B )
=> ( member_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_71_insertI1,axiom,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).
% insertI1
thf(fact_72_insertI1,axiom,
! [A: set_a,B: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B ) ) ).
% insertI1
thf(fact_73_insertE,axiom,
! [A: a,B2: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_74_insertE,axiom,
! [A: set_a,B2: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_set_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_75_DiffD2,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
=> ~ ( member_set_a @ C @ B ) ) ).
% DiffD2
thf(fact_76_DiffD2,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_77_DiffD1,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
=> ( member_set_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_78_DiffD1,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_79_DiffE,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B ) ) ) ).
% DiffE
thf(fact_80_DiffE,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_81_singleton__inject,axiom,
! [A: a,B2: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_82_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_83_doubleton__eq__iff,axiom,
! [A: a,B2: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_84_singleton__iff,axiom,
! [B2: set_a,A: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_85_singleton__iff,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_86_singletonD,axiom,
! [B2: set_a,A: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_87_singletonD,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_88_insert__Diff__if,axiom,
! [X: set_a,B: set_set_a,A2: set_set_a] :
( ( ( member_set_a @ X @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B )
= ( minus_5736297505244876581_set_a @ A2 @ B ) ) )
& ( ~ ( member_set_a @ X @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B )
= ( insert_set_a @ X @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_89_insert__Diff__if,axiom,
! [X: a,B: set_a,A2: set_a] :
( ( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) )
& ( ~ ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_90_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_91_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_92_Diff__insert2,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_93_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_94_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_95_Diff__insert,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_96_primeideal__iff__prime,axiom,
! [P: a] :
( ( member_a @ P @ ( 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 @ P ) @ r )
= ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% primeideal_iff_prime
thf(fact_97_field_Ozeromaximalideal,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( maxima2253313296322093082t_unit @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) @ R2 ) ) ).
% field.zeromaximalideal
thf(fact_98_field_Ozeromaximalideal,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R2 )
=> ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) @ R2 ) ) ).
% field.zeromaximalideal
thf(fact_99_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_100_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_101_genideal__self_H,axiom,
! [I2: a] :
( ( member_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I2 @ ( genideal_a_b @ r @ ( insert_a @ I2 @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_102_irreducible__imp__maximalideal,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P ) @ r ) ) ) ).
% irreducible_imp_maximalideal
thf(fact_103_principal__domain_Oprimeness__condition,axiom,
! [R2: partia2175431115845679010xt_a_b,P: a] :
( ( ring_p8803135361686045600in_a_b @ R2 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_r999134135267193926le_a_b @ R2 @ P )
= ( ring_ring_prime_a_b @ R2 @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_104_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_105_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_106_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_107_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_108_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_109_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_110_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_111_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_112_cgenideal__self,axiom,
! [I2: a] :
( ( member_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I2 @ ( cgenid547466209912283029xt_a_b @ r @ I2 ) ) ) ).
% cgenideal_self
thf(fact_113_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_114_ring__irreducibleE_I4_J,axiom,
! [R: a] :
( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R )
=> ~ ( member_a @ R @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_115_cgenideal__is__principalideal,axiom,
! [I2: a] :
( ( member_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I2 ) @ r ) ) ).
% cgenideal_is_principalideal
thf(fact_116_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_117_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_118_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_119_cgenideal__eq__genideal,axiom,
! [I2: a] :
( ( member_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( cgenid547466209912283029xt_a_b @ r @ I2 )
= ( genideal_a_b @ r @ ( insert_a @ I2 @ bot_bot_set_a ) ) ) ) ).
% cgenideal_eq_genideal
thf(fact_120_field__intro2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) ) )
=> ( field_a_b @ r ) ) ) ).
% field_intro2
thf(fact_121_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_122_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_123_domain_Oring__irreducibleE_I4_J,axiom,
! [R2: partia6043505979758434576t_unit,R: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ R @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( ring_r7790391342995787508t_unit @ R2 @ R )
=> ~ ( member_set_a @ R @ ( units_2471184348132832486t_unit @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_124_domain_Oring__irreducibleE_I4_J,axiom,
! [R2: partia2175431115845679010xt_a_b,R: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ R @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_r999134135267193926le_a_b @ R2 @ R )
=> ~ ( member_a @ R @ ( units_a_ring_ext_a_b @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_125_principal__domain_Oaxioms_I1_J,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( ring_p2862007038493914190t_unit @ R2 )
=> ( domain4236798911309298543t_unit @ R2 ) ) ).
% principal_domain.axioms(1)
thf(fact_126_principal__domain_Oaxioms_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_p8803135361686045600in_a_b @ R2 )
=> ( domain_a_b @ R2 ) ) ).
% principal_domain.axioms(1)
thf(fact_127_noetherian__domain_Oaxioms_I2_J,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( ring_n3212398840814694743t_unit @ R2 )
=> ( domain4236798911309298543t_unit @ R2 ) ) ).
% noetherian_domain.axioms(2)
thf(fact_128_noetherian__domain_Oaxioms_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_n4045954140777738665in_a_b @ R2 )
=> ( domain_a_b @ R2 ) ) ).
% noetherian_domain.axioms(2)
thf(fact_129_factorial__domain_Oaxioms_I1_J,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( ring_f6820247627256571077t_unit @ R2 )
=> ( domain4236798911309298543t_unit @ R2 ) ) ).
% factorial_domain.axioms(1)
thf(fact_130_factorial__domain_Oaxioms_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_f5272581269873410839in_a_b @ R2 )
=> ( domain_a_b @ R2 ) ) ).
% factorial_domain.axioms(1)
thf(fact_131_domain_Ozero__is__prime_I1_J,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( prime_4522187476880896870t_unit @ R2 @ ( zero_s2174465271003423091t_unit @ R2 ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_132_domain_Ozero__is__prime_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R2 )
=> ( prime_a_ring_ext_a_b @ R2 @ ( zero_a_b @ R2 ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_133_noetherian__domain__def,axiom,
( ring_n3212398840814694743t_unit
= ( ^ [R3: partia6043505979758434576t_unit] :
( ( ring_n5014428767265248323t_unit @ R3 )
& ( domain4236798911309298543t_unit @ R3 ) ) ) ) ).
% noetherian_domain_def
thf(fact_134_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_135_noetherian__domain_Ointro,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( ring_n5014428767265248323t_unit @ R2 )
=> ( ( domain4236798911309298543t_unit @ R2 )
=> ( ring_n3212398840814694743t_unit @ R2 ) ) ) ).
% noetherian_domain.intro
thf(fact_136_noetherian__domain_Ointro,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_n3639167112692572309ng_a_b @ R2 )
=> ( ( domain_a_b @ R2 )
=> ( ring_n4045954140777738665in_a_b @ R2 ) ) ) ).
% noetherian_domain.intro
thf(fact_137_domain_Oring__irreducibleE_I1_J,axiom,
! [R2: partia6043505979758434576t_unit,R: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ R @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( ring_r7790391342995787508t_unit @ R2 @ R )
=> ( R
!= ( zero_s2174465271003423091t_unit @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_138_domain_Oring__irreducibleE_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,R: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ R @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_r999134135267193926le_a_b @ R2 @ R )
=> ( R
!= ( zero_a_b @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_139_domain_Oring__primeE_I1_J,axiom,
! [R2: partia6043505979758434576t_unit,P: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ P @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( ring_r6795642478576035723t_unit @ R2 @ P )
=> ( P
!= ( zero_s2174465271003423091t_unit @ R2 ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_140_domain_Oring__primeE_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_ring_prime_a_b @ R2 @ P )
=> ( P
!= ( zero_a_b @ R2 ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_141_domain_Oring__primeE_I3_J,axiom,
! [R2: partia6043505979758434576t_unit,P: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ P @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( ring_r6795642478576035723t_unit @ R2 @ P )
=> ( prime_4522187476880896870t_unit @ R2 @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_142_domain_Oring__primeE_I3_J,axiom,
! [R2: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_ring_prime_a_b @ R2 @ P )
=> ( prime_a_ring_ext_a_b @ R2 @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_143_domain_Ozeroprimeideal,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( primei7645216761534224334t_unit @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) @ R2 ) ) ).
% domain.zeroprimeideal
thf(fact_144_domain_Ozeroprimeideal,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R2 )
=> ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) @ R2 ) ) ).
% domain.zeroprimeideal
thf(fact_145_domain_Oprimeideal__iff__prime,axiom,
! [R2: partia6043505979758434576t_unit,P: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ P @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R2 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) )
=> ( ( primei7645216761534224334t_unit @ ( cgenid6682780793756002467t_unit @ R2 @ P ) @ R2 )
= ( ring_r6795642478576035723t_unit @ R2 @ P ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_146_domain_Oprimeideal__iff__prime,axiom,
! [R2: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ P @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
=> ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ R2 @ P ) @ R2 )
= ( ring_ring_prime_a_b @ R2 @ P ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_147_principalideal_Ogenerate,axiom,
! [I: set_a,R2: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I @ R2 )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
& ( I
= ( genideal_a_b @ R2 @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_148_primeideal_Oprimeideal,axiom,
! [I: set_a,R2: partia2175431115845679010xt_a_b] :
( ( primeideal_a_b @ I @ R2 )
=> ( primeideal_a_b @ I @ R2 ) ) ).
% primeideal.primeideal
thf(fact_149_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R2: partia2175431115845679010xt_a_b,P: a] :
( ( ring_p8803135361686045600in_a_b @ R2 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_r999134135267193926le_a_b @ R2 @ P )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ R2 @ P ) @ R2 ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_150_maximalideal_Ois__maximalideal,axiom,
! [I: set_a,R2: partia2175431115845679010xt_a_b] :
( ( maximalideal_a_b @ I @ R2 )
=> ( maximalideal_a_b @ I @ R2 ) ) ).
% maximalideal.is_maximalideal
thf(fact_151_principalideal_Ois__principalideal,axiom,
! [I: set_a,R2: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I @ R2 )
=> ( principalideal_a_b @ I @ R2 ) ) ).
% principalideal.is_principalideal
thf(fact_152_primeideal_OI__notcarr,axiom,
! [I: set_a,R2: partia2175431115845679010xt_a_b] :
( ( primeideal_a_b @ I @ R2 )
=> ( ( partia707051561876973205xt_a_b @ R2 )
!= I ) ) ).
% primeideal.I_notcarr
thf(fact_153_maximalideal_OI__notcarr,axiom,
! [I: set_a,R2: partia2175431115845679010xt_a_b] :
( ( maximalideal_a_b @ I @ R2 )
=> ( ( partia707051561876973205xt_a_b @ R2 )
!= I ) ) ).
% maximalideal.I_notcarr
thf(fact_154_noetherian__domain_Oaxioms_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_n4045954140777738665in_a_b @ R2 )
=> ( ring_n3639167112692572309ng_a_b @ R2 ) ) ).
% noetherian_domain.axioms(1)
thf(fact_155_ring__prime__def,axiom,
( ring_ring_prime_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,A3: a] :
( ( A3
!= ( zero_a_b @ R3 ) )
& ( prime_a_ring_ext_a_b @ R3 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_156_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_157_Ring_Ofield__Units,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( ( units_2471184348132832486t_unit @ R2 )
= ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R2 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) ) ) ).
% Ring.field_Units
thf(fact_158_Ring_Ofield__Units,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R2 )
=> ( ( units_a_ring_ext_a_b @ R2 )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) ) ) ).
% Ring.field_Units
thf(fact_159_ring__irreducibleI,axiom,
! [R: a] :
( ( member_a @ R @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R @ ( units_a_ring_ext_a_b @ r ) )
=> ( ! [A4: a,B4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R
= ( mult_a_ring_ext_a_b @ r @ A4 @ B4 ) )
=> ( ( member_a @ A4 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B4 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ r @ R ) ) ) ) ).
% ring_irreducibleI
thf(fact_160_finite__domain__units,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_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 ) ) ) ) ).
% finite_domain_units
thf(fact_161_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 ) )
=> ~ ! [B4: a] :
( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ B4 )
=> ~ ( factor8216151070175719842xt_a_b @ r @ B4 @ A ) ) ) ) ) ).
% exists_irreducible_divisor
thf(fact_162_f__comm__group__2,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
!= ( zero_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% f_comm_group_2
thf(fact_163_Idl__subset__ideal_H,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( 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 @ B2 @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ r @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_164_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A4
!= ( zero_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A4 @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_165_ring__irreducibleE_I5_J,axiom,
! [R: a,A: a,B2: a] :
( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R
= ( mult_a_ring_ext_a_b @ r @ A @ B2 ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_166_Units__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_167_m__lcomm,axiom,
! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( 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 @ Y2 @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_168_m__comm,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X ) ) ) ) ).
% m_comm
thf(fact_169_m__assoc,axiom,
! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( 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 @ Y2 ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y2 @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_170_divides__trans,axiom,
! [A: a,B2: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B2 )
=> ( ( factor8216151070175719842xt_a_b @ r @ B2 @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ C ) ) ) ) ).
% divides_trans
thf(fact_171_zero__divides,axiom,
! [A: a] :
( ( factor8216151070175719842xt_a_b @ r @ ( zero_a_b @ r ) @ A )
= ( A
= ( zero_a_b @ r ) ) ) ).
% zero_divides
thf(fact_172_m__rcancel,axiom,
! [A: a,B2: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B2 @ A )
= ( mult_a_ring_ext_a_b @ r @ C @ A ) )
= ( B2 = C ) ) ) ) ) ) ).
% m_rcancel
thf(fact_173_m__lcancel,axiom,
! [A: a,B2: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B2 )
= ( mult_a_ring_ext_a_b @ r @ A @ C ) )
= ( B2 = C ) ) ) ) ) ) ).
% m_lcancel
thf(fact_174_integral__iff,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B2 )
= ( zero_a_b @ r ) )
= ( ( A
= ( zero_a_b @ r ) )
| ( B2
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_175_local_Ointegral,axiom,
! [A: a,B2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ B2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( zero_a_b @ r ) )
| ( B2
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_176_f__comm__group__1,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
!= ( zero_a_b @ r ) )
=> ( ( Y2
!= ( zero_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% f_comm_group_1
thf(fact_177_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X3 )
= X3 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_178_inv__unique,axiom,
! [Y2: a,X: a,Y3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y2 = Y3 ) ) ) ) ) ) ).
% inv_unique
thf(fact_179_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_180_unit__factor,axiom,
! [A: a,B2: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_181_prod__unit__r,axiom,
! [A: a,B2: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_182_prod__unit__l,axiom,
! [A: a,B2: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) @ ( 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 @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_183_divides__prod__r,axiom,
! [A: a,B2: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B2 )
=> ( ( 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 @ B2 @ C ) ) ) ) ) ).
% divides_prod_r
thf(fact_184_divides__prod__l,axiom,
! [A: a,B2: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B2 )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ C @ B2 ) ) ) ) ) ) ).
% divides_prod_l
thf(fact_185_local_Odivides__mult,axiom,
! [A: a,C: a,B2: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B2 )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B2 ) ) ) ) ) ).
% local.divides_mult
thf(fact_186_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_187_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_188_subsetI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_189_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_190_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_191_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_192_Units__inv__comm,axiom,
! [X: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_inv_comm
thf(fact_193_subset__Idl__subset,axiom,
! [I: set_a,H: set_a] :
( ( ord_less_eq_set_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H @ I )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H ) @ ( genideal_a_b @ r @ I ) ) ) ) ).
% subset_Idl_subset
thf(fact_194_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_195_Units__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_196_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_197_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_198_to__contain__is__to__divide,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( cgenid547466209912283029xt_a_b @ r @ B2 ) @ ( cgenid547466209912283029xt_a_b @ r @ A ) )
= ( factor8216151070175719842xt_a_b @ r @ A @ B2 ) ) ) ) ).
% to_contain_is_to_divide
thf(fact_199_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_200_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_201_insert__subset,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A2 ) @ B )
= ( ( member_set_a @ X @ B )
& ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_202_insert__subset,axiom,
! [X: a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_203_mult__divides,axiom,
! [A: a,B2: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( 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 @ B2 ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ B2 ) ) ) ) ) ).
% mult_divides
thf(fact_204_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_205_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B2: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_206_singleton__insert__inj__eq,axiom,
! [B2: a,A: a,A2: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_207_Diff__eq__empty__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_208_m__closed,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_209_divides__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ A ) ) ).
% divides_refl
thf(fact_210_Units__m__closed,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y2 ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_211_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_212_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_213_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_214_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_215_Units__l__cancel,axiom,
! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
= ( Y2 = Z ) ) ) ) ) ).
% Units_l_cancel
thf(fact_216_divides__mult__rI,axiom,
! [A: a,B2: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( 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 @ B2 @ C ) ) ) ) ) ) ).
% divides_mult_rI
thf(fact_217_divides__mult__lI,axiom,
! [A: a,B2: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B2 )
=> ( ( 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 @ B2 ) ) ) ) ) ).
% divides_mult_lI
thf(fact_218_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_219_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_220_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_221_Collect__mono__iff,axiom,
! [P2: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_222_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_223_subset__trans,axiom,
! [A2: set_a,B: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C3 )
=> ( ord_less_eq_set_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_224_Collect__mono,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_225_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_226_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A5 )
=> ( member_set_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_227_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_228_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_229_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_230_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B5: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A5 )
=> ( member_set_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_231_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ( member_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_232_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_233_subsetD,axiom,
! [A2: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_234_subsetD,axiom,
! [A2: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_235_in__mono,axiom,
! [A2: set_set_a,B: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_236_in__mono,axiom,
! [A2: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_237_subset__insertI2,axiom,
! [A2: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_238_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).
% subset_insertI
thf(fact_239_subset__insert,axiom,
! [X: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_240_subset__insert,axiom,
! [X: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_241_insert__mono,axiom,
! [C3: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C3 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_242_double__diff,axiom,
! [A2: set_a,B: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C3 )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_243_Diff__subset,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_244_Diff__mono,axiom,
! [A2: set_a,C3: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_245_domain_Ointegral__iff,axiom,
! [R2: partia6043505979758434576t_unit,A: set_a,B2: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ B2 @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( ( mult_s7930653359683758801t_unit @ R2 @ A @ B2 )
= ( zero_s2174465271003423091t_unit @ R2 ) )
= ( ( A
= ( zero_s2174465271003423091t_unit @ R2 ) )
| ( B2
= ( zero_s2174465271003423091t_unit @ R2 ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_246_domain_Ointegral__iff,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B2: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ( mult_a_ring_ext_a_b @ R2 @ A @ B2 )
= ( zero_a_b @ R2 ) )
= ( ( A
= ( zero_a_b @ R2 ) )
| ( B2
= ( zero_a_b @ R2 ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_247_domain_Om__rcancel,axiom,
! [R2: partia6043505979758434576t_unit,A: set_a,B2: set_a,C: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( A
!= ( zero_s2174465271003423091t_unit @ R2 ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ B2 @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( ( mult_s7930653359683758801t_unit @ R2 @ B2 @ A )
= ( mult_s7930653359683758801t_unit @ R2 @ C @ A ) )
= ( B2 = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_248_domain_Om__rcancel,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B2: a,C: a] :
( ( domain_a_b @ R2 )
=> ( ( A
!= ( zero_a_b @ R2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ( mult_a_ring_ext_a_b @ R2 @ B2 @ A )
= ( mult_a_ring_ext_a_b @ R2 @ C @ A ) )
= ( B2 = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_249_domain_Om__lcancel,axiom,
! [R2: partia6043505979758434576t_unit,A: set_a,B2: set_a,C: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( A
!= ( zero_s2174465271003423091t_unit @ R2 ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ B2 @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( ( mult_s7930653359683758801t_unit @ R2 @ A @ B2 )
= ( mult_s7930653359683758801t_unit @ R2 @ A @ C ) )
= ( B2 = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_250_domain_Om__lcancel,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B2: a,C: a] :
( ( domain_a_b @ R2 )
=> ( ( A
!= ( zero_a_b @ R2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ( mult_a_ring_ext_a_b @ R2 @ A @ B2 )
= ( mult_a_ring_ext_a_b @ R2 @ A @ C ) )
= ( B2 = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_251_domain_Ointegral,axiom,
! [R2: partia6043505979758434576t_unit,A: set_a,B2: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( ( mult_s7930653359683758801t_unit @ R2 @ A @ B2 )
= ( zero_s2174465271003423091t_unit @ R2 ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ B2 @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( A
= ( zero_s2174465271003423091t_unit @ R2 ) )
| ( B2
= ( zero_s2174465271003423091t_unit @ R2 ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_252_domain_Ointegral,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B2: a] :
( ( domain_a_b @ R2 )
=> ( ( ( mult_a_ring_ext_a_b @ R2 @ A @ B2 )
= ( zero_a_b @ R2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( A
= ( zero_a_b @ R2 ) )
| ( B2
= ( zero_a_b @ R2 ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_253_Ring_Ointegral,axiom,
! [R2: partia6043505979758434576t_unit,A: set_a,B2: set_a] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( ( ( mult_s7930653359683758801t_unit @ R2 @ A @ B2 )
= ( zero_s2174465271003423091t_unit @ R2 ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ B2 @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( A
= ( zero_s2174465271003423091t_unit @ R2 ) )
| ( B2
= ( zero_s2174465271003423091t_unit @ R2 ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_254_Ring_Ointegral,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B2: a] :
( ( field_a_b @ R2 )
=> ( ( ( mult_a_ring_ext_a_b @ R2 @ A @ B2 )
= ( zero_a_b @ R2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( A
= ( zero_a_b @ R2 ) )
| ( B2
= ( zero_a_b @ R2 ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_255_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_256_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_257_primeideal_OI__prime,axiom,
! [I: set_a,R2: partia2175431115845679010xt_a_b,A: a,B2: a] :
( ( primeideal_a_b @ I @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ R2 @ A @ B2 ) @ I )
=> ( ( member_a @ A @ I )
| ( member_a @ B2 @ I ) ) ) ) ) ) ).
% primeideal.I_prime
thf(fact_258_subset__Diff__insert,axiom,
! [A2: set_set_a,B: set_set_a,X: set_a,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B @ ( insert_set_a @ X @ C3 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B @ C3 ) )
& ~ ( member_set_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_259_subset__Diff__insert,axiom,
! [A2: set_a,B: set_a,X: a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ ( insert_a @ X @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ C3 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_260_domain_Omult__divides,axiom,
! [R2: partia6043505979758434576t_unit,A: set_a,B2: set_a,C: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ B2 @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R2 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) )
=> ( ( factor5460682277579321776t_unit @ R2 @ ( mult_s7930653359683758801t_unit @ R2 @ C @ A ) @ ( mult_s7930653359683758801t_unit @ R2 @ C @ B2 ) )
=> ( factor5460682277579321776t_unit @ R2 @ A @ B2 ) ) ) ) ) ) ).
% domain.mult_divides
thf(fact_261_domain_Omult__divides,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B2: a,C: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ C @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
=> ( ( factor8216151070175719842xt_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ C @ A ) @ ( mult_a_ring_ext_a_b @ R2 @ C @ B2 ) )
=> ( factor8216151070175719842xt_a_b @ R2 @ A @ B2 ) ) ) ) ) ) ).
% domain.mult_divides
thf(fact_262_Diff__single__insert,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_263_subset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B ) )
= ( ( ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_264_subset__insert__iff,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_265_domain_Oring__irreducibleE_I5_J,axiom,
! [R2: partia6043505979758434576t_unit,R: set_a,A: set_a,B2: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ R @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( ring_r7790391342995787508t_unit @ R2 @ R )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ B2 @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( R
= ( mult_s7930653359683758801t_unit @ R2 @ A @ B2 ) )
=> ( ( member_set_a @ A @ ( units_2471184348132832486t_unit @ R2 ) )
| ( member_set_a @ B2 @ ( units_2471184348132832486t_unit @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_266_domain_Oring__irreducibleE_I5_J,axiom,
! [R2: partia2175431115845679010xt_a_b,R: a,A: a,B2: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ R @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_r999134135267193926le_a_b @ R2 @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( R
= ( mult_a_ring_ext_a_b @ R2 @ A @ B2 ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ R2 ) )
| ( member_a @ B2 @ ( units_a_ring_ext_a_b @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_267_principal__domain_Odomain__iff__prime,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a] :
( ( ring_p8803135361686045600in_a_b @ R2 )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ R2 @ ( cgenid547466209912283029xt_a_b @ R2 @ A ) ) )
= ( ring_ring_prime_a_b @ R2 @ A ) ) ) ) ).
% principal_domain.domain_iff_prime
thf(fact_268_field_Oaxioms_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R2 )
=> ( domain_a_b @ R2 ) ) ).
% field.axioms(1)
thf(fact_269_field_Oaxioms_I1_J,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( domain4236798911309298543t_unit @ R2 ) ) ).
% field.axioms(1)
thf(fact_270_noetherian__domain_Oexists__irreducible__divisor,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a] :
( ( ring_n4045954140777738665in_a_b @ R2 )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ R2 ) )
=> ~ ! [B4: a] :
( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_r999134135267193926le_a_b @ R2 @ B4 )
=> ~ ( factor8216151070175719842xt_a_b @ R2 @ B4 @ A ) ) ) ) ) ) ).
% noetherian_domain.exists_irreducible_divisor
thf(fact_271_domain_Oring__irreducibleI,axiom,
! [R2: partia6043505979758434576t_unit,R: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ R @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R2 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) )
=> ( ~ ( member_set_a @ R @ ( units_2471184348132832486t_unit @ R2 ) )
=> ( ! [A4: set_a,B4: set_a] :
( ( member_set_a @ A4 @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ B4 @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( R
= ( mult_s7930653359683758801t_unit @ R2 @ A4 @ B4 ) )
=> ( ( member_set_a @ A4 @ ( units_2471184348132832486t_unit @ R2 ) )
| ( member_set_a @ B4 @ ( units_2471184348132832486t_unit @ R2 ) ) ) ) ) )
=> ( ring_r7790391342995787508t_unit @ R2 @ R ) ) ) ) ) ).
% domain.ring_irreducibleI
thf(fact_272_domain_Oring__irreducibleI,axiom,
! [R2: partia2175431115845679010xt_a_b,R: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ R @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R @ ( units_a_ring_ext_a_b @ R2 ) )
=> ( ! [A4: a,B4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( R
= ( mult_a_ring_ext_a_b @ R2 @ A4 @ B4 ) )
=> ( ( member_a @ A4 @ ( units_a_ring_ext_a_b @ R2 ) )
| ( member_a @ B4 @ ( units_a_ring_ext_a_b @ R2 ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ R2 @ R ) ) ) ) ) ).
% domain.ring_irreducibleI
thf(fact_273_principal__domain_Ofield__iff__prime,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a] :
( ( ring_p8803135361686045600in_a_b @ R2 )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
=> ( ( field_6045675692312731021t_unit @ ( factRing_a_b @ R2 @ ( cgenid547466209912283029xt_a_b @ R2 @ A ) ) )
= ( ring_ring_prime_a_b @ R2 @ A ) ) ) ) ).
% principal_domain.field_iff_prime
thf(fact_274_domain_Oone__not__zero,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( one_se211549098623999037t_unit @ R2 )
!= ( zero_s2174465271003423091t_unit @ R2 ) ) ) ).
% domain.one_not_zero
thf(fact_275_domain_Oone__not__zero,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R2 )
=> ( ( one_a_ring_ext_a_b @ R2 )
!= ( zero_a_b @ R2 ) ) ) ).
% domain.one_not_zero
thf(fact_276_domain_Ozero__not__one,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( zero_s2174465271003423091t_unit @ R2 )
!= ( one_se211549098623999037t_unit @ R2 ) ) ) ).
% domain.zero_not_one
thf(fact_277_domain_Ozero__not__one,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R2 )
=> ( ( zero_a_b @ R2 )
!= ( one_a_ring_ext_a_b @ R2 ) ) ) ).
% domain.zero_not_one
thf(fact_278_Ring_Oone__not__zero,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( ( one_se211549098623999037t_unit @ R2 )
!= ( zero_s2174465271003423091t_unit @ R2 ) ) ) ).
% Ring.one_not_zero
thf(fact_279_Ring_Oone__not__zero,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R2 )
=> ( ( one_a_ring_ext_a_b @ R2 )
!= ( zero_a_b @ R2 ) ) ) ).
% Ring.one_not_zero
thf(fact_280_isgcd__divides__r,axiom,
! [B2: a,A: a] :
( ( factor8216151070175719842xt_a_b @ r @ B2 @ A )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ B2 @ A @ B2 ) ) ) ) ).
% isgcd_divides_r
thf(fact_281_isgcd__divides__l,axiom,
! [A: a,B2: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ A @ A @ B2 ) ) ) ) ).
% isgcd_divides_l
thf(fact_282_field_Of__comm__group__2,axiom,
! [R2: partia6043505979758434576t_unit,X: set_a] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( X
!= ( zero_s2174465271003423091t_unit @ R2 ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R2 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) )
& ( ( mult_s7930653359683758801t_unit @ R2 @ X3 @ X )
= ( one_se211549098623999037t_unit @ R2 ) ) ) ) ) ) ).
% field.f_comm_group_2
thf(fact_283_field_Of__comm__group__2,axiom,
! [R2: partia2175431115845679010xt_a_b,X: a] :
( ( field_a_b @ R2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( X
!= ( zero_a_b @ R2 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
& ( ( mult_a_ring_ext_a_b @ R2 @ X3 @ X )
= ( one_a_ring_ext_a_b @ R2 ) ) ) ) ) ) ).
% field.f_comm_group_2
thf(fact_284_domain_Ofinite__domain__units,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( finite_finite_set_a @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( units_2471184348132832486t_unit @ R2 )
= ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R2 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) ) ) ) ).
% domain.finite_domain_units
thf(fact_285_domain_Ofinite__domain__units,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R2 )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( units_a_ring_ext_a_b @ R2 )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) ) ) ) ).
% domain.finite_domain_units
thf(fact_286_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_287_dividesI_H,axiom,
! [B2: a,G: partia2175431115845679010xt_a_b,A: a,C: a] :
( ( B2
= ( mult_a_ring_ext_a_b @ G @ A @ C ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( factor8216151070175719842xt_a_b @ G @ A @ B2 ) ) ) ).
% dividesI'
thf(fact_288_dividesI_H,axiom,
! [B2: a,G: partia8223610829204095565t_unit,A: a,C: a] :
( ( B2
= ( mult_a_Product_unit @ G @ A @ C ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ A @ B2 ) ) ) ).
% dividesI'
thf(fact_289_monoid__cancelI,axiom,
( ! [A4: a,B4: a,C4: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C4 @ A4 )
= ( mult_a_ring_ext_a_b @ r @ C4 @ B4 ) )
=> ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A4 = B4 ) ) ) ) )
=> ( ! [A4: a,B4: a,C4: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A4 @ C4 )
= ( mult_a_ring_ext_a_b @ r @ B4 @ C4 ) )
=> ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A4 = B4 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_290_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_291_Divisibility_Oprime__def,axiom,
( prime_a_ring_ext_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b,P3: a] :
( ~ ( member_a @ P3 @ ( units_a_ring_ext_a_b @ G2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ P3 @ ( mult_a_ring_ext_a_b @ G2 @ X2 @ Y5 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ P3 @ X2 )
| ( factor8216151070175719842xt_a_b @ G2 @ P3 @ Y5 ) ) ) ) ) ) ) ) ).
% Divisibility.prime_def
thf(fact_292_Divisibility_Oprime__def,axiom,
( prime_a_Product_unit
= ( ^ [G2: partia8223610829204095565t_unit,P3: a] :
( ~ ( member_a @ P3 @ ( units_a_Product_unit @ G2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( factor3040189038382604065t_unit @ G2 @ P3 @ ( mult_a_Product_unit @ G2 @ X2 @ Y5 ) )
=> ( ( factor3040189038382604065t_unit @ G2 @ P3 @ X2 )
| ( factor3040189038382604065t_unit @ G2 @ P3 @ Y5 ) ) ) ) ) ) ) ) ).
% Divisibility.prime_def
thf(fact_293_polynomial__ring__assms,axiom,
subfield_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% polynomial_ring_assms
thf(fact_294_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_295_subring__props_I6_J,axiom,
! [K: set_a,H1: a,H2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H2 @ K )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H2 ) @ K ) ) ) ) ).
% subring_props(6)
thf(fact_296_subring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( K != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_297_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_298_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_299_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_300_finite__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_301_finite__Diff2,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_302_monoid__cancel_Ois__monoid__cancel,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( monoid5798828371819920185xt_a_b @ G ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_303_monoid__cancel_Ois__monoid__cancel,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid1999574367301118026t_unit @ G )
=> ( monoid1999574367301118026t_unit @ G ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_304_monoid__cancel_Ol__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,C: a,A: a,B2: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ C @ A )
= ( mult_a_ring_ext_a_b @ G @ C @ B2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A = B2 ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_305_monoid__cancel_Ol__cancel,axiom,
! [G: partia8223610829204095565t_unit,C: a,A: a,B2: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ C @ A )
= ( mult_a_Product_unit @ G @ C @ B2 ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( A = B2 ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_306_monoid__cancel_Or__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,C: a,B2: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ A @ C )
= ( mult_a_ring_ext_a_b @ G @ B2 @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A = B2 ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_307_monoid__cancel_Or__cancel,axiom,
! [G: partia8223610829204095565t_unit,A: a,C: a,B2: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ A @ C )
= ( mult_a_Product_unit @ G @ B2 @ C ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( A = B2 ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_308_isgcd__def,axiom,
( isgcd_a_ring_ext_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b,X2: a,A3: a,B6: a] :
( ( factor8216151070175719842xt_a_b @ G2 @ X2 @ A3 )
& ( factor8216151070175719842xt_a_b @ G2 @ X2 @ B6 )
& ! [Y5: a] :
( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( ( factor8216151070175719842xt_a_b @ G2 @ Y5 @ A3 )
& ( factor8216151070175719842xt_a_b @ G2 @ Y5 @ B6 ) )
=> ( factor8216151070175719842xt_a_b @ G2 @ Y5 @ X2 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_309_isgcd__def,axiom,
( isgcd_a_Product_unit
= ( ^ [G2: partia8223610829204095565t_unit,X2: a,A3: a,B6: a] :
( ( factor3040189038382604065t_unit @ G2 @ X2 @ A3 )
& ( factor3040189038382604065t_unit @ G2 @ X2 @ B6 )
& ! [Y5: a] :
( ( member_a @ Y5 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( ( factor3040189038382604065t_unit @ G2 @ Y5 @ A3 )
& ( factor3040189038382604065t_unit @ G2 @ Y5 @ B6 ) )
=> ( factor3040189038382604065t_unit @ G2 @ Y5 @ X2 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_310_monoid__cancel_Odivides__mult__l,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B2: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ C @ A ) @ ( mult_a_ring_ext_a_b @ G @ C @ B2 ) )
= ( factor8216151070175719842xt_a_b @ G @ A @ B2 ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_311_monoid__cancel_Odivides__mult__l,axiom,
! [G: partia8223610829204095565t_unit,A: a,B2: a,C: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ ( mult_a_Product_unit @ G @ C @ A ) @ ( mult_a_Product_unit @ G @ C @ B2 ) )
= ( factor3040189038382604065t_unit @ G @ A @ B2 ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_312_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_313_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_314_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_315_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_316_rev__finite__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_317_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_318_finite__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_319_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_320_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_321_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_322_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_323_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_324_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_325_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_326_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_327_dividesD,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B2: a] :
( ( factor8216151070175719842xt_a_b @ G @ A @ B2 )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
& ( B2
= ( mult_a_ring_ext_a_b @ G @ A @ X3 ) ) ) ) ).
% dividesD
thf(fact_328_dividesD,axiom,
! [G: partia8223610829204095565t_unit,A: a,B2: a] :
( ( factor3040189038382604065t_unit @ G @ A @ B2 )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
& ( B2
= ( mult_a_Product_unit @ G @ A @ X3 ) ) ) ) ).
% dividesD
thf(fact_329_dividesE,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B2: a] :
( ( factor8216151070175719842xt_a_b @ G @ A @ B2 )
=> ~ ! [C4: a] :
( ( B2
= ( mult_a_ring_ext_a_b @ G @ A @ C4 ) )
=> ~ ( member_a @ C4 @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% dividesE
thf(fact_330_dividesE,axiom,
! [G: partia8223610829204095565t_unit,A: a,B2: a] :
( ( factor3040189038382604065t_unit @ G @ A @ B2 )
=> ~ ! [C4: a] :
( ( B2
= ( mult_a_Product_unit @ G @ A @ C4 ) )
=> ~ ( member_a @ C4 @ ( partia6735698275553448452t_unit @ G ) ) ) ) ).
% dividesE
thf(fact_331_dividesI,axiom,
! [C: a,G: partia2175431115845679010xt_a_b,B2: a,A: a] :
( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( B2
= ( mult_a_ring_ext_a_b @ G @ A @ C ) )
=> ( factor8216151070175719842xt_a_b @ G @ A @ B2 ) ) ) ).
% dividesI
thf(fact_332_dividesI,axiom,
! [C: a,G: partia8223610829204095565t_unit,B2: a,A: a] :
( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( B2
= ( mult_a_Product_unit @ G @ A @ C ) )
=> ( factor3040189038382604065t_unit @ G @ A @ B2 ) ) ) ).
% dividesI
thf(fact_333_factor__def,axiom,
( factor8216151070175719842xt_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b,A3: a,B6: a] :
? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G2 ) )
& ( B6
= ( mult_a_ring_ext_a_b @ G2 @ A3 @ X2 ) ) ) ) ) ).
% factor_def
thf(fact_334_factor__def,axiom,
( factor3040189038382604065t_unit
= ( ^ [G2: partia8223610829204095565t_unit,A3: a,B6: a] :
? [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ G2 ) )
& ( B6
= ( mult_a_Product_unit @ G2 @ A3 @ X2 ) ) ) ) ) ).
% factor_def
thf(fact_335_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A6: set_a] :
( ? [A4: a] :
( A
= ( insert_a @ A4 @ A6 ) )
=> ~ ( finite_finite_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_336_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A3: set_a] :
( ( A3 = bot_bot_set_a )
| ? [A5: set_a,B6: a] :
( ( A3
= ( insert_a @ B6 @ A5 ) )
& ( finite_finite_a @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_337_finite__induct,axiom,
! [F: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ~ ( member_set_a @ X3 @ F2 )
=> ( ( P2 @ F2 )
=> ( P2 @ ( insert_set_a @ X3 @ F2 ) ) ) ) )
=> ( P2 @ F ) ) ) ) ).
% finite_induct
thf(fact_338_finite__induct,axiom,
! [F: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [X3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X3 @ F2 )
=> ( ( P2 @ F2 )
=> ( P2 @ ( insert_a @ X3 @ F2 ) ) ) ) )
=> ( P2 @ F ) ) ) ) ).
% finite_induct
thf(fact_339_finite__ne__induct,axiom,
! [F: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( F != bot_bot_set_set_a )
=> ( ! [X3: set_a] : ( P2 @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
=> ( ! [X3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( F2 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X3 @ F2 )
=> ( ( P2 @ F2 )
=> ( P2 @ ( insert_set_a @ X3 @ F2 ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_340_finite__ne__induct,axiom,
! [F: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( F != bot_bot_set_a )
=> ( ! [X3: a] : ( P2 @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F2 )
=> ( ( P2 @ F2 )
=> ( P2 @ ( insert_a @ X3 @ F2 ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_341_infinite__finite__induct,axiom,
! [P2: set_set_a > $o,A2: set_set_a] :
( ! [A6: set_set_a] :
( ~ ( finite_finite_set_a @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ~ ( member_set_a @ X3 @ F2 )
=> ( ( P2 @ F2 )
=> ( P2 @ ( insert_set_a @ X3 @ F2 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_342_infinite__finite__induct,axiom,
! [P2: set_a > $o,A2: set_a] :
( ! [A6: set_a] :
( ~ ( finite_finite_a @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [X3: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X3 @ F2 )
=> ( ( P2 @ F2 )
=> ( P2 @ ( insert_a @ X3 @ F2 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_343_finite__subset__induct,axiom,
! [F: set_set_a,A2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ F @ A2 )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [A4: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( member_set_a @ A4 @ A2 )
=> ( ~ ( member_set_a @ A4 @ F2 )
=> ( ( P2 @ F2 )
=> ( P2 @ ( insert_set_a @ A4 @ F2 ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_344_finite__subset__induct,axiom,
! [F: set_a,A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A4: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A4 @ A2 )
=> ( ~ ( member_a @ A4 @ F2 )
=> ( ( P2 @ F2 )
=> ( P2 @ ( insert_a @ A4 @ F2 ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_345_finite__subset__induct_H,axiom,
! [F: set_set_a,A2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ F )
=> ( ( ord_le3724670747650509150_set_a @ F @ A2 )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [A4: set_a,F2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( member_set_a @ A4 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ~ ( member_set_a @ A4 @ F2 )
=> ( ( P2 @ F2 )
=> ( P2 @ ( insert_set_a @ A4 @ F2 ) ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_346_finite__subset__induct_H,axiom,
! [F: set_a,A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A4: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A4 @ A2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ~ ( member_a @ A4 @ F2 )
=> ( ( P2 @ F2 )
=> ( P2 @ ( insert_a @ A4 @ F2 ) ) ) ) ) ) )
=> ( P2 @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_347_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_348_infinite__coinduct,axiom,
! [X5: set_a > $o,A2: set_a] :
( ( X5 @ A2 )
=> ( ! [A6: set_a] :
( ( X5 @ A6 )
=> ? [X4: a] :
( ( member_a @ X4 @ A6 )
& ( ( X5 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_349_finite__empty__induct,axiom,
! [A2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A4: set_a,A6: set_set_a] :
( ( finite_finite_set_a @ A6 )
=> ( ( member_set_a @ A4 @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_5736297505244876581_set_a @ A6 @ ( insert_set_a @ A4 @ bot_bot_set_set_a ) ) ) ) ) )
=> ( P2 @ bot_bot_set_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_350_finite__empty__induct,axiom,
! [A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A4: a,A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( member_a @ A4 @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_minus_set_a @ A6 @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ) )
=> ( P2 @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_351_field_Of__comm__group__1,axiom,
! [R2: partia6043505979758434576t_unit,X: set_a,Y2: set_a] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ Y2 @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( X
!= ( zero_s2174465271003423091t_unit @ R2 ) )
=> ( ( Y2
!= ( zero_s2174465271003423091t_unit @ R2 ) )
=> ( ( mult_s7930653359683758801t_unit @ R2 @ X @ Y2 )
!= ( zero_s2174465271003423091t_unit @ R2 ) ) ) ) ) ) ) ).
% field.f_comm_group_1
thf(fact_352_field_Of__comm__group__1,axiom,
! [R2: partia2175431115845679010xt_a_b,X: a,Y2: a] :
( ( field_a_b @ R2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( X
!= ( zero_a_b @ R2 ) )
=> ( ( Y2
!= ( zero_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ X @ Y2 )
!= ( zero_a_b @ R2 ) ) ) ) ) ) ) ).
% field.f_comm_group_1
thf(fact_353_remove__induct,axiom,
! [P2: set_set_a > $o,B: set_set_a] :
( ( P2 @ bot_bot_set_set_a )
=> ( ( ~ ( finite_finite_set_a @ B )
=> ( P2 @ B ) )
=> ( ! [A6: set_set_a] :
( ( finite_finite_set_a @ A6 )
=> ( ( A6 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A6 @ B )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A6 )
=> ( P2 @ ( minus_5736297505244876581_set_a @ A6 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% remove_induct
thf(fact_354_remove__induct,axiom,
! [P2: set_a > $o,B: set_a] :
( ( P2 @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P2 @ B ) )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( P2 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% remove_induct
thf(fact_355_finite__remove__induct,axiom,
! [B: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ B )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [A6: set_set_a] :
( ( finite_finite_set_a @ A6 )
=> ( ( A6 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A6 @ B )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A6 )
=> ( P2 @ ( minus_5736297505244876581_set_a @ A6 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_356_finite__remove__induct,axiom,
! [B: set_a,P2: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( P2 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_357_primeE,axiom,
! [G: partia2175431115845679010xt_a_b,P: a] :
( ( prime_a_ring_ext_a_b @ G @ P )
=> ~ ( ~ ( member_a @ P @ ( units_a_ring_ext_a_b @ G ) )
=> ~ ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ P @ ( mult_a_ring_ext_a_b @ G @ X4 @ Xa ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ P @ X4 )
| ( factor8216151070175719842xt_a_b @ G @ P @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_358_primeE,axiom,
! [G: partia8223610829204095565t_unit,P: a] :
( ( prime_a_Product_unit @ G @ P )
=> ~ ( ~ ( member_a @ P @ ( units_a_Product_unit @ G ) )
=> ~ ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ P @ ( mult_a_Product_unit @ G @ X4 @ Xa ) )
=> ( ( factor3040189038382604065t_unit @ G @ P @ X4 )
| ( factor3040189038382604065t_unit @ G @ P @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_359_Divisibility_OprimeI,axiom,
! [P: a,G: partia2175431115845679010xt_a_b] :
( ~ ( member_a @ P @ ( units_a_ring_ext_a_b @ G ) )
=> ( ! [A4: a,B4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ P @ ( mult_a_ring_ext_a_b @ G @ A4 @ B4 ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ P @ A4 )
| ( factor8216151070175719842xt_a_b @ G @ P @ B4 ) ) ) ) )
=> ( prime_a_ring_ext_a_b @ G @ P ) ) ) ).
% Divisibility.primeI
thf(fact_360_Divisibility_OprimeI,axiom,
! [P: a,G: partia8223610829204095565t_unit] :
( ~ ( member_a @ P @ ( units_a_Product_unit @ G ) )
=> ( ! [A4: a,B4: a] :
( ( member_a @ A4 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B4 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ P @ ( mult_a_Product_unit @ G @ A4 @ B4 ) )
=> ( ( factor3040189038382604065t_unit @ G @ P @ A4 )
| ( factor3040189038382604065t_unit @ G @ P @ B4 ) ) ) ) )
=> ( prime_a_Product_unit @ G @ P ) ) ) ).
% Divisibility.primeI
thf(fact_361_ring__primeI_H,axiom,
! [P: a] :
( ( member_a @ P @ ( 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 ) @ P )
=> ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% ring_primeI'
thf(fact_362_divides__imp__divides__mult,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B2 )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B2 ) ) ) ) ).
% divides_imp_divides_mult
thf(fact_363_line__extension__smult__closed,axiom,
! [K: set_a,E: set_a,A: a,K2: a,U: a] :
( ( subfield_a_b @ K @ r )
=> ( ! [K3: a,V: a] :
( ( member_a @ K3 @ K )
=> ( ( member_a @ V @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ V ) @ 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_364_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_365_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_366_space__subgroup__props_I6_J,axiom,
! [K: set_a,N: nat,E: set_a,K2: a,A: a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ 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_367_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_368_mult__of_Omonoid__cancel__axioms,axiom,
monoid1999574367301118026t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.monoid_cancel_axioms
thf(fact_369_mult__of_Ogcdof__exists,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [C4: a] :
( ( member_a @ C4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ C4 @ A @ B2 ) ) ) ) ).
% mult_of.gcdof_exists
thf(fact_370_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_371_mult__of_OUnits,axiom,
ord_less_eq_set_a @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ).
% mult_of.Units
thf(fact_372_mult__of_Omonoid__cancelI,axiom,
( ! [A4: a,B4: a,C4: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C4 @ A4 )
= ( mult_a_ring_ext_a_b @ r @ C4 @ B4 ) )
=> ( ( member_a @ A4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A4 = B4 ) ) ) ) )
=> ( ! [A4: a,B4: a,C4: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A4 @ C4 )
= ( mult_a_ring_ext_a_b @ r @ B4 @ C4 ) )
=> ( ( member_a @ A4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A4 = B4 ) ) ) ) )
=> ( monoid1999574367301118026t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.monoid_cancelI
thf(fact_373_mult__of_Or__cancel,axiom,
! [A: a,C: a,B2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ C )
= ( mult_a_ring_ext_a_b @ r @ B2 @ C ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A = B2 ) ) ) ) ) ).
% mult_of.r_cancel
thf(fact_374_mult__of_Om__lcomm,axiom,
! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( 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 @ Y2 @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% mult_of.m_lcomm
thf(fact_375_mult__of_Om__comm,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X ) ) ) ) ).
% mult_of.m_comm
thf(fact_376_mult__of_Om__assoc,axiom,
! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( 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 @ Y2 ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y2 @ Z ) ) ) ) ) ) ).
% mult_of.m_assoc
thf(fact_377_mult__of_Ol__cancel,axiom,
! [C: a,A: a,B2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C @ A )
= ( mult_a_ring_ext_a_b @ r @ C @ B2 ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A = B2 ) ) ) ) ) ).
% mult_of.l_cancel
thf(fact_378_mult__of_Ounit__factor,axiom,
! [A: a,B2: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( 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_379_mult__of_Oprod__unit__r,axiom,
! [A: a,B2: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( 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_380_mult__of_Oprod__unit__l,axiom,
! [A: a,B2: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) @ ( 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 @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ B2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.prod_unit_l
thf(fact_381_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_382_mult__of_Oisgcd__divides__l,axiom,
! [A: a,B2: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B2 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A @ A @ B2 ) ) ) ) ).
% mult_of.isgcd_divides_l
thf(fact_383_mult__of_Oisgcd__divides__r,axiom,
! [B2: a,A: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ B2 @ A )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ B2 @ A @ B2 ) ) ) ) ).
% mult_of.isgcd_divides_r
thf(fact_384_mult__of_Odivides__trans,axiom,
! [A: a,B2: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B2 )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ B2 @ 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_385_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_386_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_387_dimension__is__inj,axiom,
! [K: set_a,N: nat,E: set_a,M2: nat] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M2 @ K @ E )
=> ( N = M2 ) ) ) ) ).
% dimension_is_inj
thf(fact_388_inv__eq__imp__eq,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( m_inv_a_ring_ext_a_b @ r @ Y2 ) )
=> ( X = Y2 ) ) ) ) ).
% inv_eq_imp_eq
thf(fact_389_mult__of_Oinv__comm,axiom,
! [X: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.inv_comm
thf(fact_390_mult__of_Oinv__unique,axiom,
! [Y2: a,X: a,Y3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( Y2 = Y3 ) ) ) ) ) ) ).
% mult_of.inv_unique
thf(fact_391_mult__of_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.l_inv_ex
thf(fact_392_mult__of_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X3 )
= X3 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.one_unique
thf(fact_393_mult__of_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.r_inv_ex
thf(fact_394_mult__of_OUnits__inv__comm,axiom,
! [X: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.Units_inv_comm
thf(fact_395_mult__of_OUnits__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.Units_l_inv_ex
thf(fact_396_mult__of_OUnits__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.Units_r_inv_ex
thf(fact_397_mult__of_Odivides__prod__r,axiom,
! [A: a,B2: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B2 )
=> ( ( 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 @ B2 @ C ) ) ) ) ) ).
% mult_of.divides_prod_r
thf(fact_398_mult__of_Odivides__prod__l,axiom,
! [A: a,B2: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( 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 @ B2 )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( mult_a_ring_ext_a_b @ r @ C @ B2 ) ) ) ) ) ) ).
% mult_of.divides_prod_l
thf(fact_399_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_400_space__subgroup__props_I2_J,axiom,
! [K: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( member_a @ ( zero_a_b @ r ) @ E ) ) ) ).
% space_subgroup_props(2)
thf(fact_401_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_402_space__subgroup__props_I5_J,axiom,
! [K: set_a,N: nat,E: set_a,K2: a,V2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ V2 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ V2 ) @ E ) ) ) ) ) ).
% space_subgroup_props(5)
thf(fact_403_a__l__coset__subset__G,axiom,
! [H: set_a,X: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_404_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_405_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_406_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_407_space__subgroup__props_I1_J,axiom,
! [K: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% space_subgroup_props(1)
thf(fact_408_inv__unique_H,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( Y2
= ( m_inv_a_ring_ext_a_b @ r @ X ) ) ) ) ) ) ).
% inv_unique'
thf(fact_409_inv__char,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ X )
= Y2 ) ) ) ) ) ).
% inv_char
thf(fact_410_comm__inv__char,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ X )
= Y2 ) ) ) ) ).
% comm_inv_char
thf(fact_411_mult__of_Oprime__divides,axiom,
! [A: a,B2: a,P: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ A )
| ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ B2 ) ) ) ) ) ) ).
% mult_of.prime_divides
thf(fact_412_ring__primeE_I2_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ) ).
% ring_primeE(2)
thf(fact_413_prime__eq__prime__mult,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P )
= ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ) ).
% prime_eq_prime_mult
thf(fact_414_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_415_Ring__Divisibility_Omult__mult__of,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( mult_a_Product_unit @ ( ring_mult_of_a_b @ R2 ) )
= ( mult_a_ring_ext_a_b @ R2 ) ) ).
% Ring_Divisibility.mult_mult_of
thf(fact_416_mult__of_Olcmof__exists,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [C4: a] :
( ( member_a @ C4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( islcm_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ C4 @ A @ B2 ) ) ) ) ).
% mult_of.lcmof_exists
thf(fact_417_mult__of_OUnits__eq,axiom,
( ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) )
= ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ).
% mult_of.Units_eq
thf(fact_418_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_419_mult__of_Om__closed,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.m_closed
thf(fact_420_mult__of_Oright__cancel,axiom,
! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( 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 @ Y2 @ X )
= ( mult_a_ring_ext_a_b @ r @ Z @ X ) )
= ( Y2 = Z ) ) ) ) ) ).
% mult_of.right_cancel
thf(fact_421_mult__of_OUnits__m__closed,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y2 ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.Units_m_closed
thf(fact_422_mult__of_OUnits__l__cancel,axiom,
! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( 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 @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
= ( Y2 = Z ) ) ) ) ) ).
% mult_of.Units_l_cancel
thf(fact_423_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_424_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_425_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_426_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_427_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_428_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_429_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_430_mult__of_Ol__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ A )
= X )
= ( A
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.l_cancel_one
thf(fact_431_mult__of_Ol__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( X
= ( mult_a_ring_ext_a_b @ r @ X @ A ) )
= ( A
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.l_cancel_one'
thf(fact_432_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_433_mult__of_Or__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ X )
= X )
= ( A
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.r_cancel_one
thf(fact_434_mult__of_Or__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( X
= ( mult_a_ring_ext_a_b @ r @ A @ X ) )
= ( A
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.r_cancel_one'
thf(fact_435_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_436_mult__of_Odivides__mult__rI,axiom,
! [A: a,B2: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B2 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( 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 @ B2 @ C ) ) ) ) ) ) ).
% mult_of.divides_mult_rI
thf(fact_437_mult__of_Odivides__mult__r,axiom,
! [A: a,B2: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( 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 @ B2 @ C ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B2 ) ) ) ) ) ).
% mult_of.divides_mult_r
thf(fact_438_mult__of_Odivides__mult__lI,axiom,
! [A: a,B2: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B2 )
=> ( ( 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 @ B2 ) ) ) ) ) ).
% mult_of.divides_mult_lI
thf(fact_439_mult__of_Odivides__mult__l,axiom,
! [A: a,B2: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( 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 @ B2 ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B2 ) ) ) ) ) ).
% mult_of.divides_mult_l
thf(fact_440_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_441_Ring__Divisibility_Ocarrier__mult__of,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ R2 ) )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) ) ).
% Ring_Divisibility.carrier_mult_of
thf(fact_442_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_443_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_444_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_445_mult__of_Oprimeness__condition__monoid__axioms,axiom,
primen965786292471834261t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.primeness_condition_monoid_axioms
thf(fact_446_divides__mult__imp__divides,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B2: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ R2 ) @ A @ B2 )
=> ( factor8216151070175719842xt_a_b @ R2 @ A @ B2 ) ) ).
% divides_mult_imp_divides
thf(fact_447_domain_Ozero__is__prime_I2_J,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( prime_8576247383786985867t_unit @ ( ring_m2800496791135293897t_unit @ R2 ) @ ( zero_s2174465271003423091t_unit @ R2 ) ) ) ).
% domain.zero_is_prime(2)
thf(fact_448_domain_Ozero__is__prime_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R2 )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ R2 ) @ ( zero_a_b @ R2 ) ) ) ).
% domain.zero_is_prime(2)
thf(fact_449_domain_OUnits__mult__eq__Units,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( units_3682061850359937035t_unit @ ( ring_m2800496791135293897t_unit @ R2 ) )
= ( units_2471184348132832486t_unit @ R2 ) ) ) ).
% domain.Units_mult_eq_Units
thf(fact_450_domain_OUnits__mult__eq__Units,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R2 )
=> ( ( units_a_Product_unit @ ( ring_mult_of_a_b @ R2 ) )
= ( units_a_ring_ext_a_b @ R2 ) ) ) ).
% domain.Units_mult_eq_Units
thf(fact_451_domain_Odivides__mult__zero,axiom,
! [R2: partia6043505979758434576t_unit,A: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( factor8582526991245238721t_unit @ ( ring_m2800496791135293897t_unit @ R2 ) @ A @ ( zero_s2174465271003423091t_unit @ R2 ) )
=> ( A
= ( zero_s2174465271003423091t_unit @ R2 ) ) ) ) ) ).
% domain.divides_mult_zero
thf(fact_452_domain_Odivides__mult__zero,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ R2 ) @ A @ ( zero_a_b @ R2 ) )
=> ( A
= ( zero_a_b @ R2 ) ) ) ) ) ).
% domain.divides_mult_zero
thf(fact_453_domain_Oring__primeE_I2_J,axiom,
! [R2: partia6043505979758434576t_unit,P: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ P @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( ring_r6795642478576035723t_unit @ R2 @ P )
=> ( prime_8576247383786985867t_unit @ ( ring_m2800496791135293897t_unit @ R2 ) @ P ) ) ) ) ).
% domain.ring_primeE(2)
thf(fact_454_domain_Oring__primeE_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_ring_prime_a_b @ R2 @ P )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ R2 ) @ P ) ) ) ) ).
% domain.ring_primeE(2)
thf(fact_455_domain_Oprime__eq__prime__mult,axiom,
! [R2: partia6043505979758434576t_unit,P: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ P @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( prime_4522187476880896870t_unit @ R2 @ P )
= ( prime_8576247383786985867t_unit @ ( ring_m2800496791135293897t_unit @ R2 ) @ P ) ) ) ) ).
% domain.prime_eq_prime_mult
thf(fact_456_domain_Oprime__eq__prime__mult,axiom,
! [R2: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( prime_a_ring_ext_a_b @ R2 @ P )
= ( prime_a_Product_unit @ ( ring_mult_of_a_b @ R2 ) @ P ) ) ) ) ).
% domain.prime_eq_prime_mult
thf(fact_457_domain_Odivides__imp__divides__mult,axiom,
! [R2: partia6043505979758434576t_unit,A: set_a,B2: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ B2 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R2 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) )
=> ( ( factor5460682277579321776t_unit @ R2 @ A @ B2 )
=> ( factor8582526991245238721t_unit @ ( ring_m2800496791135293897t_unit @ R2 ) @ A @ B2 ) ) ) ) ) ).
% domain.divides_imp_divides_mult
thf(fact_458_domain_Odivides__imp__divides__mult,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B2: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
=> ( ( factor8216151070175719842xt_a_b @ R2 @ A @ B2 )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ R2 ) @ A @ B2 ) ) ) ) ) ).
% domain.divides_imp_divides_mult
thf(fact_459_domain_Oring__primeI_H,axiom,
! [R2: partia6043505979758434576t_unit,P: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ P @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R2 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) )
=> ( ( prime_8576247383786985867t_unit @ ( ring_m2800496791135293897t_unit @ R2 ) @ P )
=> ( ring_r6795642478576035723t_unit @ R2 @ P ) ) ) ) ).
% domain.ring_primeI'
thf(fact_460_domain_Oring__primeI_H,axiom,
! [R2: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ P @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
=> ( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ R2 ) @ P )
=> ( ring_ring_prime_a_b @ R2 @ P ) ) ) ) ).
% domain.ring_primeI'
thf(fact_461_mult__of_Odivides__fcount,axiom,
! [A: a,B2: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B2 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( 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 ) @ B2 ) ) ) ) ) ).
% mult_of.divides_fcount
thf(fact_462_mult__of_Otrivial__group__subset,axiom,
( ( elemen1145482699608675729t_unit @ ( ring_mult_of_a_b @ r ) )
= ( ord_less_eq_set_a @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% mult_of.trivial_group_subset
thf(fact_463_mult__of_Otrivial__group__alt,axiom,
( ( elemen1145482699608675729t_unit @ ( ring_mult_of_a_b @ r ) )
= ( ? [A3: a] : ( ord_less_eq_set_a @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% mult_of.trivial_group_alt
thf(fact_464_mult__of_Otrivial__group,axiom,
( ( elemen1145482699608675729t_unit @ ( ring_mult_of_a_b @ r ) )
= ( ? [A3: a] :
( ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) )
= ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% mult_of.trivial_group
thf(fact_465_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_466_mult__of_Oderived__eq__singleton,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( genera353947490595344117t_unit @ ( ring_mult_of_a_b @ r ) @ H )
= ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% mult_of.derived_eq_singleton
thf(fact_467_ring__irreducibleI_H,axiom,
! [R: a] :
( ( member_a @ R @ ( 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 ) @ R )
=> ( ring_r999134135267193926le_a_b @ r @ R ) ) ) ).
% ring_irreducibleI'
thf(fact_468_zero__is__irreducible__mult,axiom,
irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( zero_a_b @ r ) ).
% zero_is_irreducible_mult
thf(fact_469_mult__of_Omono__derived,axiom,
! [K: set_a,H: set_a] :
( ( ord_less_eq_set_a @ K @ H )
=> ( ord_less_eq_set_a @ ( genera353947490595344117t_unit @ ( ring_mult_of_a_b @ r ) @ K ) @ ( genera353947490595344117t_unit @ ( ring_mult_of_a_b @ r ) @ H ) ) ) ).
% mult_of.mono_derived
thf(fact_470_mult__of_Oprime__irreducible,axiom,
! [P: a] :
( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ).
% mult_of.prime_irreducible
thf(fact_471_mult__of_Oderived__in__carrier,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ord_less_eq_set_a @ ( genera353947490595344117t_unit @ ( ring_mult_of_a_b @ r ) @ H ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.derived_in_carrier
thf(fact_472_ring__irreducibleE_I3_J,axiom,
! [R: a] :
( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ R ) ) ) ).
% ring_irreducibleE(3)
thf(fact_473_mult__of_Oirreducible__prime,axiom,
! [P: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ( member_a @ P @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ) ).
% mult_of.irreducible_prime
thf(fact_474_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_475_mult__of_Oirreducible__prodE,axiom,
! [A: a,B2: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A )
=> ~ ( member_a @ B2 @ ( 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 ) @ B2 ) ) ) ) ) ) ).
% mult_of.irreducible_prodE
thf(fact_476_mult__of_Oirreducible__prod__lI,axiom,
! [B2: a,A: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ B2 )
=> ( ( 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 @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) ) ) ) ) ) ).
% mult_of.irreducible_prod_lI
thf(fact_477_mult__of_Oirreducible__prod__rI,axiom,
! [A: a,B2: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A )
=> ( ( member_a @ B2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) ) ) ) ) ) ).
% mult_of.irreducible_prod_rI
thf(fact_478_primeness__condition__monoid_Oirreducible__prime,axiom,
! [G: partia2175431115845679010xt_a_b,A: a] :
( ( primen9005823089519874350xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( irredu6211895646901577903xt_a_b @ G @ A )
=> ( prime_a_ring_ext_a_b @ G @ A ) ) ) ) ).
% primeness_condition_monoid.irreducible_prime
thf(fact_479_primeness__condition__monoid_Oirreducible__prime,axiom,
! [G: partia8223610829204095565t_unit,A: a] :
( ( primen965786292471834261t_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( irredu4023057619401689684t_unit @ G @ A )
=> ( prime_a_Product_unit @ G @ A ) ) ) ) ).
% primeness_condition_monoid.irreducible_prime
thf(fact_480_domain_Ozero__is__irreducible__mult,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( irredu8646402277169070324t_unit @ ( ring_m2800496791135293897t_unit @ R2 ) @ ( zero_s2174465271003423091t_unit @ R2 ) ) ) ).
% domain.zero_is_irreducible_mult
thf(fact_481_domain_Ozero__is__irreducible__mult,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R2 )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R2 ) @ ( zero_a_b @ R2 ) ) ) ).
% domain.zero_is_irreducible_mult
thf(fact_482_domain_Oring__irreducibleE_I3_J,axiom,
! [R2: partia6043505979758434576t_unit,R: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ R @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( ring_r7790391342995787508t_unit @ R2 @ R )
=> ( irredu8646402277169070324t_unit @ ( ring_m2800496791135293897t_unit @ R2 ) @ R ) ) ) ) ).
% domain.ring_irreducibleE(3)
thf(fact_483_domain_Oring__irreducibleE_I3_J,axiom,
! [R2: partia2175431115845679010xt_a_b,R: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ R @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_r999134135267193926le_a_b @ R2 @ R )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R2 ) @ R ) ) ) ) ).
% domain.ring_irreducibleE(3)
thf(fact_484_islcm__def,axiom,
( islcm_a_ring_ext_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b,X2: a,A3: a,B6: a] :
( ( factor8216151070175719842xt_a_b @ G2 @ A3 @ X2 )
& ( factor8216151070175719842xt_a_b @ G2 @ B6 @ X2 )
& ! [Y5: a] :
( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( ( factor8216151070175719842xt_a_b @ G2 @ A3 @ Y5 )
& ( factor8216151070175719842xt_a_b @ G2 @ B6 @ Y5 ) )
=> ( factor8216151070175719842xt_a_b @ G2 @ X2 @ Y5 ) ) ) ) ) ) ).
% islcm_def
thf(fact_485_islcm__def,axiom,
( islcm_a_Product_unit
= ( ^ [G2: partia8223610829204095565t_unit,X2: a,A3: a,B6: a] :
( ( factor3040189038382604065t_unit @ G2 @ A3 @ X2 )
& ( factor3040189038382604065t_unit @ G2 @ B6 @ X2 )
& ! [Y5: a] :
( ( member_a @ Y5 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( ( factor3040189038382604065t_unit @ G2 @ A3 @ Y5 )
& ( factor3040189038382604065t_unit @ G2 @ B6 @ Y5 ) )
=> ( factor3040189038382604065t_unit @ G2 @ X2 @ Y5 ) ) ) ) ) ) ).
% islcm_def
thf(fact_486_domain_Oring__irreducibleI_H,axiom,
! [R2: partia6043505979758434576t_unit,R: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ R @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R2 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) )
=> ( ( irredu8646402277169070324t_unit @ ( ring_m2800496791135293897t_unit @ R2 ) @ R )
=> ( ring_r7790391342995787508t_unit @ R2 @ R ) ) ) ) ).
% domain.ring_irreducibleI'
thf(fact_487_domain_Oring__irreducibleI_H,axiom,
! [R2: partia2175431115845679010xt_a_b,R: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ R @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
=> ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R2 ) @ R )
=> ( ring_r999134135267193926le_a_b @ R2 @ R ) ) ) ) ).
% domain.ring_irreducibleI'
thf(fact_488_diff__ge__0__iff__ge,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
= ( ord_less_eq_int @ B2 @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_489_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 ) ) )
=> ~ ! [V: a,E2: set_a,N2: nat] :
( ( A1
= ( suc @ N2 ) )
=> ( ( A32
= ( embedd971793762689825387on_a_b @ r @ A22 @ V @ E2 ) )
=> ( ( member_a @ V @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ V @ E2 )
=> ~ ( embedd2795209813406577254on_a_b @ r @ N2 @ A22 @ E2 ) ) ) ) ) ) ) ).
% dimension.cases
thf(fact_490_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 ) ) )
| ? [V3: a,E3: set_a,N3: nat,K4: set_a] :
( ( A1
= ( suc @ N3 ) )
& ( A22 = K4 )
& ( A32
= ( embedd971793762689825387on_a_b @ r @ K4 @ V3 @ E3 ) )
& ( member_a @ V3 @ ( partia707051561876973205xt_a_b @ r ) )
& ~ ( member_a @ V3 @ E3 )
& ( embedd2795209813406577254on_a_b @ r @ N3 @ K4 @ E3 ) ) ) ) ).
% dimension.simps
thf(fact_491_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_492_ring__irreducibleE_I2_J,axiom,
! [R: a] :
( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R )
=> ( irredu6211895646901577903xt_a_b @ r @ R ) ) ) ).
% ring_irreducibleE(2)
thf(fact_493_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_494_Suc__dim,axiom,
! [V2: a,E: set_a,N: nat,K: set_a] :
( ( member_a @ V2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ V2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( embedd2795209813406577254on_a_b @ r @ ( suc @ N ) @ K @ ( embedd971793762689825387on_a_b @ r @ K @ V2 @ E ) ) ) ) ) ).
% Suc_dim
thf(fact_495_irreducible__prod__rI,axiom,
! [A: a,B2: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( ( member_a @ B2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) ) ) ) ) ) ).
% irreducible_prod_rI
thf(fact_496_irreducible__prod__lI,axiom,
! [B2: a,A: a] :
( ( irredu6211895646901577903xt_a_b @ r @ B2 )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) ) ) ) ) ) ).
% irreducible_prod_lI
thf(fact_497_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_498_dimension__backwards,axiom,
! [K: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ ( suc @ N ) @ K @ E )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ? [E4: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E4 )
& ~ ( member_a @ X3 @ E4 )
& ( E
= ( embedd971793762689825387on_a_b @ r @ K @ X3 @ E4 ) ) ) ) ) ) ).
% dimension_backwards
thf(fact_499_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_500_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_501_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_502_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_503_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_504_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_505_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_506_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_507_ring__irreducible__def,axiom,
( ring_r999134135267193926le_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,A3: a] :
( ( A3
!= ( zero_a_b @ R3 ) )
& ( irredu6211895646901577903xt_a_b @ R3 @ A3 ) ) ) ) ).
% ring_irreducible_def
thf(fact_508_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_509_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 )
= ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_510_diff__eq__diff__eq,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B2 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_511_domain_Oring__irreducibleE_I2_J,axiom,
! [R2: partia6043505979758434576t_unit,R: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ R @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( ring_r7790391342995787508t_unit @ R2 @ R )
=> ( irredu5346329325703585725t_unit @ R2 @ R ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_512_domain_Oring__irreducibleE_I2_J,axiom,
! [R2: partia2175431115845679010xt_a_b,R: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ R @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_r999134135267193926le_a_b @ R2 @ R )
=> ( irredu6211895646901577903xt_a_b @ R2 @ R ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_513_domain_Ozero__is__irreducible__iff__field,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( irredu5346329325703585725t_unit @ R2 @ ( zero_s2174465271003423091t_unit @ R2 ) )
= ( field_6045675692312731021t_unit @ R2 ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_514_domain_Ozero__is__irreducible__iff__field,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R2 )
=> ( ( irredu6211895646901577903xt_a_b @ R2 @ ( zero_a_b @ R2 ) )
= ( field_a_b @ R2 ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_515_domain_Oirreducible__imp__irreducible__mult,axiom,
! [R2: partia6043505979758434576t_unit,A: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( irredu5346329325703585725t_unit @ R2 @ A )
=> ( irredu8646402277169070324t_unit @ ( ring_m2800496791135293897t_unit @ R2 ) @ A ) ) ) ) ).
% domain.irreducible_imp_irreducible_mult
thf(fact_516_domain_Oirreducible__imp__irreducible__mult,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( irredu6211895646901577903xt_a_b @ R2 @ A )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R2 ) @ A ) ) ) ) ).
% domain.irreducible_imp_irreducible_mult
thf(fact_517_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_518_diff__eq__diff__less__eq,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B2 )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_519_diff__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_520_diff__left__mono,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_521_diff__mono,axiom,
! [A: int,B2: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_mono
thf(fact_522_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A3: int,B6: int] :
( ( minus_minus_int @ A3 @ B6 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_523_domain_Oirreducible__mult__imp__irreducible,axiom,
! [R2: partia6043505979758434576t_unit,A: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ A @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R2 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) )
=> ( ( irredu8646402277169070324t_unit @ ( ring_m2800496791135293897t_unit @ R2 ) @ A )
=> ( irredu5346329325703585725t_unit @ R2 @ A ) ) ) ) ).
% domain.irreducible_mult_imp_irreducible
thf(fact_524_domain_Oirreducible__mult__imp__irreducible,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
=> ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R2 ) @ A )
=> ( irredu6211895646901577903xt_a_b @ R2 @ A ) ) ) ) ).
% domain.irreducible_mult_imp_irreducible
thf(fact_525_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B6: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B6 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_526_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_527_mult__of_Orcosets__finite,axiom,
! [R2: set_a,H: set_a] :
( ( member_set_a @ R2 @ ( rCOSET407642731378740692t_unit @ ( ring_mult_of_a_b @ r ) @ H ) )
=> ( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( finite_finite_a @ H )
=> ( finite_finite_a @ R2 ) ) ) ) ).
% mult_of.rcosets_finite
thf(fact_528_mult__of_Ocosets__finite,axiom,
! [C: set_a,H: set_a] :
( ( member_set_a @ C @ ( rCOSET407642731378740692t_unit @ ( ring_mult_of_a_b @ r ) @ H ) )
=> ( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( finite_finite_a @ C ) ) ) ) ).
% mult_of.cosets_finite
thf(fact_529_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_530_mult__of_Olcos__mult__one,axiom,
! [M: set_a] :
( ( ord_less_eq_set_a @ M @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( l_cose374570025740779235t_unit @ ( ring_mult_of_a_b @ r ) @ ( one_a_ring_ext_a_b @ r ) @ M )
= M ) ) ).
% mult_of.lcos_mult_one
thf(fact_531_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_532_mult__of_Ol__coset__subset__G,axiom,
! [H: set_a,X: a] :
( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ord_less_eq_set_a @ ( l_cose374570025740779235t_unit @ ( ring_mult_of_a_b @ r ) @ X @ H ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.l_coset_subset_G
thf(fact_533_mult__of_Olcos__m__assoc,axiom,
! [M: set_a,G3: a,H3: a] :
( ( ord_less_eq_set_a @ M @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ G3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ H3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( l_cose374570025740779235t_unit @ ( ring_mult_of_a_b @ r ) @ G3 @ ( l_cose374570025740779235t_unit @ ( ring_mult_of_a_b @ r ) @ H3 @ M ) )
= ( l_cose374570025740779235t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ G3 @ H3 ) @ M ) ) ) ) ) ).
% mult_of.lcos_m_assoc
thf(fact_534_Multiplicative__Group_Ocarrier__mult__of,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( partia6735698275553448452t_unit @ ( multip3210463924028840165of_a_b @ R2 ) )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) ) ).
% Multiplicative_Group.carrier_mult_of
thf(fact_535_mult__of__is__Units,axiom,
( ( multip3210463924028840165of_a_b @ r )
= ( units_8174867845824275201xt_a_b @ r ) ) ).
% mult_of_is_Units
thf(fact_536_Multiplicative__Group_Omult__mult__of,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( mult_a_Product_unit @ ( multip3210463924028840165of_a_b @ R2 ) )
= ( mult_a_ring_ext_a_b @ R2 ) ) ).
% Multiplicative_Group.mult_mult_of
thf(fact_537_field_OsubfieldI_H,axiom,
! [R2: partia6043505979758434576t_unit,K: set_set_a] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( ( subrin1511138061850335568t_unit @ K @ R2 )
=> ( ! [K3: set_a] :
( ( member_set_a @ K3 @ ( minus_5736297505244876581_set_a @ K @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) )
=> ( member_set_a @ ( m_inv_7491079437187478987t_unit @ R2 @ K3 ) @ K ) )
=> ( subfie5224850075530046424t_unit @ K @ R2 ) ) ) ) ).
% field.subfieldI'
thf(fact_538_field_OsubfieldI_H,axiom,
! [R2: partia2175431115845679010xt_a_b,K: set_a] :
( ( field_a_b @ R2 )
=> ( ( subring_a_b @ K @ R2 )
=> ( ! [K3: a] :
( ( member_a @ K3 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ R2 @ K3 ) @ K ) )
=> ( subfield_a_b @ K @ R2 ) ) ) ) ).
% field.subfieldI'
thf(fact_539_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_540_mult__of_Osubgroup__self,axiom,
subgro3222307229058429633t_unit @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) @ ( ring_mult_of_a_b @ r ) ).
% mult_of.subgroup_self
thf(fact_541_mult__of_Osolve__equation,axiom,
! [H: set_a,X: a,Y2: a] :
( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( ( member_a @ X @ H )
=> ( ( member_a @ Y2 @ H )
=> ? [X3: a] :
( ( member_a @ X3 @ H )
& ( Y2
= ( mult_a_ring_ext_a_b @ r @ X3 @ X ) ) ) ) ) ) ).
% mult_of.solve_equation
thf(fact_542_mult__of_OsubgroupE_I4_J,axiom,
! [H: set_a,A: a,B2: a] :
( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( ( member_a @ A @ H )
=> ( ( member_a @ B2 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) @ H ) ) ) ) ).
% mult_of.subgroupE(4)
thf(fact_543_mult__of_OsubgroupE_I2_J,axiom,
! [H: set_a] :
( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( H != bot_bot_set_a ) ) ).
% mult_of.subgroupE(2)
thf(fact_544_mult__of_OsubgroupE_I1_J,axiom,
! [H: set_a] :
( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.subgroupE(1)
thf(fact_545_mult__of_Oderived__incl,axiom,
! [K: set_a,H: set_a] :
( ( ord_less_eq_set_a @ K @ H )
=> ( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( genera353947490595344117t_unit @ ( ring_mult_of_a_b @ r ) @ K ) @ H ) ) ) ).
% mult_of.derived_incl
thf(fact_546_mult__of_Ocoset__join3,axiom,
! [X: a,H: set_a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( ( member_a @ X @ H )
=> ( ( l_cose374570025740779235t_unit @ ( ring_mult_of_a_b @ r ) @ X @ H )
= H ) ) ) ) ).
% mult_of.coset_join3
thf(fact_547_mult__of_Ol__coset__carrier,axiom,
! [Y2: a,X: a,H: set_a] :
( ( member_a @ Y2 @ ( l_cose374570025740779235t_unit @ ( ring_mult_of_a_b @ r ) @ X @ H ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.l_coset_carrier
thf(fact_548_mult__of_Ol__coset__swap,axiom,
! [Y2: a,X: a,H: set_a] :
( ( member_a @ Y2 @ ( l_cose374570025740779235t_unit @ ( ring_mult_of_a_b @ r ) @ X @ H ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( member_a @ X @ ( l_cose374570025740779235t_unit @ ( ring_mult_of_a_b @ r ) @ Y2 @ H ) ) ) ) ) ).
% mult_of.l_coset_swap
thf(fact_549_mult__of_Ol__repr__independence,axiom,
! [Y2: a,X: a,H: set_a] :
( ( member_a @ Y2 @ ( l_cose374570025740779235t_unit @ ( ring_mult_of_a_b @ r ) @ X @ H ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( ( l_cose374570025740779235t_unit @ ( ring_mult_of_a_b @ r ) @ X @ H )
= ( l_cose374570025740779235t_unit @ ( ring_mult_of_a_b @ r ) @ Y2 @ H ) ) ) ) ) ).
% mult_of.l_repr_independence
thf(fact_550_mult__of_Oderived__is__subgroup,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( subgro3222307229058429633t_unit @ ( genera353947490595344117t_unit @ ( ring_mult_of_a_b @ r ) @ H ) @ ( ring_mult_of_a_b @ r ) ) ) ).
% mult_of.derived_is_subgroup
thf(fact_551_field_Omult__of__is__Units,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( ( multip3774352783277980819t_unit @ R2 )
= ( units_1455294149231095823t_unit @ R2 ) ) ) ).
% field.mult_of_is_Units
thf(fact_552_field_Omult__of__is__Units,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R2 )
=> ( ( multip3210463924028840165of_a_b @ R2 )
= ( units_8174867845824275201xt_a_b @ R2 ) ) ) ).
% field.mult_of_is_Units
thf(fact_553_field_Osubgroup__mult__of,axiom,
! [R2: partia6043505979758434576t_unit,K: set_set_a] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( ( subfie5224850075530046424t_unit @ K @ R2 )
=> ( subgro7904897551812261217t_unit @ ( minus_5736297505244876581_set_a @ K @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) @ ( multip3774352783277980819t_unit @ R2 ) ) ) ) ).
% field.subgroup_mult_of
thf(fact_554_field_Osubgroup__mult__of,axiom,
! [R2: partia2175431115845679010xt_a_b,K: set_a] :
( ( field_a_b @ R2 )
=> ( ( subfield_a_b @ K @ R2 )
=> ( subgro3222307229058429633t_unit @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) @ ( multip3210463924028840165of_a_b @ R2 ) ) ) ) ).
% field.subgroup_mult_of
thf(fact_555_subringE_I2_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R2 )
=> ( member_a @ ( zero_a_b @ R2 ) @ H ) ) ).
% subringE(2)
thf(fact_556_subringE_I4_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R2 )
=> ( H != bot_bot_set_a ) ) ).
% subringE(4)
thf(fact_557_subfieldE_I4_J,axiom,
! [K: set_a,R2: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K @ R2 )
=> ( ( member_a @ K1 @ K )
=> ( ( member_a @ K22 @ K )
=> ( ( mult_a_ring_ext_a_b @ R2 @ K1 @ K22 )
= ( mult_a_ring_ext_a_b @ R2 @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_558_subringE_I6_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subring_a_b @ H @ R2 )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R2 @ H1 @ H2 ) @ H ) ) ) ) ).
% subringE(6)
thf(fact_559_subringE_I3_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R2 )
=> ( member_a @ ( one_a_ring_ext_a_b @ R2 ) @ H ) ) ).
% subringE(3)
thf(fact_560_subfieldE_I1_J,axiom,
! [K: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R2 )
=> ( subring_a_b @ K @ R2 ) ) ).
% subfieldE(1)
thf(fact_561_subfieldE_I3_J,axiom,
! [K: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R2 )
=> ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ).
% subfieldE(3)
thf(fact_562_subringE_I1_J,axiom,
! [H: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R2 )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ).
% subringE(1)
thf(fact_563_subfieldE_I5_J,axiom,
! [K: set_a,R2: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K @ R2 )
=> ( ( member_a @ K1 @ K )
=> ( ( member_a @ K22 @ K )
=> ( ( ( mult_a_ring_ext_a_b @ R2 @ K1 @ K22 )
= ( zero_a_b @ R2 ) )
=> ( ( K1
= ( zero_a_b @ R2 ) )
| ( K22
= ( zero_a_b @ R2 ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_564_subfieldE_I6_J,axiom,
! [K: set_a,R2: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R2 )
=> ( ( one_a_ring_ext_a_b @ R2 )
!= ( zero_a_b @ R2 ) ) ) ).
% subfieldE(6)
thf(fact_565_field_Ocarrier__is__subfield,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( subfie5224850075530046424t_unit @ ( partia5907974310037520643t_unit @ R2 ) @ R2 ) ) ).
% field.carrier_is_subfield
thf(fact_566_field_Ocarrier__is__subfield,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R2 )
=> ( subfield_a_b @ ( partia707051561876973205xt_a_b @ R2 ) @ R2 ) ) ).
% field.carrier_is_subfield
thf(fact_567_field_Ofinite__mult__of,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( ( finite_finite_set_a @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( finite_finite_set_a @ ( partia8299590604543202116t_unit @ ( multip3774352783277980819t_unit @ R2 ) ) ) ) ) ).
% field.finite_mult_of
thf(fact_568_field_Ofinite__mult__of,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R2 )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( multip3210463924028840165of_a_b @ R2 ) ) ) ) ) ).
% field.finite_mult_of
thf(fact_569_units__of__units,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( units_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G ) )
= ( units_a_ring_ext_a_b @ G ) ) ).
% units_of_units
thf(fact_570_units__of__units,axiom,
! [G: partia8223610829204095565t_unit] :
( ( units_a_Product_unit @ ( units_7501539392726747778t_unit @ G ) )
= ( units_a_Product_unit @ G ) ) ).
% units_of_units
thf(fact_571_subgroup__def,axiom,
( subgro1816942748394427906xt_a_b
= ( ^ [H4: set_a,G2: partia2175431115845679010xt_a_b] :
( ( ord_less_eq_set_a @ H4 @ ( partia707051561876973205xt_a_b @ G2 ) )
& ! [X2: a,Y5: a] :
( ( member_a @ X2 @ H4 )
=> ( ( member_a @ Y5 @ H4 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G2 @ X2 @ Y5 ) @ H4 ) ) )
& ( member_a @ ( one_a_ring_ext_a_b @ G2 ) @ H4 )
& ! [X2: a] :
( ( member_a @ X2 @ H4 )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ G2 @ X2 ) @ H4 ) ) ) ) ) ).
% subgroup_def
thf(fact_572_subgroup__def,axiom,
( subgro3222307229058429633t_unit
= ( ^ [H4: set_a,G2: partia8223610829204095565t_unit] :
( ( ord_less_eq_set_a @ H4 @ ( partia6735698275553448452t_unit @ G2 ) )
& ! [X2: a,Y5: a] :
( ( member_a @ X2 @ H4 )
=> ( ( member_a @ Y5 @ H4 )
=> ( member_a @ ( mult_a_Product_unit @ G2 @ X2 @ Y5 ) @ H4 ) ) )
& ( member_a @ ( one_a_Product_unit @ G2 ) @ H4 )
& ! [X2: a] :
( ( member_a @ X2 @ H4 )
=> ( member_a @ ( m_inv_a_Product_unit @ G2 @ X2 ) @ H4 ) ) ) ) ) ).
% subgroup_def
thf(fact_573_subgroup_Ointro,axiom,
! [H: set_a,G: partia2175431115845679010xt_a_b] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X3: a,Y: a] :
( ( member_a @ X3 @ H )
=> ( ( member_a @ Y @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y ) @ H ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ H )
=> ( ! [X3: a] :
( ( member_a @ X3 @ H )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ G @ X3 ) @ H ) )
=> ( subgro1816942748394427906xt_a_b @ H @ G ) ) ) ) ) ).
% subgroup.intro
thf(fact_574_subgroup_Ointro,axiom,
! [H: set_a,G: partia8223610829204095565t_unit] :
( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ G ) )
=> ( ! [X3: a,Y: a] :
( ( member_a @ X3 @ H )
=> ( ( member_a @ Y @ H )
=> ( member_a @ ( mult_a_Product_unit @ G @ X3 @ Y ) @ H ) ) )
=> ( ( member_a @ ( one_a_Product_unit @ G ) @ H )
=> ( ! [X3: a] :
( ( member_a @ X3 @ H )
=> ( member_a @ ( m_inv_a_Product_unit @ G @ X3 ) @ H ) )
=> ( subgro3222307229058429633t_unit @ H @ G ) ) ) ) ) ).
% subgroup.intro
thf(fact_575_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_576_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_577_mult__of_OsubgroupE_I3_J,axiom,
! [H: set_a,A: a] :
( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( ( member_a @ A @ H )
=> ( member_a @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A ) @ H ) ) ) ).
% mult_of.subgroupE(3)
thf(fact_578_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_579_mult__of_Oinv__eq__imp__eq,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( 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 ) @ Y2 ) )
=> ( X = Y2 ) ) ) ) ).
% mult_of.inv_eq_imp_eq
thf(fact_580_mult__of_Oinv__mult,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ X @ Y2 ) )
= ( mult_a_ring_ext_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 ) @ Y2 ) ) ) ) ) ).
% mult_of.inv_mult
thf(fact_581_mult__of_Oinv__mult__group,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ X @ Y2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Y2 ) @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X ) ) ) ) ) ).
% mult_of.inv_mult_group
thf(fact_582_mult__of_Oinv__solve__left,axiom,
! [A: a,B2: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ B2 ) @ C ) )
= ( C
= ( mult_a_ring_ext_a_b @ r @ B2 @ A ) ) ) ) ) ) ).
% mult_of.inv_solve_left
thf(fact_583_mult__of_Oinv__solve__left_H,axiom,
! [A: a,B2: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_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 ) @ B2 ) @ C )
= A )
= ( C
= ( mult_a_ring_ext_a_b @ r @ B2 @ A ) ) ) ) ) ) ).
% mult_of.inv_solve_left'
thf(fact_584_mult__of_Oinv__solve__right,axiom,
! [A: a,B2: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ r @ B2 @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ C ) ) )
= ( B2
= ( mult_a_ring_ext_a_b @ r @ A @ C ) ) ) ) ) ) ).
% mult_of.inv_solve_right
thf(fact_585_mult__of_Oinv__solve__right_H,axiom,
! [A: a,B2: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B2 @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ C ) )
= A )
= ( B2
= ( mult_a_ring_ext_a_b @ r @ A @ C ) ) ) ) ) ) ).
% mult_of.inv_solve_right'
thf(fact_586_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_587_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_588_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_589_mult__of_Oinv__unique_H,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( Y2
= ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X ) ) ) ) ) ) ).
% mult_of.inv_unique'
thf(fact_590_mult__of_Oinv__equality,axiom,
! [Y2: a,X: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X )
= Y2 ) ) ) ) ).
% mult_of.inv_equality
thf(fact_591_mult__of_Oinv__char,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X )
= Y2 ) ) ) ) ) ).
% mult_of.inv_char
thf(fact_592_mult__of_Ocomm__inv__char,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X )
= Y2 ) ) ) ) ).
% mult_of.comm_inv_char
thf(fact_593_mult__of_Odiff__neutralizes,axiom,
! [H: set_a,R2: set_a,R1: a,R22: a] :
( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( ( member_set_a @ R2 @ ( rCOSET407642731378740692t_unit @ ( ring_mult_of_a_b @ r ) @ H ) )
=> ( ( member_a @ R1 @ R2 )
=> ( ( member_a @ R22 @ R2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ R1 @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ R22 ) ) @ H ) ) ) ) ) ).
% mult_of.diff_neutralizes
thf(fact_594_mult__of_Oone__in__subset,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( H != bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ H )
=> ( member_a @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X3 ) @ H ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ H )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X3 @ Xa2 ) @ H ) ) )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ H ) ) ) ) ) ).
% mult_of.one_in_subset
thf(fact_595_mult__of_OsubgroupI,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( H != bot_bot_set_a )
=> ( ! [A4: a] :
( ( member_a @ A4 @ H )
=> ( member_a @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A4 ) @ H ) )
=> ( ! [A4: a,B4: a] :
( ( member_a @ A4 @ H )
=> ( ( member_a @ B4 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ A4 @ B4 ) @ H ) ) )
=> ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.subgroupI
thf(fact_596_Ring__Divisibility_Oone__mult__of,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( one_a_Product_unit @ ( ring_mult_of_a_b @ R2 ) )
= ( one_a_ring_ext_a_b @ R2 ) ) ).
% Ring_Divisibility.one_mult_of
thf(fact_597_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_598_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_599_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_600_Multiplicative__Group_Oone__mult__of,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( one_a_Product_unit @ ( multip3210463924028840165of_a_b @ R2 ) )
= ( one_a_ring_ext_a_b @ R2 ) ) ).
% Multiplicative_Group.one_mult_of
thf(fact_601_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_602_mult__of_Oinv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_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.inv_closed
thf(fact_603_mult__of_Oinv__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_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.inv_inv
thf(fact_604_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_605_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_606_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_607_mult__of_Oinv__eq__1__iff,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_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_1_iff
thf(fact_608_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_609_mult__of_Or__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_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.r_inv
thf(fact_610_mult__of_Ol__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_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.l_inv
thf(fact_611_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_612_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_613_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_614_le__trans,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I2 @ K2 ) ) ) ).
% le_trans
thf(fact_615_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_616_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_617_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_618_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K2: nat,B2: nat] :
( ( P2 @ K2 )
=> ( ! [Y: nat] :
( ( P2 @ Y )
=> ( ord_less_eq_nat @ Y @ B2 ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_619_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M2 @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_620_le__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_621_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_622_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_623_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_624_le__diff__iff_H,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_625_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_626_field_Om__inv__mult__of,axiom,
! [R2: partia6043505979758434576t_unit,X: set_a] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( ( member_set_a @ X @ ( partia8299590604543202116t_unit @ ( multip3774352783277980819t_unit @ R2 ) ) )
=> ( ( m_inv_3738623195918084710t_unit @ ( multip3774352783277980819t_unit @ R2 ) @ X )
= ( m_inv_7491079437187478987t_unit @ R2 @ X ) ) ) ) ).
% field.m_inv_mult_of
thf(fact_627_field_Om__inv__mult__of,axiom,
! [R2: partia2175431115845679010xt_a_b,X: a] :
( ( field_a_b @ R2 )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( multip3210463924028840165of_a_b @ R2 ) ) )
=> ( ( m_inv_a_Product_unit @ ( multip3210463924028840165of_a_b @ R2 ) @ X )
= ( m_inv_a_ring_ext_a_b @ R2 @ X ) ) ) ) ).
% field.m_inv_mult_of
thf(fact_628_subgroup_Omem__carrier,axiom,
! [H: set_a,G: partia2175431115845679010xt_a_b,X: a] :
( ( subgro1816942748394427906xt_a_b @ H @ G )
=> ( ( member_a @ X @ H )
=> ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% subgroup.mem_carrier
thf(fact_629_subgroup_Omem__carrier,axiom,
! [H: set_a,G: partia8223610829204095565t_unit,X: a] :
( ( subgro3222307229058429633t_unit @ H @ G )
=> ( ( member_a @ X @ H )
=> ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) ) ) ) ).
% subgroup.mem_carrier
thf(fact_630_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_631_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_632_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_633_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_634_subgroup__nonempty,axiom,
! [G: partia8223610829204095565t_unit] :
~ ( subgro3222307229058429633t_unit @ bot_bot_set_a @ G ) ).
% subgroup_nonempty
thf(fact_635_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_636_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y: nat,Z3: nat] :
( ( R2 @ X3 @ Y )
=> ( ( R2 @ Y @ Z3 )
=> ( R2 @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_637_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P2 @ M2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_638_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( P2 @ M3 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_639_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_640_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_641_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_642_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_643_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_644_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_645_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_646_subgroup_Om__closed,axiom,
! [H: set_a,G: partia8223610829204095565t_unit,X: a,Y2: a] :
( ( subgro3222307229058429633t_unit @ H @ G )
=> ( ( member_a @ X @ H )
=> ( ( member_a @ Y2 @ H )
=> ( member_a @ ( mult_a_Product_unit @ G @ X @ Y2 ) @ H ) ) ) ) ).
% subgroup.m_closed
thf(fact_647_subgroup_Om__closed,axiom,
! [H: set_a,G: partia2175431115845679010xt_a_b,X: a,Y2: a] :
( ( subgro1816942748394427906xt_a_b @ H @ G )
=> ( ( member_a @ X @ H )
=> ( ( member_a @ Y2 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y2 ) @ H ) ) ) ) ).
% subgroup.m_closed
thf(fact_648_subgroup_Oone__closed,axiom,
! [H: set_a,G: partia8223610829204095565t_unit] :
( ( subgro3222307229058429633t_unit @ H @ G )
=> ( member_a @ ( one_a_Product_unit @ G ) @ H ) ) ).
% subgroup.one_closed
thf(fact_649_subgroup_Oone__closed,axiom,
! [H: set_a,G: partia2175431115845679010xt_a_b] :
( ( subgro1816942748394427906xt_a_b @ H @ G )
=> ( member_a @ ( one_a_ring_ext_a_b @ G ) @ H ) ) ).
% subgroup.one_closed
thf(fact_650_subgroup_Om__inv__closed,axiom,
! [H: set_a,G: partia2175431115845679010xt_a_b,X: a] :
( ( subgro1816942748394427906xt_a_b @ H @ G )
=> ( ( member_a @ X @ H )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ G @ X ) @ H ) ) ) ).
% subgroup.m_inv_closed
thf(fact_651_subgroup_Om__inv__closed,axiom,
! [H: set_a,G: partia8223610829204095565t_unit,X: a] :
( ( subgro3222307229058429633t_unit @ H @ G )
=> ( ( member_a @ X @ H )
=> ( member_a @ ( m_inv_a_Product_unit @ G @ X ) @ H ) ) ) ).
% subgroup.m_inv_closed
thf(fact_652_units__of__one,axiom,
! [G: partia8223610829204095565t_unit] :
( ( one_a_Product_unit @ ( units_7501539392726747778t_unit @ G ) )
= ( one_a_Product_unit @ G ) ) ).
% units_of_one
thf(fact_653_units__of__one,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( one_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G ) )
= ( one_a_ring_ext_a_b @ G ) ) ).
% units_of_one
thf(fact_654_subgroup_Osubset,axiom,
! [H: set_a,G: partia2175431115845679010xt_a_b] :
( ( subgro1816942748394427906xt_a_b @ H @ G )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ G ) ) ) ).
% subgroup.subset
thf(fact_655_subgroup_Osubset,axiom,
! [H: set_a,G: partia8223610829204095565t_unit] :
( ( subgro3222307229058429633t_unit @ H @ G )
=> ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ G ) ) ) ).
% subgroup.subset
thf(fact_656_lift__Suc__mono__le,axiom,
! [F3: nat > set_a,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_a @ ( F3 @ N ) @ ( F3 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_657_lift__Suc__mono__le,axiom,
! [F3: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F3 @ N ) @ ( F3 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_658_lift__Suc__antimono__le,axiom,
! [F3: nat > set_a,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_a @ ( F3 @ N4 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_659_lift__Suc__antimono__le,axiom,
! [F3: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F3 @ N4 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_660_units__of__carrier,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( partia6735698275553448452t_unit @ ( units_8174867845824275201xt_a_b @ G ) )
= ( units_a_ring_ext_a_b @ G ) ) ).
% units_of_carrier
thf(fact_661_units__of__carrier,axiom,
! [G: partia8223610829204095565t_unit] :
( ( partia6735698275553448452t_unit @ ( units_7501539392726747778t_unit @ G ) )
= ( units_a_Product_unit @ G ) ) ).
% units_of_carrier
thf(fact_662_units__of__mult,axiom,
! [G: partia8223610829204095565t_unit] :
( ( mult_a_Product_unit @ ( units_7501539392726747778t_unit @ G ) )
= ( mult_a_Product_unit @ G ) ) ).
% units_of_mult
thf(fact_663_units__of__mult,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( mult_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G ) )
= ( mult_a_ring_ext_a_b @ G ) ) ).
% units_of_mult
thf(fact_664_mult__of_Osubmonoid__subgroupI,axiom,
! [H: set_a] :
( ( submon7992156029481901626t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ H )
=> ( member_a @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A4 ) @ H ) )
=> ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.submonoid_subgroupI
thf(fact_665_set__add__zero,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ A2 )
= A2 ) ) ).
% set_add_zero
thf(fact_666_mult__of_Otrivial__group__subgroup__generated,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) )
=> ( elemen1145482699608675729t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ S ) ) ) ).
% mult_of.trivial_group_subgroup_generated
thf(fact_667_mult__of_Ogcd__isgcd,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( 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 @ B2 ) @ A @ B2 ) ) ) ).
% mult_of.gcd_isgcd
thf(fact_668_mult__of_Osubgroup__generated__group__carrier,axiom,
( ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
= ( ring_mult_of_a_b @ r ) ) ).
% mult_of.subgroup_generated_group_carrier
thf(fact_669_setadd__subset__G,axiom,
! [H: set_a,K: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H @ K ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% setadd_subset_G
thf(fact_670_set__add__comm,axiom,
! [I: set_a,J2: set_a] :
( ( ord_less_eq_set_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ J2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ I @ J2 )
= ( set_add_a_b @ r @ J2 @ I ) ) ) ) ).
% set_add_comm
thf(fact_671_set__add__closed,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A2 @ B ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_add_closed
thf(fact_672_mult__of_Ogcd__exists,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.gcd_exists
thf(fact_673_mult__of_Ogcd__closed,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.gcd_closed
thf(fact_674_mult__of_Osubgroup__generated__subset__carrier__subset,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ord_less_eq_set_a @ S @ ( partia6735698275553448452t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ S ) ) ) ) ).
% mult_of.subgroup_generated_subset_carrier_subset
thf(fact_675_mult__of_Ocarrier__subgroup__generated__subset,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ ( partia6735698275553448452t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ A2 ) ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ).
% mult_of.carrier_subgroup_generated_subset
thf(fact_676_mult__of_Osubgroup__subgroup__generated,axiom,
! [S: set_a] : ( subgro3222307229058429633t_unit @ ( partia6735698275553448452t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ S ) ) @ ( ring_mult_of_a_b @ r ) ) ).
% mult_of.subgroup_subgroup_generated
thf(fact_677_mult__of_Osubgroup__of__subgroup__generated,axiom,
! [H: set_a,B: set_a] :
( ( ord_less_eq_set_a @ H @ B )
=> ( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( subgro3222307229058429633t_unit @ H @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ B ) ) ) ) ).
% mult_of.subgroup_of_subgroup_generated
thf(fact_678_mult__of_Ogcd__divides,axiom,
! [Z: a,X: a,Y2: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Z @ X )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Z @ Y2 )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( 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 @ Y2 ) ) ) ) ) ) ) ).
% mult_of.gcd_divides
thf(fact_679_mult__of_Ogcd__divides__l,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( 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 @ B2 ) @ A ) ) ) ).
% mult_of.gcd_divides_l
thf(fact_680_mult__of_Ogcd__divides__r,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( 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 @ B2 ) @ B2 ) ) ) ).
% mult_of.gcd_divides_r
thf(fact_681_mult__of_Osubgroup__subgroup__generated__iff,axiom,
! [H: set_a,B: set_a] :
( ( subgro3222307229058429633t_unit @ H @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ B ) )
= ( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
& ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ B ) ) ) ) ) ).
% mult_of.subgroup_subgroup_generated_iff
thf(fact_682_mult__of_Osubgroup__generated__minimal,axiom,
! [H: set_a,S: set_a] :
( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ S @ H )
=> ( ord_less_eq_set_a @ ( partia6735698275553448452t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ S ) ) @ H ) ) ) ).
% mult_of.subgroup_generated_minimal
thf(fact_683_ideal__sum__iff__gcd,axiom,
! [A: a,B2: a,D: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ D @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( cgenid547466209912283029xt_a_b @ r @ D )
= ( set_add_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ ( cgenid547466209912283029xt_a_b @ r @ B2 ) ) )
= ( isgcd_a_ring_ext_a_b @ r @ D @ A @ B2 ) ) ) ) ) ).
% ideal_sum_iff_gcd
thf(fact_684_mult__of_Osubgroup__generated2,axiom,
! [S: set_a] :
( ( genera8815471607677139784t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ S ) @ S )
= ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ S ) ) ).
% mult_of.subgroup_generated2
thf(fact_685_mult__of_Oinv__subgroup__generated,axiom,
! [F3: a,S: set_a] :
( ( member_a @ F3 @ ( partia6735698275553448452t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ S ) ) )
=> ( ( m_inv_a_Product_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ S ) @ F3 )
= ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ F3 ) ) ) ).
% mult_of.inv_subgroup_generated
thf(fact_686_submonoid_Omem__carrier,axiom,
! [H: set_a,G: partia2175431115845679010xt_a_b,X: a] :
( ( submon8907322713594755401xt_a_b @ H @ G )
=> ( ( member_a @ X @ H )
=> ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% submonoid.mem_carrier
thf(fact_687_submonoid_Omem__carrier,axiom,
! [H: set_a,G: partia8223610829204095565t_unit,X: a] :
( ( submon7992156029481901626t_unit @ H @ G )
=> ( ( member_a @ X @ H )
=> ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) ) ) ) ).
% submonoid.mem_carrier
thf(fact_688_submonoid__nonempty,axiom,
! [G: partia8223610829204095565t_unit] :
~ ( submon7992156029481901626t_unit @ bot_bot_set_a @ G ) ).
% submonoid_nonempty
thf(fact_689_submonoid_Om__closed,axiom,
! [H: set_a,G: partia8223610829204095565t_unit,X: a,Y2: a] :
( ( submon7992156029481901626t_unit @ H @ G )
=> ( ( member_a @ X @ H )
=> ( ( member_a @ Y2 @ H )
=> ( member_a @ ( mult_a_Product_unit @ G @ X @ Y2 ) @ H ) ) ) ) ).
% submonoid.m_closed
thf(fact_690_submonoid_Om__closed,axiom,
! [H: set_a,G: partia2175431115845679010xt_a_b,X: a,Y2: a] :
( ( submon8907322713594755401xt_a_b @ H @ G )
=> ( ( member_a @ X @ H )
=> ( ( member_a @ Y2 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y2 ) @ H ) ) ) ) ).
% submonoid.m_closed
thf(fact_691_submonoid_Oone__closed,axiom,
! [H: set_a,G: partia8223610829204095565t_unit] :
( ( submon7992156029481901626t_unit @ H @ G )
=> ( member_a @ ( one_a_Product_unit @ G ) @ H ) ) ).
% submonoid.one_closed
thf(fact_692_submonoid_Oone__closed,axiom,
! [H: set_a,G: partia2175431115845679010xt_a_b] :
( ( submon8907322713594755401xt_a_b @ H @ G )
=> ( member_a @ ( one_a_ring_ext_a_b @ G ) @ H ) ) ).
% submonoid.one_closed
thf(fact_693_submonoid_Osubset,axiom,
! [H: set_a,G: partia2175431115845679010xt_a_b] :
( ( submon8907322713594755401xt_a_b @ H @ G )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ G ) ) ) ).
% submonoid.subset
thf(fact_694_submonoid_Osubset,axiom,
! [H: set_a,G: partia8223610829204095565t_unit] :
( ( submon7992156029481901626t_unit @ H @ G )
=> ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ G ) ) ) ).
% submonoid.subset
thf(fact_695_principal__domain_Oideal__sum__iff__gcd,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B2: a,D: a] :
( ( ring_p8803135361686045600in_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ D @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ( cgenid547466209912283029xt_a_b @ R2 @ D )
= ( set_add_a_b @ R2 @ ( cgenid547466209912283029xt_a_b @ R2 @ A ) @ ( cgenid547466209912283029xt_a_b @ R2 @ B2 ) ) )
= ( isgcd_a_ring_ext_a_b @ R2 @ D @ A @ B2 ) ) ) ) ) ) ).
% principal_domain.ideal_sum_iff_gcd
thf(fact_696_submonoid__def,axiom,
( submon8907322713594755401xt_a_b
= ( ^ [H4: set_a,G2: partia2175431115845679010xt_a_b] :
( ( ord_less_eq_set_a @ H4 @ ( partia707051561876973205xt_a_b @ G2 ) )
& ! [X2: a,Y5: a] :
( ( member_a @ X2 @ H4 )
=> ( ( member_a @ Y5 @ H4 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G2 @ X2 @ Y5 ) @ H4 ) ) )
& ( member_a @ ( one_a_ring_ext_a_b @ G2 ) @ H4 ) ) ) ) ).
% submonoid_def
thf(fact_697_submonoid__def,axiom,
( submon7992156029481901626t_unit
= ( ^ [H4: set_a,G2: partia8223610829204095565t_unit] :
( ( ord_less_eq_set_a @ H4 @ ( partia6735698275553448452t_unit @ G2 ) )
& ! [X2: a,Y5: a] :
( ( member_a @ X2 @ H4 )
=> ( ( member_a @ Y5 @ H4 )
=> ( member_a @ ( mult_a_Product_unit @ G2 @ X2 @ Y5 ) @ H4 ) ) )
& ( member_a @ ( one_a_Product_unit @ G2 ) @ H4 ) ) ) ) ).
% submonoid_def
thf(fact_698_submonoid_Ointro,axiom,
! [H: set_a,G: partia2175431115845679010xt_a_b] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X3: a,Y: a] :
( ( member_a @ X3 @ H )
=> ( ( member_a @ Y @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X3 @ Y ) @ H ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ H )
=> ( submon8907322713594755401xt_a_b @ H @ G ) ) ) ) ).
% submonoid.intro
thf(fact_699_submonoid_Ointro,axiom,
! [H: set_a,G: partia8223610829204095565t_unit] :
( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ G ) )
=> ( ! [X3: a,Y: a] :
( ( member_a @ X3 @ H )
=> ( ( member_a @ Y @ H )
=> ( member_a @ ( mult_a_Product_unit @ G @ X3 @ Y ) @ H ) ) )
=> ( ( member_a @ ( one_a_Product_unit @ G ) @ H )
=> ( submon7992156029481901626t_unit @ H @ G ) ) ) ) ).
% submonoid.intro
thf(fact_700_mult__of_Ocyclic__group__generated,axiom,
! [X: a] : ( elemen5394956844934664649t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% mult_of.cyclic_group_generated
thf(fact_701_mult__of_Ocyclic__group__alt,axiom,
( ( elemen5394956844934664649t_unit @ ( ring_mult_of_a_b @ r ) )
= ( ? [X2: a] :
( ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.cyclic_group_alt
thf(fact_702_bezout__identity,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ ( cgenid547466209912283029xt_a_b @ r @ B2 ) )
= ( cgenid547466209912283029xt_a_b @ r @ ( somegc1600592057159103747xt_a_b @ r @ A @ B2 ) ) ) ) ) ).
% bezout_identity
thf(fact_703_mult__of_Otrivial__group__subgroup__generated__eq,axiom,
! [S2: set_a] :
( ( elemen1145482699608675729t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ S2 ) )
= ( ord_less_eq_set_a @ ( inf_inf_set_a @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) @ S2 ) @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% mult_of.trivial_group_subgroup_generated_eq
thf(fact_704_subring__inter,axiom,
! [I: set_a,J2: set_a] :
( ( subring_a_b @ I @ r )
=> ( ( subring_a_b @ J2 @ r )
=> ( subring_a_b @ ( inf_inf_set_a @ I @ J2 ) @ r ) ) ) ).
% subring_inter
thf(fact_705_IntI,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_706_IntI,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_707_Int__iff,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) )
= ( ( member_set_a @ C @ A2 )
& ( member_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_708_Int__iff,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_709_mult__of_Osubgroups__Inter__pair,axiom,
! [I: set_a,J2: set_a] :
( ( subgro3222307229058429633t_unit @ I @ ( ring_mult_of_a_b @ r ) )
=> ( ( subgro3222307229058429633t_unit @ J2 @ ( ring_mult_of_a_b @ r ) )
=> ( subgro3222307229058429633t_unit @ ( inf_inf_set_a @ I @ J2 ) @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.subgroups_Inter_pair
thf(fact_710_Int__subset__iff,axiom,
! [C3: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( ord_less_eq_set_a @ C3 @ A2 )
& ( ord_less_eq_set_a @ C3 @ B ) ) ) ).
% Int_subset_iff
thf(fact_711_Int__insert__right__if1,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_712_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_713_Int__insert__right__if0,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_714_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_715_insert__inter__insert,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_716_Int__insert__left__if1,axiom,
! [A: set_a,C3: set_set_a,B: set_set_a] :
( ( member_set_a @ A @ C3 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C3 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_717_Int__insert__left__if1,axiom,
! [A: a,C3: set_a,B: set_a] :
( ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C3 )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_718_Int__insert__left__if0,axiom,
! [A: set_a,C3: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ C3 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C3 )
= ( inf_inf_set_set_a @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_719_Int__insert__left__if0,axiom,
! [A: a,C3: set_a,B: set_a] :
( ~ ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C3 )
= ( inf_inf_set_a @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_720_finite__Int,axiom,
! [F: set_a,G: set_a] :
( ( ( finite_finite_a @ F )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F @ G ) ) ) ).
% finite_Int
thf(fact_721_disjoint__insert_I2_J,axiom,
! [A2: set_set_a,B2: set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) )
= ( ~ ( member_set_a @ B2 @ A2 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_722_disjoint__insert_I2_J,axiom,
! [A2: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_723_disjoint__insert_I1_J,axiom,
! [B: set_set_a,A: set_a,A2: set_set_a] :
( ( ( inf_inf_set_set_a @ B @ ( insert_set_a @ A @ A2 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B )
& ( ( inf_inf_set_set_a @ B @ A2 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_724_disjoint__insert_I1_J,axiom,
! [B: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_725_insert__disjoint_I2_J,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B ) )
= ( ~ ( member_set_a @ A @ B )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_726_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B ) )
= ( ~ ( member_a @ A @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_727_insert__disjoint_I1_J,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B )
& ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_728_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_729_Diff__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_730_Int__Collect__mono,axiom,
! [A2: set_set_a,B: set_set_a,P2: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ ( collect_set_a @ P2 ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_731_Int__Collect__mono,axiom,
! [A2: set_a,B: set_a,P2: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P2 ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_732_Int__greatest,axiom,
! [C3: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( ( ord_less_eq_set_a @ C3 @ B )
=> ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_733_Int__absorb2,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_734_Int__absorb1,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_735_Int__lower2,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_736_Int__lower1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_737_Int__mono,axiom,
! [A2: set_a,C3: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_738_disjoint__iff__not__equal,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y5: a] :
( ( member_a @ Y5 @ B )
=> ( X2 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_739_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_740_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_741_disjoint__iff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ~ ( member_set_a @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_742_disjoint__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_743_Int__emptyI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ~ ( member_set_a @ X3 @ B ) )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_744_Int__emptyI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_745_Int__insert__right,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_746_Int__insert__right,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_747_Int__insert__left,axiom,
! [A: set_a,C3: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A @ C3 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C3 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B @ C3 ) ) ) )
& ( ~ ( member_set_a @ A @ C3 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C3 )
= ( inf_inf_set_set_a @ B @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_748_Int__insert__left,axiom,
! [A: a,C3: set_a,B: set_a] :
( ( ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C3 )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C3 ) ) ) )
& ( ~ ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C3 )
= ( inf_inf_set_a @ B @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_749_IntE,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ~ ( member_set_a @ C @ B ) ) ) ).
% IntE
thf(fact_750_IntE,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_751_IntD1,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C @ A2 ) ) ).
% IntD1
thf(fact_752_IntD1,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_753_IntD2,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C @ B ) ) ).
% IntD2
thf(fact_754_IntD2,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_755_Int__assoc,axiom,
! [A2: set_a,B: set_a,C3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C3 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C3 ) ) ) ).
% Int_assoc
thf(fact_756_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_757_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_758_Int__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_759_Int__left__commute,axiom,
! [A2: set_a,B: set_a,C3: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C3 ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A2 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_760_Int__Diff,axiom,
! [A2: set_a,B: set_a,C3: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C3 )
= ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B @ C3 ) ) ) ).
% Int_Diff
thf(fact_761_Diff__Int2,axiom,
! [A2: set_a,C3: set_a,B: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C3 ) @ ( inf_inf_set_a @ B @ C3 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C3 ) @ B ) ) ).
% Diff_Int2
thf(fact_762_Diff__Diff__Int,axiom,
! [A2: set_a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_763_Diff__Int__distrib,axiom,
! [C3: set_a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ C3 @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C3 @ A2 ) @ ( inf_inf_set_a @ C3 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_764_Diff__Int__distrib2,axiom,
! [A2: set_a,B: set_a,C3: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C3 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C3 ) @ ( inf_inf_set_a @ B @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_765_Diff__triv,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_766_Int__Diff__disjoint,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( minus_minus_set_a @ A2 @ B ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_767_principal__domain_Obezout__identity,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B2: a] :
( ( ring_p8803135361686045600in_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( set_add_a_b @ R2 @ ( cgenid547466209912283029xt_a_b @ R2 @ A ) @ ( cgenid547466209912283029xt_a_b @ R2 @ B2 ) )
= ( cgenid547466209912283029xt_a_b @ R2 @ ( somegc1600592057159103747xt_a_b @ R2 @ A @ B2 ) ) ) ) ) ) ).
% principal_domain.bezout_identity
thf(fact_768_subgroup__generated__restrict,axiom,
! [G: partia2175431115845679010xt_a_b,S: set_a] :
( ( genera8625346715478425275xt_a_b @ G @ ( inf_inf_set_a @ ( partia707051561876973205xt_a_b @ G ) @ S ) )
= ( genera8625346715478425275xt_a_b @ G @ S ) ) ).
% subgroup_generated_restrict
thf(fact_769_subgroup__generated__restrict,axiom,
! [G: partia8223610829204095565t_unit,S: set_a] :
( ( genera8815471607677139784t_unit @ G @ ( inf_inf_set_a @ ( partia6735698275553448452t_unit @ G ) @ S ) )
= ( genera8815471607677139784t_unit @ G @ S ) ) ).
% subgroup_generated_restrict
thf(fact_770_dimension__direct__sum__space,axiom,
! [K: set_a,N: nat,E: set_a,M2: nat,F: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M2 @ K @ F )
=> ( ( ( inf_inf_set_a @ E @ F )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( embedd2795209813406577254on_a_b @ r @ ( plus_plus_nat @ N @ M2 ) @ K @ ( set_add_a_b @ r @ E @ F ) ) ) ) ) ) ).
% dimension_direct_sum_space
thf(fact_771_mult__of_Ofinite__cyclic__subgroup,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
= ( ? [N3: nat] :
( ( N3 != zero_zero_nat )
& ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% mult_of.finite_cyclic_subgroup
thf(fact_772_dimension__sum__space,axiom,
! [K: set_a,N: nat,E: set_a,M2: nat,F: set_a,K2: nat] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M2 @ K @ F )
=> ( ( embedd2795209813406577254on_a_b @ r @ K2 @ K @ ( inf_inf_set_a @ E @ F ) )
=> ( embedd2795209813406577254on_a_b @ r @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ K2 ) @ K @ ( set_add_a_b @ r @ E @ F ) ) ) ) ) ) ).
% dimension_sum_space
thf(fact_773_mult__of_OUnits__pow__closed,axiom,
! [X: a,D: 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 @ D ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.Units_pow_closed
thf(fact_774_mult__of_Opow__mult__distrib,axiom,
! [X: a,Y2: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( 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 @ Y2 ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ Y2 @ N ) ) ) ) ) ) ).
% mult_of.pow_mult_distrib
thf(fact_775_mult__of_Onat__pow__distrib,axiom,
! [X: a,Y2: a,N: nat] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( 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 @ Y2 ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ Y2 @ N ) ) ) ) ) ).
% mult_of.nat_pow_distrib
thf(fact_776_mult__of_Onat__pow__comm,axiom,
! [X: a,N: 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 @ N ) @ ( 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 @ N ) ) ) ) ).
% mult_of.nat_pow_comm
thf(fact_777_mult__of_Ogroup__commutes__pow,axiom,
! [X: a,Y2: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( 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 @ N ) @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N ) ) ) ) ) ) ).
% mult_of.group_commutes_pow
thf(fact_778_mult__of_Onat__pow__inv,axiom,
! [X: a,I2: nat] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X ) @ I2 )
= ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ I2 ) ) ) ) ).
% mult_of.nat_pow_inv
thf(fact_779_mult__of_Ounits__of__pow,axiom,
! [X: a,N: 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 @ N )
= ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N ) ) ) ).
% mult_of.units_of_pow
thf(fact_780_mult__of_Onat__pow__Suc2,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ ( suc @ N ) )
= ( mult_a_ring_ext_a_b @ r @ X @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N ) ) ) ) ).
% mult_of.nat_pow_Suc2
thf(fact_781_mult__of_Onat__pow__mult,axiom,
! [X: a,N: 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 @ N ) @ ( 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 @ N @ M2 ) ) ) ) ).
% mult_of.nat_pow_mult
thf(fact_782_mult__of_Opow__eq__div2,axiom,
! [X: a,M2: nat,N: nat] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_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 @ N ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ ( minus_minus_nat @ M2 @ N ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.pow_eq_div2
thf(fact_783_add__le__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_eq_int @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_784_add__le__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_785_add__le__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_eq_int @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_786_add__le__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_787_mult__of_Oprime__pow__divides__iff,axiom,
! [P: a,A: a,B2: a,N: nat] :
( ( member_a @ P @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ~ ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ A )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ P @ N ) @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ P @ N ) @ B2 ) ) ) ) ) ) ) ).
% mult_of.prime_pow_divides_iff
thf(fact_788_add__diff__cancel__right_H,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel_right'
thf(fact_789_add__diff__cancel__right_H,axiom,
! [A: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel_right'
thf(fact_790_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_791_add__diff__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( minus_minus_int @ A @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_792_add__diff__cancel__left_H,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ A )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_793_add__diff__cancel__left_H,axiom,
! [A: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ A )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_794_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_795_add__diff__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
= ( minus_minus_int @ A @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_796_diff__add__cancel,axiom,
! [A: int,B2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
= A ) ).
% diff_add_cancel
thf(fact_797_add__diff__cancel,axiom,
! [A: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel
thf(fact_798_nat__add__left__cancel__le,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_799_mult__subgroup__generated,axiom,
! [G: partia2175431115845679010xt_a_b,S: set_a] :
( ( mult_a_ring_ext_a_b @ ( genera8625346715478425275xt_a_b @ G @ S ) )
= ( mult_a_ring_ext_a_b @ G ) ) ).
% mult_subgroup_generated
thf(fact_800_mult__subgroup__generated,axiom,
! [G: partia8223610829204095565t_unit,S: set_a] :
( ( mult_a_Product_unit @ ( genera8815471607677139784t_unit @ G @ S ) )
= ( mult_a_Product_unit @ G ) ) ).
% mult_subgroup_generated
thf(fact_801_one__subgroup__generated,axiom,
! [G: partia2175431115845679010xt_a_b,S: set_a] :
( ( one_a_ring_ext_a_b @ ( genera8625346715478425275xt_a_b @ G @ S ) )
= ( one_a_ring_ext_a_b @ G ) ) ).
% one_subgroup_generated
thf(fact_802_one__subgroup__generated,axiom,
! [G: partia8223610829204095565t_unit,S: set_a] :
( ( one_a_Product_unit @ ( genera8815471607677139784t_unit @ G @ S ) )
= ( one_a_Product_unit @ G ) ) ).
% one_subgroup_generated
thf(fact_803_mult__of_Oinfinite__cyclic__subgroup,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ~ ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) )
= ( ! [M6: nat,N3: nat] :
( ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ M6 )
= ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N3 ) )
=> ( M6 = N3 ) ) ) ) ) ).
% mult_of.infinite_cyclic_subgroup
thf(fact_804_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_805_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_806_le__add__same__cancel2,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B2 @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_807_le__add__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_808_le__add__same__cancel1,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_809_le__add__same__cancel1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_810_add__le__same__cancel2,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_811_add__le__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_812_add__le__same__cancel1,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_813_add__le__same__cancel1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_814_diff__add__zero,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_815_Group_Onat__pow__Suc,axiom,
! [G: partia8223610829204095565t_unit,X: a,N: nat] :
( ( pow_a_1875594501834816709it_nat @ G @ X @ ( suc @ N ) )
= ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) @ X ) ) ).
% Group.nat_pow_Suc
thf(fact_816_Group_Onat__pow__Suc,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,N: nat] :
( ( pow_a_1026414303147256608_b_nat @ G @ X @ ( suc @ N ) )
= ( mult_a_ring_ext_a_b @ G @ ( pow_a_1026414303147256608_b_nat @ G @ X @ N ) @ X ) ) ).
% Group.nat_pow_Suc
thf(fact_817_Group_Onat__pow__0,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( pow_a_1875594501834816709it_nat @ G @ X @ zero_zero_nat )
= ( one_a_Product_unit @ G ) ) ).
% Group.nat_pow_0
thf(fact_818_Group_Onat__pow__0,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( pow_a_1026414303147256608_b_nat @ G @ X @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ G ) ) ).
% Group.nat_pow_0
thf(fact_819_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_820_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_821_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_822_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_823_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_824_mult__of_Onat__pow__closed,axiom,
! [X: a,N: 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 @ N ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.nat_pow_closed
thf(fact_825_mult__of_Onat__pow__one,axiom,
! [N: nat] :
( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( one_a_ring_ext_a_b @ r ) ) ).
% mult_of.nat_pow_one
thf(fact_826_mult__of_Onat__pow__Suc,axiom,
! [X: a,N: nat] :
( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ ( suc @ N ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N ) @ X ) ) ).
% mult_of.nat_pow_Suc
thf(fact_827_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_828_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N3: nat] :
? [K5: nat] :
( N3
= ( plus_plus_nat @ M6 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_829_trans__le__add2,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_830_trans__le__add1,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_831_add__le__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_832_add__le__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_833_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K2 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_834_add__leD2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_835_add__leD1,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_836_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_837_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_838_add__leE,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_839_group__cancel_Osub1,axiom,
! [A2: int,K2: int,A: int,B2: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( minus_minus_int @ A2 @ B2 )
= ( plus_plus_int @ K2 @ ( minus_minus_int @ A @ B2 ) ) ) ) ).
% group_cancel.sub1
thf(fact_840_diff__eq__eq,axiom,
! [A: int,B2: int,C: int] :
( ( ( minus_minus_int @ A @ B2 )
= C )
= ( A
= ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_eq_eq
thf(fact_841_eq__diff__eq,axiom,
! [A: int,C: int,B2: int] :
( ( A
= ( minus_minus_int @ C @ B2 ) )
= ( ( plus_plus_int @ A @ B2 )
= C ) ) ).
% eq_diff_eq
thf(fact_842_add__diff__eq,axiom,
! [A: int,B2: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% add_diff_eq
thf(fact_843_diff__diff__eq2,axiom,
! [A: int,B2: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).
% diff_diff_eq2
thf(fact_844_diff__add__eq,axiom,
! [A: int,B2: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).
% diff_add_eq
thf(fact_845_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B2: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_846_add__implies__diff,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A )
=> ( C
= ( minus_minus_nat @ A @ B2 ) ) ) ).
% add_implies_diff
thf(fact_847_add__implies__diff,axiom,
! [C: int,B2: int,A: int] :
( ( ( plus_plus_int @ C @ B2 )
= A )
=> ( C
= ( minus_minus_int @ A @ B2 ) ) ) ).
% add_implies_diff
thf(fact_848_diff__diff__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_849_diff__diff__eq,axiom,
! [A: int,B2: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_850_add__le__imp__le__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_eq_int @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_851_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_852_add__le__imp__le__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_eq_int @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_853_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_854_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B6: nat] :
? [C5: nat] :
( B6
= ( plus_plus_nat @ A3 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_855_add__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_856_add__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_857_less__eqE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ~ ! [C4: nat] :
( B2
!= ( plus_plus_nat @ A @ C4 ) ) ) ).
% less_eqE
thf(fact_858_add__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_859_add__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_860_add__mono,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_861_add__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_862_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_863_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_864_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( I2 = J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_865_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_866_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_867_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_868_add__nonpos__eq__0__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y2 @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y2 )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_869_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_870_add__nonneg__eq__0__iff,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( ( plus_plus_int @ X @ Y2 )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y2 = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_871_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_872_add__nonpos__nonpos,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_873_add__nonpos__nonpos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_874_add__nonneg__nonneg,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_875_add__nonneg__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_876_add__increasing2,axiom,
! [C: int,B2: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B2 @ A )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_877_add__increasing2,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_878_add__decreasing2,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_879_add__decreasing2,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_880_add__increasing,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_881_add__increasing,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_882_add__decreasing,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_883_add__decreasing,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_884_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A )
= C )
= ( B2
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_885_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B2 @ A ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_886_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_887_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_888_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_889_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_890_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_891_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_892_le__add__diff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_893_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ A )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_894_le__diff__eq,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B2 ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% le_diff_eq
thf(fact_895_diff__le__eq,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B2 ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_le_eq
thf(fact_896_le__diff__conv,axiom,
! [J: nat,K2: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K2 ) ) ) ).
% le_diff_conv
thf(fact_897_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_898_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K2 )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_899_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_900_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_901_subgroup_Ocarrier__subgroup__generated__subgroup,axiom,
! [H: set_a,G: partia2175431115845679010xt_a_b] :
( ( subgro1816942748394427906xt_a_b @ H @ G )
=> ( ( partia707051561876973205xt_a_b @ ( genera8625346715478425275xt_a_b @ G @ H ) )
= H ) ) ).
% subgroup.carrier_subgroup_generated_subgroup
thf(fact_902_subgroup_Ocarrier__subgroup__generated__subgroup,axiom,
! [H: set_a,G: partia8223610829204095565t_unit] :
( ( subgro3222307229058429633t_unit @ H @ G )
=> ( ( partia6735698275553448452t_unit @ ( genera8815471607677139784t_unit @ G @ H ) )
= H ) ) ).
% subgroup.carrier_subgroup_generated_subgroup
thf(fact_903_mult__of_Omultiplicity__ge__iff,axiom,
! [D: a,A: a,K2: nat] :
( ( member_a @ D @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ D )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_nat @ K2 @ ( finite3319336104485747662t_unit @ ( ring_mult_of_a_b @ r ) @ D @ A ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ D @ K2 ) @ A ) ) ) ) ) ).
% mult_of.multiplicity_ge_iff
thf(fact_904_mult__of_Opow__order__eq__1,axiom,
! [A: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ A @ ( order_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% mult_of.pow_order_eq_1
thf(fact_905_le__add__diff__inverse,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_906_le__add__diff__inverse,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_907_Units__pow__closed,axiom,
! [X: a,D: nat] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X @ D ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_pow_closed
thf(fact_908_pow__non__zero,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
!= ( zero_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ N )
!= ( zero_a_b @ r ) ) ) ) ).
% pow_non_zero
thf(fact_909_group__commutes__pow,axiom,
! [X: a,Y2: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_910_nat__pow__comm,axiom,
! [X: a,N: nat,M2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M2 ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_911_nat__pow__distrib,axiom,
! [X: a,Y2: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y2 ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y2 @ N ) ) ) ) ) ).
% nat_pow_distrib
thf(fact_912_pow__mult__distrib,axiom,
! [X: a,Y2: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y2 ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y2 @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_913_nat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( zero_a_b @ r ) @ N )
= ( zero_a_b @ r ) ) ) ).
% nat_pow_zero
thf(fact_914_nat__pow__Suc2,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ ( suc @ N ) )
= ( mult_a_ring_ext_a_b @ r @ X @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ).
% nat_pow_Suc2
thf(fact_915_nat__pow__mult,axiom,
! [X: a,N: nat,M2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M2 ) )
= ( pow_a_1026414303147256608_b_nat @ r @ X @ ( plus_plus_nat @ N @ M2 ) ) ) ) ).
% nat_pow_mult
thf(fact_916_units__of__pow,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( units_8174867845824275201xt_a_b @ r ) @ X @ N )
= ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ).
% units_of_pow
thf(fact_917_nat__pow__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% nat_pow_closed
thf(fact_918_nat__pow__one,axiom,
! [N: nat] :
( ( pow_a_1026414303147256608_b_nat @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( one_a_ring_ext_a_b @ r ) ) ).
% nat_pow_one
thf(fact_919_le__add__diff__inverse2,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_920_le__add__diff__inverse2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B2 ) @ B2 )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_921_local_Onat__pow__Suc,axiom,
! [X: a,N: nat] :
( ( pow_a_1026414303147256608_b_nat @ r @ X @ ( suc @ N ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ X ) ) ).
% local.nat_pow_Suc
thf(fact_922_local_Onat__pow__0,axiom,
! [X: a] :
( ( pow_a_1026414303147256608_b_nat @ r @ X @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ r ) ) ).
% local.nat_pow_0
thf(fact_923_mult__of_Odivides__iff__mult__mono,axiom,
! [A: a,B2: a,R2: set_a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( finite6556546261911251790t_unit @ ( ring_mult_of_a_b @ r ) @ R2 )
=> ( ! [D3: a] :
( ( member_a @ D3 @ R2 )
=> ( ord_less_eq_nat @ ( finite3319336104485747662t_unit @ ( ring_mult_of_a_b @ r ) @ D3 @ A ) @ ( finite3319336104485747662t_unit @ ( ring_mult_of_a_b @ r ) @ D3 @ B2 ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B2 ) ) ) ) ) ).
% mult_of.divides_iff_mult_mono
thf(fact_924_Ring__Divisibility_Onat__pow__mult__of,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ R2 ) )
= ( pow_a_1026414303147256608_b_nat @ R2 ) ) ).
% Ring_Divisibility.nat_pow_mult_of
thf(fact_925_domain_Opow__non__zero,axiom,
! [R2: partia6043505979758434576t_unit,X: set_a,N: nat] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( X
!= ( zero_s2174465271003423091t_unit @ R2 ) )
=> ( ( pow_se2949424662122293806it_nat @ R2 @ X @ N )
!= ( zero_s2174465271003423091t_unit @ R2 ) ) ) ) ) ).
% domain.pow_non_zero
thf(fact_926_domain_Opow__non__zero,axiom,
! [R2: partia2175431115845679010xt_a_b,X: a,N: nat] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( X
!= ( zero_a_b @ R2 ) )
=> ( ( pow_a_1026414303147256608_b_nat @ R2 @ X @ N )
!= ( zero_a_b @ R2 ) ) ) ) ) ).
% domain.pow_non_zero
thf(fact_927_add__le__add__imp__diff__le,axiom,
! [I2: int,K2: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_928_add__le__add__imp__diff__le,axiom,
! [I2: nat,K2: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_929_add__le__imp__le__diff,axiom,
! [I2: int,K2: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ N )
=> ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_930_add__le__imp__le__diff,axiom,
! [I2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_931_mult__of_Omultiplicity__gt__0__iff,axiom,
! [D: a,A: a] :
( ( member_a @ D @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ D )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite3319336104485747662t_unit @ ( ring_mult_of_a_b @ r ) @ D @ A ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ D @ A ) ) ) ) ) ).
% mult_of.multiplicity_gt_0_iff
thf(fact_932_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_933_mult__of_Ofinite__cyclic__subgroup__order,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
= ( ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ X )
!= zero_zero_nat ) ) ) ).
% mult_of.finite_cyclic_subgroup_order
thf(fact_934_mult__of_Oord__eq__0,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ X )
= zero_zero_nat )
= ( ! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N3 )
!= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% mult_of.ord_eq_0
thf(fact_935_mult__of_Oord__le__group__order,axiom,
! [A: a] :
( ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ord_less_eq_nat @ ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ A ) @ ( order_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.ord_le_group_order
thf(fact_936_mult__of_Ocyclic__order__is__ord,axiom,
! [G3: a] :
( ( member_a @ G3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ G3 )
= ( order_a_Product_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ ( insert_a @ G3 @ bot_bot_set_a ) ) ) ) ) ).
% mult_of.cyclic_order_is_ord
thf(fact_937_mult__of_Oinfinite__cyclic__subgroup__order,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ~ ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) )
= ( ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ X )
= zero_zero_nat ) ) ) ).
% mult_of.infinite_cyclic_subgroup_order
thf(fact_938_diff__gt__0__iff__gt,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
= ( ord_less_int @ B2 @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_939_mult__of_Oord__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X ) )
= ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ X ) ) ) ).
% mult_of.ord_inv
thf(fact_940_mult__of_Opow__ord__eq__1,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 @ ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ X ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% mult_of.pow_ord_eq_1
thf(fact_941_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B2: int] :
( ~ ( ord_less_int @ A @ B2 )
=> ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_942_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B2: nat] :
( ~ ( ord_less_nat @ A @ B2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_943_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_944_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_945_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_946_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_947_add__le__less__mono,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_948_add__le__less__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_949_add__less__le__mono,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_950_add__less__le__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_951_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B6: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B6 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_952_diff__less__eq,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B2 ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_less_eq
thf(fact_953_less__diff__eq,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B2 ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% less_diff_eq
thf(fact_954_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F3 @ I3 ) @ ( F3 @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F3 @ I2 ) @ ( F3 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_955_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_956_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_957_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
| ( M6 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_958_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_959_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N3: nat] :
( ( ord_less_eq_nat @ M6 @ N3 )
& ( M6 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_960_diff__strict__mono,axiom,
! [A: int,B2: int,D: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_961_diff__eq__diff__less,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B2 )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_962_diff__strict__left__mono,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_963_diff__strict__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_964_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_965_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_966_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_967_dec__induct,axiom,
! [I2: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P2 @ I2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P2 @ N2 )
=> ( P2 @ ( suc @ N2 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_968_inc__induct,axiom,
! [I2: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P2 @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P2 @ ( suc @ N2 ) )
=> ( P2 @ N2 ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% inc_induct
thf(fact_969_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_970_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_971_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_972_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_973_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_974_less__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_975_diff__less__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_976_mono__nat__linear__lb,axiom,
! [F3: nat > nat,M2: nat,K2: nat] :
( ! [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( ord_less_nat @ ( F3 @ M5 ) @ ( F3 @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F3 @ M2 ) @ K2 ) @ ( F3 @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_977_add__strict__increasing2,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_978_add__strict__increasing2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_979_add__strict__increasing,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_980_add__strict__increasing,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_981_add__pos__nonneg,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_982_add__pos__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_983_add__nonpos__neg,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_984_add__nonpos__neg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_985_add__nonneg__pos,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_986_add__nonneg__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_987_add__neg__nonpos,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_988_add__neg__nonpos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_989_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_990_less__diff__conv2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I2 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_991_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_992_boundD__carrier,axiom,
! [N: nat,F3: nat > a,M2: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F3 )
=> ( ( ord_less_nat @ N @ M2 )
=> ( member_a @ ( F3 @ M2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_993_euclidean__function,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B2 @ ( 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 @ B2 @ Q2 ) @ R4 ) )
& ( ( R4
= ( zero_a_b @ r ) )
| ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ) ) ).
% euclidean_function
thf(fact_994_add_Ol__cancel,axiom,
! [C: a,A: a,B2: a] :
( ( ( add_a_b @ r @ C @ A )
= ( add_a_b @ r @ C @ B2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B2 ) ) ) ) ) ).
% add.l_cancel
thf(fact_995_add_Or__cancel,axiom,
! [A: a,C: a,B2: a] :
( ( ( add_a_b @ r @ A @ C )
= ( add_a_b @ r @ B2 @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B2 ) ) ) ) ) ).
% add.r_cancel
thf(fact_996_a__assoc,axiom,
! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y2 ) @ Z )
= ( add_a_b @ r @ X @ ( add_a_b @ r @ Y2 @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_997_a__comm,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ Y2 )
= ( add_a_b @ r @ Y2 @ X ) ) ) ) ).
% a_comm
thf(fact_998_a__lcomm,axiom,
! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y2 @ Z ) )
= ( add_a_b @ r @ Y2 @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_999_subring__props_I7_J,axiom,
! [K: set_a,H1: a,H2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H2 @ K )
=> ( member_a @ ( add_a_b @ r @ H1 @ H2 ) @ K ) ) ) ) ).
% subring_props(7)
thf(fact_1000_add_Oinv__comm,axiom,
! [X: a,Y2: a] :
( ( ( add_a_b @ r @ X @ Y2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y2 @ X )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_1001_add_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X3 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_1002_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X3 )
= X3 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_1003_add_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X3 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_1004_local_Ominus__unique,axiom,
! [Y2: a,X: a,Y3: a] :
( ( ( add_a_b @ r @ Y2 @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y3 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y2 = Y3 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_1005_l__distr,axiom,
! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y2 ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y2 @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_1006_r__distr,axiom,
! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y2 ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y2 ) ) ) ) ) ) ).
% r_distr
thf(fact_1007_div__sum,axiom,
! [A: a,B2: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B2 )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ C )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( add_a_b @ r @ B2 @ C ) ) ) ) ) ) ) ).
% div_sum
thf(fact_1008_div__sum__iff,axiom,
! [A: a,B2: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B2 )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ ( add_a_b @ r @ B2 @ C ) )
= ( factor8216151070175719842xt_a_b @ r @ A @ C ) ) ) ) ) ) ).
% div_sum_iff
thf(fact_1009_psubsetI,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_a @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_1010_space__subgroup__props_I3_J,axiom,
! [K: set_a,N: nat,E: set_a,V1: a,V22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ 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_1011_line__extension__mem__iff,axiom,
! [U: a,K: set_a,A: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ K )
& ? [Y5: a] :
( ( member_a @ Y5 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ A ) @ Y5 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_1012_a__lcos__m__assoc,axiom,
! [M: set_a,G3: a,H3: a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G3 @ ( a_l_coset_a_b @ r @ H3 @ M ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G3 @ H3 ) @ M ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_1013_local_Oadd_Oright__cancel,axiom,
! [X: a,Y2: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y2 @ X )
= ( add_a_b @ r @ Z @ X ) )
= ( Y2 = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_1014_a__closed,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X @ Y2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_1015_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_1016_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_1017_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_1018_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_1019_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_1020_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_1021_euclidean__domainI,axiom,
! [Phi: a > nat] :
( ! [A4: a,B4: a] :
( ( member_a @ A4 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B4 @ ( 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 ) )
& ( A4
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ B4 @ Q3 ) @ R5 ) )
& ( ( R5
= ( zero_a_b @ r ) )
| ( ord_less_nat @ ( Phi @ R5 ) @ ( Phi @ B4 ) ) ) ) ) )
=> ( ring_e8745995371659049232in_a_b @ r @ Phi ) ) ).
% euclidean_domainI
thf(fact_1022_psubset__imp__ex__mem,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B )
=> ? [B4: set_a] : ( member_set_a @ B4 @ ( minus_5736297505244876581_set_a @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1023_psubset__imp__ex__mem,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ? [B4: a] : ( member_a @ B4 @ ( minus_minus_set_a @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1024_psubsetE,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% psubsetE
thf(fact_1025_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_1026_psubset__imp__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% psubset_imp_subset
thf(fact_1027_psubset__subset__trans,axiom,
! [A2: set_a,B: set_a,C3: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C3 )
=> ( ord_less_set_a @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_1028_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ~ ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1029_subset__psubset__trans,axiom,
! [A2: set_a,B: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_set_a @ B @ C3 )
=> ( ord_less_set_a @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_1030_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_set_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1031_not__psubset__empty,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_1032_finite__psubset__induct,axiom,
! [A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ! [B7: set_a] :
( ( ord_less_set_a @ B7 @ A6 )
=> ( P2 @ B7 ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_1033_domain_Odiv__sum__iff,axiom,
! [R2: partia6043505979758434576t_unit,A: set_a,B2: set_a,C: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ B2 @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( factor5460682277579321776t_unit @ R2 @ A @ B2 )
=> ( ( factor5460682277579321776t_unit @ R2 @ A @ ( add_se3735415688806051380t_unit @ R2 @ B2 @ C ) )
= ( factor5460682277579321776t_unit @ R2 @ A @ C ) ) ) ) ) ) ) ).
% domain.div_sum_iff
thf(fact_1034_domain_Odiv__sum__iff,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B2: a,C: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( factor8216151070175719842xt_a_b @ R2 @ A @ B2 )
=> ( ( factor8216151070175719842xt_a_b @ R2 @ A @ ( add_a_b @ R2 @ B2 @ C ) )
= ( factor8216151070175719842xt_a_b @ R2 @ A @ C ) ) ) ) ) ) ) ).
% domain.div_sum_iff
thf(fact_1035_domain_Odiv__sum,axiom,
! [R2: partia6043505979758434576t_unit,A: set_a,B2: set_a,C: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ B2 @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( factor5460682277579321776t_unit @ R2 @ A @ B2 )
=> ( ( factor5460682277579321776t_unit @ R2 @ A @ C )
=> ( factor5460682277579321776t_unit @ R2 @ A @ ( add_se3735415688806051380t_unit @ R2 @ B2 @ C ) ) ) ) ) ) ) ) ).
% domain.div_sum
thf(fact_1036_domain_Odiv__sum,axiom,
! [R2: partia2175431115845679010xt_a_b,A: a,B2: a,C: a] :
( ( domain_a_b @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( factor8216151070175719842xt_a_b @ R2 @ A @ B2 )
=> ( ( factor8216151070175719842xt_a_b @ R2 @ A @ C )
=> ( factor8216151070175719842xt_a_b @ R2 @ A @ ( add_a_b @ R2 @ B2 @ C ) ) ) ) ) ) ) ) ).
% domain.div_sum
thf(fact_1037_psubset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A2 @ ( insert_set_a @ X @ B ) )
= ( ( ( member_set_a @ X @ B )
=> ( ord_less_set_set_a @ A2 @ B ) )
& ( ~ ( member_set_a @ X @ B )
=> ( ( ( member_set_a @ X @ A2 )
=> ( ord_less_set_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1038_psubset__insert__iff,axiom,
! [A2: set_a,X: a,B: set_a] :
( ( ord_less_set_a @ A2 @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ B )
=> ( ord_less_set_a @ A2 @ B ) )
& ( ~ ( member_a @ X @ B )
=> ( ( ( member_a @ X @ A2 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1039_finite__induct__select,axiom,
! [S: set_a,P2: set_a > $o] :
( ( finite_finite_a @ S )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [T3: set_a] :
( ( ord_less_set_a @ T3 @ S )
=> ( ( P2 @ T3 )
=> ? [X4: a] :
( ( member_a @ X4 @ ( minus_minus_set_a @ S @ T3 ) )
& ( P2 @ ( insert_a @ X4 @ T3 ) ) ) ) )
=> ( P2 @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1040_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_1041_mult__of_Oord__ge__1,axiom,
! [A: a] :
( ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ A ) ) ) ) ).
% mult_of.ord_ge_1
thf(fact_1042_telescopic__base__aux,axiom,
! [K: set_a,F: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( subfield_a_b @ F @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ F )
=> ( ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ F @ E )
=> ( embedd2795209813406577254on_a_b @ r @ N @ K @ E ) ) ) ) ) ).
% telescopic_base_aux
thf(fact_1043_mult__of_Oord__eq__1,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ X )
= one_one_nat )
= ( X
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.ord_eq_1
thf(fact_1044_nat__pow__eone,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ one_one_nat )
= X ) ) ).
% nat_pow_eone
thf(fact_1045_dimension__one,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ K @ K ) ) ).
% dimension_one
thf(fact_1046_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_1047_mult__of_Oord__id,axiom,
( ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ ( one_a_ring_ext_a_b @ r ) )
= one_one_nat ) ).
% mult_of.ord_id
thf(fact_1048_psubsetD,axiom,
! [A2: set_a,B: set_a,C: a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_1049_psubsetD,axiom,
! [A2: set_set_a,B: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A2 @ B )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B ) ) ) ).
% psubsetD
thf(fact_1050_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_1051_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1052_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1053_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1054_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_1055_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1056_euclidean__domain_Oaxioms_I1_J,axiom,
! [R2: partia6043505979758434576t_unit,Phi: set_a > nat] :
( ( ring_e187967263881214398t_unit @ R2 @ Phi )
=> ( domain4236798911309298543t_unit @ R2 ) ) ).
% euclidean_domain.axioms(1)
thf(fact_1057_euclidean__domain_Oaxioms_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b,Phi: a > nat] :
( ( ring_e8745995371659049232in_a_b @ R2 @ Phi )
=> ( domain_a_b @ R2 ) ) ).
% euclidean_domain.axioms(1)
thf(fact_1058_field_Oorder__mult__of,axiom,
! [R2: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( ( finite_finite_set_a @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( order_6731527964810494035t_unit @ ( multip3774352783277980819t_unit @ R2 ) )
= ( minus_minus_nat @ ( order_3761338894132136606t_unit @ R2 ) @ one_one_nat ) ) ) ) ).
% field.order_mult_of
thf(fact_1059_field_Oorder__mult__of,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R2 )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( order_a_Product_unit @ ( multip3210463924028840165of_a_b @ R2 ) )
= ( minus_minus_nat @ ( order_a_ring_ext_a_b @ R2 ) @ one_one_nat ) ) ) ) ).
% field.order_mult_of
thf(fact_1060_euclidean__domain_Oeuclidean__function,axiom,
! [R2: partia2175431115845679010xt_a_b,Phi: a > nat,A: a,B2: a] :
( ( ring_e8745995371659049232in_a_b @ R2 @ Phi )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
=> ? [Q2: a,R4: a] :
( ( member_a @ Q2 @ ( partia707051561876973205xt_a_b @ R2 ) )
& ( member_a @ R4 @ ( partia707051561876973205xt_a_b @ R2 ) )
& ( A
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ B2 @ Q2 ) @ R4 ) )
& ( ( R4
= ( zero_a_b @ R2 ) )
| ( ord_less_nat @ ( Phi @ R4 ) @ ( Phi @ B2 ) ) ) ) ) ) ) ).
% euclidean_domain.euclidean_function
thf(fact_1061_domain_Oeuclidean__domainI,axiom,
! [R2: partia6043505979758434576t_unit,Phi: set_a > nat] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ! [A4: set_a,B4: set_a] :
( ( member_set_a @ A4 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R2 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) )
=> ( ( member_set_a @ B4 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R2 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) )
=> ? [Q3: set_a,R5: set_a] :
( ( member_set_a @ Q3 @ ( partia5907974310037520643t_unit @ R2 ) )
& ( member_set_a @ R5 @ ( partia5907974310037520643t_unit @ R2 ) )
& ( A4
= ( add_se3735415688806051380t_unit @ R2 @ ( mult_s7930653359683758801t_unit @ R2 @ B4 @ Q3 ) @ R5 ) )
& ( ( R5
= ( zero_s2174465271003423091t_unit @ R2 ) )
| ( ord_less_nat @ ( Phi @ R5 ) @ ( Phi @ B4 ) ) ) ) ) )
=> ( ring_e187967263881214398t_unit @ R2 @ Phi ) ) ) ).
% domain.euclidean_domainI
thf(fact_1062_domain_Oeuclidean__domainI,axiom,
! [R2: partia2175431115845679010xt_a_b,Phi: a > nat] :
( ( domain_a_b @ R2 )
=> ( ! [A4: a,B4: a] :
( ( member_a @ A4 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B4 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
=> ? [Q3: a,R5: a] :
( ( member_a @ Q3 @ ( partia707051561876973205xt_a_b @ R2 ) )
& ( member_a @ R5 @ ( partia707051561876973205xt_a_b @ R2 ) )
& ( A4
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ B4 @ Q3 ) @ R5 ) )
& ( ( R5
= ( zero_a_b @ R2 ) )
| ( ord_less_nat @ ( Phi @ R5 ) @ ( Phi @ B4 ) ) ) ) ) )
=> ( ring_e8745995371659049232in_a_b @ R2 @ Phi ) ) ) ).
% domain.euclidean_domainI
thf(fact_1063_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1064_freshmans__dream,axiom,
! [X: a,Y2: a] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( add_a_b @ r @ X @ Y2 ) @ ( ring_char_a_b @ r ) )
= ( add_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ ( ring_char_a_b @ r ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y2 @ ( ring_char_a_b @ r ) ) ) ) ) ) ) ).
% freshmans_dream
thf(fact_1065_finite__carr__imp__char__ge__0,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) ) ) ).
% finite_carr_imp_char_ge_0
thf(fact_1066_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_1067_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1068_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_1069_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_1070_domain_Ofreshmans__dream,axiom,
! [R2: partia6043505979758434576t_unit,X: set_a,Y2: set_a] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( ring_c1190685578609594780t_unit @ R2 ) )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ Y2 @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( pow_se2949424662122293806it_nat @ R2 @ ( add_se3735415688806051380t_unit @ R2 @ X @ Y2 ) @ ( ring_c1190685578609594780t_unit @ R2 ) )
= ( add_se3735415688806051380t_unit @ R2 @ ( pow_se2949424662122293806it_nat @ R2 @ X @ ( ring_c1190685578609594780t_unit @ R2 ) ) @ ( pow_se2949424662122293806it_nat @ R2 @ Y2 @ ( ring_c1190685578609594780t_unit @ R2 ) ) ) ) ) ) ) ) ).
% domain.freshmans_dream
thf(fact_1071_domain_Ofreshmans__dream,axiom,
! [R2: partia2175431115845679010xt_a_b,X: a,Y2: a] :
( ( domain_a_b @ R2 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ R2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( pow_a_1026414303147256608_b_nat @ R2 @ ( add_a_b @ R2 @ X @ Y2 ) @ ( ring_char_a_b @ R2 ) )
= ( add_a_b @ R2 @ ( pow_a_1026414303147256608_b_nat @ R2 @ X @ ( ring_char_a_b @ R2 ) ) @ ( pow_a_1026414303147256608_b_nat @ R2 @ Y2 @ ( ring_char_a_b @ R2 ) ) ) ) ) ) ) ) ).
% domain.freshmans_dream
thf(fact_1072_freshmans__dream__ext,axiom,
! [X: a,Y2: a,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( add_a_b @ r @ X @ Y2 ) @ ( power_power_nat @ ( ring_char_a_b @ r ) @ M2 ) )
= ( add_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ ( power_power_nat @ ( ring_char_a_b @ r ) @ M2 ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y2 @ ( power_power_nat @ ( ring_char_a_b @ r ) @ M2 ) ) ) ) ) ) ) ).
% freshmans_dream_ext
thf(fact_1073_mult__of_Ofinite__cyclic__subgroup__int,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
= ( ? [I5: int] :
( ( I5 != zero_zero_int )
& ( ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ X @ I5 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% mult_of.finite_cyclic_subgroup_int
thf(fact_1074_mult__of_Osubgroup__int__pow__closed,axiom,
! [H: set_a,H3: a,K2: int] :
( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( ( member_a @ H3 @ H )
=> ( member_a @ ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ H3 @ K2 ) @ H ) ) ) ).
% mult_of.subgroup_int_pow_closed
thf(fact_1075_mult__of_Oint__pow__mult__distrib,axiom,
! [X: a,Y2: a,I2: int] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ X @ Y2 ) @ I2 )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ X @ I2 ) @ ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ Y2 @ I2 ) ) ) ) ) ) ).
% mult_of.int_pow_mult_distrib
thf(fact_1076_mult__of_Oint__pow__distrib,axiom,
! [X: a,Y2: a,I2: int] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ X @ Y2 ) @ I2 )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ X @ I2 ) @ ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ Y2 @ I2 ) ) ) ) ) ).
% mult_of.int_pow_distrib
thf(fact_1077_mult__of_Oint__pow__mult,axiom,
! [X: a,I2: int,J: int] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ X @ ( plus_plus_int @ I2 @ J ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ X @ I2 ) @ ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ X @ J ) ) ) ) ).
% mult_of.int_pow_mult
thf(fact_1078_mult__of_Oint__pow__inv,axiom,
! [X: a,I2: int] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X ) @ I2 )
= ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ X @ I2 ) ) ) ) ).
% mult_of.int_pow_inv
thf(fact_1079_mult__of_Oint__pow__subgroup__generated,axiom,
! [X: a,S: set_a,N: int] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ S ) ) )
=> ( ( pow_a_1873104031325766433it_int @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ S ) @ X @ N )
= ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ X @ N ) ) ) ).
% mult_of.int_pow_subgroup_generated
thf(fact_1080_mult__of_Oint__pow__diff,axiom,
! [X: a,N: int,M2: int] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ X @ ( minus_minus_int @ N @ M2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ X @ N ) @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ X @ M2 ) ) ) ) ) ).
% mult_of.int_pow_diff
thf(fact_1081_mult__of_Oinfinite__cyclic__subgroup__int,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ~ ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( genera8815471607677139784t_unit @ ( ring_mult_of_a_b @ r ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) )
= ( ! [I5: int,J4: int] :
( ( ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ X @ I5 )
= ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ X @ J4 ) )
=> ( I5 = J4 ) ) ) ) ) ).
% mult_of.infinite_cyclic_subgroup_int
thf(fact_1082_int__pow__0,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( pow_a_1873104031325766433it_int @ G @ X @ zero_zero_int )
= ( one_a_Product_unit @ G ) ) ).
% int_pow_0
thf(fact_1083_int__pow__0,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( pow_a_1023923832638206332_b_int @ G @ X @ zero_zero_int )
= ( one_a_ring_ext_a_b @ G ) ) ).
% int_pow_0
thf(fact_1084_mult__of_Oint__pow__closed,axiom,
! [X: a,I2: int] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ X @ I2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.int_pow_closed
thf(fact_1085_mult__of_Oint__pow__1,axiom,
! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ X @ one_one_int )
= X ) ) ).
% mult_of.int_pow_1
thf(fact_1086_mult__of_Oint__pow__one,axiom,
! [Z: int] :
( ( pow_a_1873104031325766433it_int @ ( ring_mult_of_a_b @ r ) @ ( one_a_ring_ext_a_b @ r ) @ Z )
= ( one_a_ring_ext_a_b @ r ) ) ).
% mult_of.int_pow_one
thf(fact_1087_domain_Ofreshmans__dream__ext,axiom,
! [R2: partia6043505979758434576t_unit,X: set_a,Y2: set_a,M2: nat] :
( ( domain4236798911309298543t_unit @ R2 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( ring_c1190685578609594780t_unit @ R2 ) )
=> ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( member_set_a @ Y2 @ ( partia5907974310037520643t_unit @ R2 ) )
=> ( ( pow_se2949424662122293806it_nat @ R2 @ ( add_se3735415688806051380t_unit @ R2 @ X @ Y2 ) @ ( power_power_nat @ ( ring_c1190685578609594780t_unit @ R2 ) @ M2 ) )
= ( add_se3735415688806051380t_unit @ R2 @ ( pow_se2949424662122293806it_nat @ R2 @ X @ ( power_power_nat @ ( ring_c1190685578609594780t_unit @ R2 ) @ M2 ) ) @ ( pow_se2949424662122293806it_nat @ R2 @ Y2 @ ( power_power_nat @ ( ring_c1190685578609594780t_unit @ R2 ) @ M2 ) ) ) ) ) ) ) ) ).
% domain.freshmans_dream_ext
thf(fact_1088_domain_Ofreshmans__dream__ext,axiom,
! [R2: partia2175431115845679010xt_a_b,X: a,Y2: a,M2: nat] :
( ( domain_a_b @ R2 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ R2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( pow_a_1026414303147256608_b_nat @ R2 @ ( add_a_b @ R2 @ X @ Y2 ) @ ( power_power_nat @ ( ring_char_a_b @ R2 ) @ M2 ) )
= ( add_a_b @ R2 @ ( pow_a_1026414303147256608_b_nat @ R2 @ X @ ( power_power_nat @ ( ring_char_a_b @ R2 ) @ M2 ) ) @ ( pow_a_1026414303147256608_b_nat @ R2 @ Y2 @ ( power_power_nat @ ( ring_char_a_b @ R2 ) @ M2 ) ) ) ) ) ) ) ) ).
% domain.freshmans_dream_ext
thf(fact_1089_power__decreasing__iff,axiom,
! [B2: int,M2: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ B2 @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B2 @ M2 ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_1090_power__decreasing__iff,axiom,
! [B2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ B2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ M2 ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_1091_power__mono__iff,axiom,
! [A: int,B2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_eq_int @ A @ B2 ) ) ) ) ) ).
% power_mono_iff
thf(fact_1092_power__mono__iff,axiom,
! [A: nat,B2: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ) ) ) ).
% power_mono_iff
thf(fact_1093_power__increasing__iff,axiom,
! [B2: int,X: nat,Y2: nat] :
( ( ord_less_int @ one_one_int @ B2 )
=> ( ( ord_less_eq_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_1094_power__increasing__iff,axiom,
! [B2: nat,X: nat,Y2: nat] :
( ( ord_less_nat @ one_one_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ) ).
% power_increasing_iff
thf(fact_1095_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1096_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1097_power__mono,axiom,
! [A: int,B2: int,N: nat] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% power_mono
thf(fact_1098_power__mono,axiom,
! [A: nat,B2: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% power_mono
thf(fact_1099_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_1100_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_1101_power__less__imp__less__base,axiom,
! [A: int,N: nat,B2: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% power_less_imp_less_base
thf(fact_1102_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B2: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% power_less_imp_less_base
thf(fact_1103_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_1104_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_1105_power__inject__base,axiom,
! [A: int,N: nat,B2: int] :
( ( ( power_power_int @ A @ ( suc @ N ) )
= ( power_power_int @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( A = B2 ) ) ) ) ).
% power_inject_base
thf(fact_1106_power__inject__base,axiom,
! [A: nat,N: nat,B2: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( A = B2 ) ) ) ) ).
% power_inject_base
thf(fact_1107_power__le__imp__le__base,axiom,
! [A: int,N: nat,B2: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ A @ B2 ) ) ) ).
% power_le_imp_le_base
thf(fact_1108_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% power_le_imp_le_base
thf(fact_1109_power__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1110_power__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1111_nat__one__le__power,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N ) ) ) ).
% nat_one_le_power
thf(fact_1112_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_1113_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_1114_power__decreasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1115_power__decreasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1116_power__le__imp__le__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1117_power__le__imp__le__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1118_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B2: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B2 @ N ) )
= ( A = B2 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1119_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B2 @ N ) )
= ( A = B2 ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1120_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B2: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B2 @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B2 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1121_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B2: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B2 @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B2 ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1122_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_1123_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_1124_power__strict__mono,axiom,
! [A: int,B2: int,N: nat] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_1125_power__strict__mono,axiom,
! [A: nat,B2: nat,N: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_1126_mult__of_Onormal__imp__commuting,axiom,
! [A2: set_a,B: set_a,X: a,Y2: a] :
( ( normal1135164612128607456t_unit @ A2 @ ( ring_mult_of_a_b @ r ) )
=> ( ( normal1135164612128607456t_unit @ B @ ( ring_mult_of_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y2 @ B )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X ) ) ) ) ) ) ) ).
% mult_of.normal_imp_commuting
thf(fact_1127_cyclic__group__def,axiom,
( elemen2247225643382758650xt_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b] :
? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G2 ) )
& ( ( genera8625346715478425275xt_a_b @ G2 @ ( insert_a @ X2 @ bot_bot_set_a ) )
= G2 ) ) ) ) ).
% cyclic_group_def
thf(fact_1128_cyclic__group__def,axiom,
( elemen5394956844934664649t_unit
= ( ^ [G2: partia8223610829204095565t_unit] :
? [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ G2 ) )
& ( ( genera8815471607677139784t_unit @ G2 @ ( insert_a @ X2 @ bot_bot_set_a ) )
= G2 ) ) ) ) ).
% cyclic_group_def
thf(fact_1129_mult__of_Onormal__self,axiom,
normal1135164612128607456t_unit @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) @ ( ring_mult_of_a_b @ r ) ).
% mult_of.normal_self
thf(fact_1130_mult__of_Onormal__invE_I1_J,axiom,
! [N5: set_a] :
( ( normal1135164612128607456t_unit @ N5 @ ( ring_mult_of_a_b @ r ) )
=> ( subgro3222307229058429633t_unit @ N5 @ ( ring_mult_of_a_b @ r ) ) ) ).
% mult_of.normal_invE(1)
thf(fact_1131_mult__of_Onormal__iff__subgroup,axiom,
! [N5: set_a] :
( ( normal1135164612128607456t_unit @ N5 @ ( ring_mult_of_a_b @ r ) )
= ( subgro3222307229058429633t_unit @ N5 @ ( ring_mult_of_a_b @ r ) ) ) ).
% mult_of.normal_iff_subgroup
thf(fact_1132_mult__of_Osubgroup__imp__normal,axiom,
! [A2: set_a] :
( ( subgro3222307229058429633t_unit @ A2 @ ( ring_mult_of_a_b @ r ) )
=> ( normal1135164612128607456t_unit @ A2 @ ( ring_mult_of_a_b @ r ) ) ) ).
% mult_of.subgroup_imp_normal
thf(fact_1133_mult__of_Oderived__is__normal,axiom,
! [H: set_a] :
( ( normal1135164612128607456t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( normal1135164612128607456t_unit @ ( genera353947490595344117t_unit @ ( ring_mult_of_a_b @ r ) @ H ) @ ( ring_mult_of_a_b @ r ) ) ) ).
% mult_of.derived_is_normal
thf(fact_1134_mult__of_Oderived__self__is__normal,axiom,
normal1135164612128607456t_unit @ ( genera353947490595344117t_unit @ ( ring_mult_of_a_b @ r ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) @ ( ring_mult_of_a_b @ r ) ).
% mult_of.derived_self_is_normal
thf(fact_1135_mult__of_Onormal__invE_I2_J,axiom,
! [N5: set_a,X: a,H3: a] :
( ( normal1135164612128607456t_unit @ N5 @ ( ring_mult_of_a_b @ r ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ H3 @ N5 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ H3 ) @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X ) ) @ N5 ) ) ) ) ).
% mult_of.normal_invE(2)
thf(fact_1136_mult__of_Oone__is__normal,axiom,
normal1135164612128607456t_unit @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) @ ( ring_mult_of_a_b @ r ) ).
% mult_of.one_is_normal
thf(fact_1137_mult__of_Onormal__inv__iff,axiom,
! [N5: set_a] :
( ( normal1135164612128607456t_unit @ N5 @ ( ring_mult_of_a_b @ r ) )
= ( ( subgro3222307229058429633t_unit @ N5 @ ( ring_mult_of_a_b @ r ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ N5 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y5 ) @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X2 ) ) @ N5 ) ) ) ) ) ).
% mult_of.normal_inv_iff
thf(fact_1138_mult__of_Onormal__invI,axiom,
! [N5: set_a] :
( ( subgro3222307229058429633t_unit @ N5 @ ( ring_mult_of_a_b @ r ) )
=> ( ! [X3: a,H5: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ H5 @ N5 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ H5 ) @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ X3 ) ) @ N5 ) ) )
=> ( normal1135164612128607456t_unit @ N5 @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.normal_invI
thf(fact_1139_trivial__imp__finite__group,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( elemen5692998889131970354xt_a_b @ G )
=> ( finite_finite_a @ ( partia707051561876973205xt_a_b @ G ) ) ) ).
% trivial_imp_finite_group
thf(fact_1140_trivial__imp__finite__group,axiom,
! [G: partia8223610829204095565t_unit] :
( ( elemen1145482699608675729t_unit @ G )
=> ( finite_finite_a @ ( partia6735698275553448452t_unit @ G ) ) ) ).
% trivial_imp_finite_group
thf(fact_1141_mult__of_Onormal__generateI,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ! [H5: a,G4: a] :
( ( member_a @ H5 @ H )
=> ( ( member_a @ G4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ G4 @ H5 ) @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ G4 ) ) @ H ) ) )
=> ( normal1135164612128607456t_unit @ ( genera4054545294875501499t_unit @ ( ring_mult_of_a_b @ r ) @ H ) @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.normal_generateI
thf(fact_1142_mult__of_Oord__unique,axiom,
! [X: a,D: nat] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ X )
= D )
= ( ! [N3: nat] :
( ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N3 )
= ( one_a_ring_ext_a_b @ r ) )
= ( dvd_dvd_nat @ D @ N3 ) ) ) ) ) ).
% mult_of.ord_unique
thf(fact_1143_mult__of_Omono__generate,axiom,
! [K: set_a,H: set_a] :
( ( ord_less_eq_set_a @ K @ H )
=> ( ord_less_eq_set_a @ ( genera4054545294875501499t_unit @ ( ring_mult_of_a_b @ r ) @ K ) @ ( genera4054545294875501499t_unit @ ( ring_mult_of_a_b @ r ) @ H ) ) ) ).
% mult_of.mono_generate
thf(fact_1144_mult__of_Ogenerate__in__carrier,axiom,
! [H: set_a,H3: a] :
( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ H3 @ ( genera4054545294875501499t_unit @ ( ring_mult_of_a_b @ r ) @ H ) )
=> ( member_a @ H3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.generate_in_carrier
thf(fact_1145_mult__of_Ogenerate__incl,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ord_less_eq_set_a @ ( genera4054545294875501499t_unit @ ( ring_mult_of_a_b @ r ) @ H ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.generate_incl
thf(fact_1146_mult__of_Ogenerate__subgroup__incl,axiom,
! [K: set_a,H: set_a] :
( ( ord_less_eq_set_a @ K @ H )
=> ( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( genera4054545294875501499t_unit @ ( ring_mult_of_a_b @ r ) @ K ) @ H ) ) ) ).
% mult_of.generate_subgroup_incl
thf(fact_1147_mult__of_OgenerateI,axiom,
! [E: set_a,H: set_a] :
( ( subgro3222307229058429633t_unit @ E @ ( ring_mult_of_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H @ E )
=> ( ! [K6: set_a] :
( ( subgro3222307229058429633t_unit @ K6 @ ( ring_mult_of_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H @ K6 )
=> ( ord_less_eq_set_a @ E @ K6 ) ) )
=> ( E
= ( genera4054545294875501499t_unit @ ( ring_mult_of_a_b @ r ) @ H ) ) ) ) ) ).
% mult_of.generateI
thf(fact_1148_mult__of_Ogenerate__is__subgroup,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( subgro3222307229058429633t_unit @ ( genera4054545294875501499t_unit @ ( ring_mult_of_a_b @ r ) @ H ) @ ( ring_mult_of_a_b @ r ) ) ) ).
% mult_of.generate_is_subgroup
thf(fact_1149_mult__of_Ogenerate__m__inv__closed,axiom,
! [H: set_a,H3: a] :
( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ H3 @ ( genera4054545294875501499t_unit @ ( ring_mult_of_a_b @ r ) @ H ) )
=> ( member_a @ ( m_inv_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ H3 ) @ ( genera4054545294875501499t_unit @ ( ring_mult_of_a_b @ r ) @ H ) ) ) ) ).
% mult_of.generate_m_inv_closed
thf(fact_1150_mult__of_Ogenerate__empty,axiom,
( ( genera4054545294875501499t_unit @ ( ring_mult_of_a_b @ r ) @ bot_bot_set_a )
= ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) ) ).
% mult_of.generate_empty
thf(fact_1151_mult__of_Ogenerate__one,axiom,
( ( genera4054545294875501499t_unit @ ( ring_mult_of_a_b @ r ) @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) )
= ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) ) ).
% mult_of.generate_one
thf(fact_1152_mult__of_Oord__dvd__group__order,axiom,
! [A: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( dvd_dvd_nat @ ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ A ) @ ( order_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.ord_dvd_group_order
thf(fact_1153_mult__of_Opow__eq__id,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N )
= ( one_a_ring_ext_a_b @ r ) )
= ( dvd_dvd_nat @ ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ X ) @ N ) ) ) ).
% mult_of.pow_eq_id
thf(fact_1154_generate_Oeng,axiom,
! [H1: a,G: partia2175431115845679010xt_a_b,H: set_a,H2: a] :
( ( member_a @ H1 @ ( genera2604884881691289736xt_a_b @ G @ H ) )
=> ( ( member_a @ H2 @ ( genera2604884881691289736xt_a_b @ G @ H ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ H1 @ H2 ) @ ( genera2604884881691289736xt_a_b @ G @ H ) ) ) ) ).
% generate.eng
thf(fact_1155_generate_Oeng,axiom,
! [H1: a,G: partia8223610829204095565t_unit,H: set_a,H2: a] :
( ( member_a @ H1 @ ( genera4054545294875501499t_unit @ G @ H ) )
=> ( ( member_a @ H2 @ ( genera4054545294875501499t_unit @ G @ H ) )
=> ( member_a @ ( mult_a_Product_unit @ G @ H1 @ H2 ) @ ( genera4054545294875501499t_unit @ G @ H ) ) ) ) ).
% generate.eng
thf(fact_1156_generate_Oone,axiom,
! [G: partia2175431115845679010xt_a_b,H: set_a] : ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( genera2604884881691289736xt_a_b @ G @ H ) ) ).
% generate.one
thf(fact_1157_generate_Oone,axiom,
! [G: partia8223610829204095565t_unit,H: set_a] : ( member_a @ ( one_a_Product_unit @ G ) @ ( genera4054545294875501499t_unit @ G @ H ) ) ).
% generate.one
thf(fact_1158_generate_Oinv,axiom,
! [H3: a,H: set_a,G: partia2175431115845679010xt_a_b] :
( ( member_a @ H3 @ H )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ G @ H3 ) @ ( genera2604884881691289736xt_a_b @ G @ H ) ) ) ).
% generate.inv
thf(fact_1159_generate_Oinv,axiom,
! [H3: a,H: set_a,G: partia8223610829204095565t_unit] :
( ( member_a @ H3 @ H )
=> ( member_a @ ( m_inv_a_Product_unit @ G @ H3 ) @ ( genera4054545294875501499t_unit @ G @ H ) ) ) ).
% generate.inv
thf(fact_1160_dvd__diff,axiom,
! [X: int,Y2: int,Z: int] :
( ( dvd_dvd_int @ X @ Y2 )
=> ( ( dvd_dvd_int @ X @ Z )
=> ( dvd_dvd_int @ X @ ( minus_minus_int @ Y2 @ Z ) ) ) ) ).
% dvd_diff
thf(fact_1161_less__eq__dvd__minus,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( dvd_dvd_nat @ M2 @ N )
= ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1162_dvd__diffD1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) )
=> ( ( dvd_dvd_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_nat @ K2 @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_1163_dvd__diffD,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) )
=> ( ( dvd_dvd_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_nat @ K2 @ M2 ) ) ) ) ).
% dvd_diffD
thf(fact_1164_le__imp__power__dvd,axiom,
! [M2: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_1165_power__le__dvd,axiom,
! [A: nat,N: nat,B2: nat,M2: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M2 ) @ B2 ) ) ) ).
% power_le_dvd
thf(fact_1166_dvd__power__le,axiom,
! [X: nat,Y2: nat,N: nat,M2: nat] :
( ( dvd_dvd_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y2 @ M2 ) ) ) ) ).
% dvd_power_le
thf(fact_1167_dvd__nat__bounds,axiom,
! [P: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ P )
=> ( ( dvd_dvd_nat @ N @ P )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ord_less_eq_nat @ N @ P ) ) ) ) ).
% dvd_nat_bounds
thf(fact_1168_dvd__imp__le,axiom,
! [K2: nat,N: nat] :
( ( dvd_dvd_nat @ K2 @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% dvd_imp_le
thf(fact_1169_generate_Osimps,axiom,
! [A: a,G: partia2175431115845679010xt_a_b,H: set_a] :
( ( member_a @ A @ ( genera2604884881691289736xt_a_b @ G @ H ) )
= ( ( A
= ( one_a_ring_ext_a_b @ G ) )
| ? [H6: a] :
( ( A = H6 )
& ( member_a @ H6 @ H ) )
| ? [H6: a] :
( ( A
= ( m_inv_a_ring_ext_a_b @ G @ H6 ) )
& ( member_a @ H6 @ H ) )
| ? [H12: a,H22: a] :
( ( A
= ( mult_a_ring_ext_a_b @ G @ H12 @ H22 ) )
& ( member_a @ H12 @ ( genera2604884881691289736xt_a_b @ G @ H ) )
& ( member_a @ H22 @ ( genera2604884881691289736xt_a_b @ G @ H ) ) ) ) ) ).
% generate.simps
thf(fact_1170_generate_Osimps,axiom,
! [A: a,G: partia8223610829204095565t_unit,H: set_a] :
( ( member_a @ A @ ( genera4054545294875501499t_unit @ G @ H ) )
= ( ( A
= ( one_a_Product_unit @ G ) )
| ? [H6: a] :
( ( A = H6 )
& ( member_a @ H6 @ H ) )
| ? [H6: a] :
( ( A
= ( m_inv_a_Product_unit @ G @ H6 ) )
& ( member_a @ H6 @ H ) )
| ? [H12: a,H22: a] :
( ( A
= ( mult_a_Product_unit @ G @ H12 @ H22 ) )
& ( member_a @ H12 @ ( genera4054545294875501499t_unit @ G @ H ) )
& ( member_a @ H22 @ ( genera4054545294875501499t_unit @ G @ H ) ) ) ) ) ).
% generate.simps
thf(fact_1171_generate_Ocases,axiom,
! [A: a,G: partia2175431115845679010xt_a_b,H: set_a] :
( ( member_a @ A @ ( genera2604884881691289736xt_a_b @ G @ H ) )
=> ( ( A
!= ( one_a_ring_ext_a_b @ G ) )
=> ( ~ ( member_a @ A @ H )
=> ( ! [H5: a] :
( ( A
= ( m_inv_a_ring_ext_a_b @ G @ H5 ) )
=> ~ ( member_a @ H5 @ H ) )
=> ~ ! [H13: a,H23: a] :
( ( A
= ( mult_a_ring_ext_a_b @ G @ H13 @ H23 ) )
=> ( ( member_a @ H13 @ ( genera2604884881691289736xt_a_b @ G @ H ) )
=> ~ ( member_a @ H23 @ ( genera2604884881691289736xt_a_b @ G @ H ) ) ) ) ) ) ) ) ).
% generate.cases
thf(fact_1172_generate_Ocases,axiom,
! [A: a,G: partia8223610829204095565t_unit,H: set_a] :
( ( member_a @ A @ ( genera4054545294875501499t_unit @ G @ H ) )
=> ( ( A
!= ( one_a_Product_unit @ G ) )
=> ( ~ ( member_a @ A @ H )
=> ( ! [H5: a] :
( ( A
= ( m_inv_a_Product_unit @ G @ H5 ) )
=> ~ ( member_a @ H5 @ H ) )
=> ~ ! [H13: a,H23: a] :
( ( A
= ( mult_a_Product_unit @ G @ H13 @ H23 ) )
=> ( ( member_a @ H13 @ ( genera4054545294875501499t_unit @ G @ H ) )
=> ~ ( member_a @ H23 @ ( genera4054545294875501499t_unit @ G @ H ) ) ) ) ) ) ) ) ).
% generate.cases
thf(fact_1173_carrier__subgroup__generated,axiom,
! [G: partia2175431115845679010xt_a_b,S: set_a] :
( ( partia707051561876973205xt_a_b @ ( genera8625346715478425275xt_a_b @ G @ S ) )
= ( genera2604884881691289736xt_a_b @ G @ ( inf_inf_set_a @ ( partia707051561876973205xt_a_b @ G ) @ S ) ) ) ).
% carrier_subgroup_generated
thf(fact_1174_carrier__subgroup__generated,axiom,
! [G: partia8223610829204095565t_unit,S: set_a] :
( ( partia6735698275553448452t_unit @ ( genera8815471607677139784t_unit @ G @ S ) )
= ( genera4054545294875501499t_unit @ G @ ( inf_inf_set_a @ ( partia6735698275553448452t_unit @ G ) @ S ) ) ) ).
% carrier_subgroup_generated
thf(fact_1175_dvd__power__iff,axiom,
! [X: int,M2: nat,N: nat] :
( ( X != zero_zero_int )
=> ( ( dvd_dvd_int @ ( power_power_int @ X @ M2 ) @ ( power_power_int @ X @ N ) )
= ( ( dvd_dvd_int @ X @ one_one_int )
| ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_1176_dvd__power__iff,axiom,
! [X: nat,M2: nat,N: nat] :
( ( X != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M2 ) @ ( power_power_nat @ X @ N ) )
= ( ( dvd_dvd_nat @ X @ one_one_nat )
| ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_1177_power__dvd__imp__le,axiom,
! [I2: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I2 @ M2 ) @ ( power_power_nat @ I2 @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I2 )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_1178_mult__of_Oord__pow,axiom,
! [X: a,K2: nat] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( dvd_dvd_nat @ K2 @ ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ X ) )
=> ( ( K2 != zero_zero_nat )
=> ( ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ K2 ) )
= ( divide_divide_nat @ ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ X ) @ K2 ) ) ) ) ) ).
% mult_of.ord_pow
thf(fact_1179_mult__of_Ogenerate__pow__card,axiom,
! [A: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ A )
= ( finite_card_a @ ( genera4054545294875501499t_unit @ ( ring_mult_of_a_b @ r ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ).
% mult_of.generate_pow_card
thf(fact_1180_mult__of_Ocard__rcosets__equal,axiom,
! [R2: set_a,H: set_a] :
( ( member_set_a @ R2 @ ( rCOSET407642731378740692t_unit @ ( ring_mult_of_a_b @ r ) @ H ) )
=> ( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( finite_card_a @ H )
= ( finite_card_a @ R2 ) ) ) ) ).
% mult_of.card_rcosets_equal
thf(fact_1181_units__power__order__eq__one,axiom,
! [A: a] :
( ( finite_finite_a @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ A @ ( finite_card_a @ ( units_a_ring_ext_a_b @ r ) ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% units_power_order_eq_one
thf(fact_1182_mult__of_Opower__order__eq__one,axiom,
! [A: a] :
( ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ A @ ( finite_card_a @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.power_order_eq_one
thf(fact_1183_div__diff,axiom,
! [C: int,A: int,B2: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( ( divide_divide_int @ ( minus_minus_int @ A @ B2 ) @ C )
= ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).
% div_diff
thf(fact_1184_card_Oempty,axiom,
( ( finite_card_set_a @ bot_bot_set_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_1185_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_1186_card_Oinfinite,axiom,
! [A2: set_set_a] :
( ~ ( finite_finite_set_a @ A2 )
=> ( ( finite_card_set_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1187_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1188_card__0__eq,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( finite_card_set_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_set_a ) ) ) ).
% card_0_eq
thf(fact_1189_card__0__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_1190_card__insert__disjoint,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ~ ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( insert_set_a @ X @ A2 ) )
= ( suc @ ( finite_card_set_a @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1191_card__insert__disjoint,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( suc @ ( finite_card_a @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1192_card__Diff__insert,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ A @ B )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1193_card__Diff__insert,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ~ ( member_a @ A @ B )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1194_card__eq__0__iff,axiom,
! [A2: set_set_a] :
( ( ( finite_card_set_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_set_a )
| ~ ( finite_finite_set_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_1195_card__eq__0__iff,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_a )
| ~ ( finite_finite_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_1196_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_set_a,C3: nat] :
( ! [G5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ G5 @ F )
=> ( ( finite_finite_set_a @ G5 )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ G5 ) @ C3 ) ) )
=> ( ( finite_finite_set_a @ F )
& ( ord_less_eq_nat @ ( finite_card_set_a @ F ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1197_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_a,C3: nat] :
( ! [G5: set_a] :
( ( ord_less_eq_set_a @ G5 @ F )
=> ( ( finite_finite_a @ G5 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G5 ) @ C3 ) ) )
=> ( ( finite_finite_a @ F )
& ( ord_less_eq_nat @ ( finite_card_a @ F ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1198_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ S ) )
=> ~ ! [T3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ T3 @ S )
=> ( ( ( finite_card_set_a @ T3 )
= N )
=> ~ ( finite_finite_set_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1199_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S )
=> ( ( ( finite_card_a @ T3 )
= N )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1200_exists__subset__between,axiom,
! [A2: set_set_a,N: nat,C3: set_set_a] :
( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ C3 ) )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ C3 )
=> ( ( finite_finite_set_a @ C3 )
=> ? [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ C3 )
& ( ( finite_card_set_a @ B3 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1201_exists__subset__between,axiom,
! [A2: set_a,N: nat,C3: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C3 ) )
=> ( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( finite_finite_a @ C3 )
=> ? [B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
& ( ord_less_eq_set_a @ B3 @ C3 )
& ( ( finite_card_a @ B3 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1202_card__seteq,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ B ) @ ( finite_card_set_a @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_1203_card__seteq,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_1204_card__mono,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B ) ) ) ) ).
% card_mono
thf(fact_1205_card__mono,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ).
% card_mono
thf(fact_1206_infinite__arbitrarily__large,axiom,
! [A2: set_set_a,N: nat] :
( ~ ( finite_finite_set_a @ A2 )
=> ? [B3: set_set_a] :
( ( finite_finite_set_a @ B3 )
& ( ( finite_card_set_a @ B3 )
= N )
& ( ord_le3724670747650509150_set_a @ B3 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1207_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ( finite_card_a @ B3 )
= N )
& ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1208_card__subset__eq,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ( finite_card_set_a @ A2 )
= ( finite_card_set_a @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_1209_card__subset__eq,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_1210_card__insert__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( insert_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_1211_card__insert__le,axiom,
! [A2: set_set_a,X: set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ ( insert_set_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_1212_card__insert__if,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( insert_set_a @ X @ A2 ) )
= ( finite_card_set_a @ A2 ) ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( insert_set_a @ X @ A2 ) )
= ( suc @ ( finite_card_set_a @ A2 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1213_card__insert__if,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( finite_card_a @ A2 ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( suc @ ( finite_card_a @ A2 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1214_card__Suc__eq__finite,axiom,
! [A2: set_set_a,K2: nat] :
( ( ( finite_card_set_a @ A2 )
= ( suc @ K2 ) )
= ( ? [B6: set_a,B5: set_set_a] :
( ( A2
= ( insert_set_a @ B6 @ B5 ) )
& ~ ( member_set_a @ B6 @ B5 )
& ( ( finite_card_set_a @ B5 )
= K2 )
& ( finite_finite_set_a @ B5 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_1215_card__Suc__eq__finite,axiom,
! [A2: set_a,K2: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K2 ) )
= ( ? [B6: a,B5: set_a] :
( ( A2
= ( insert_a @ B6 @ B5 ) )
& ~ ( member_a @ B6 @ B5 )
& ( ( finite_card_a @ B5 )
= K2 )
& ( finite_finite_a @ B5 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_1216_card__ge__0__finite,axiom,
! [A2: set_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ A2 ) )
=> ( finite_finite_set_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_1217_card__ge__0__finite,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ( finite_finite_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_1218_card__1__singletonE,axiom,
! [A2: set_set_a] :
( ( ( finite_card_set_a @ A2 )
= one_one_nat )
=> ~ ! [X3: set_a] :
( A2
!= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ).
% card_1_singletonE
thf(fact_1219_card__1__singletonE,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= one_one_nat )
=> ~ ! [X3: a] :
( A2
!= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% card_1_singletonE
thf(fact_1220_card__less__sym__Diff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B )
=> ( ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B ) )
=> ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1221_card__less__sym__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1222_card__le__sym__Diff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1223_card__le__sym__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1224_psubset__card__mono,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_less_set_set_a @ A2 @ B )
=> ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_1225_psubset__card__mono,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_set_a @ A2 @ B )
=> ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_1226_dvd__div__ge__1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( dvd_dvd_nat @ B2 @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% dvd_div_ge_1
thf(fact_1227_card__1__singleton__iff,axiom,
! [A2: set_set_a] :
( ( ( finite_card_set_a @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: set_a] :
( A2
= ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1228_card__1__singleton__iff,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: a] :
( A2
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1229_card__eq__SucD,axiom,
! [A2: set_set_a,K2: nat] :
( ( ( finite_card_set_a @ A2 )
= ( suc @ K2 ) )
=> ? [B4: set_a,B3: set_set_a] :
( ( A2
= ( insert_set_a @ B4 @ B3 ) )
& ~ ( member_set_a @ B4 @ B3 )
& ( ( finite_card_set_a @ B3 )
= K2 )
& ( ( K2 = zero_zero_nat )
=> ( B3 = bot_bot_set_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_1230_card__eq__SucD,axiom,
! [A2: set_a,K2: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K2 ) )
=> ? [B4: a,B3: set_a] :
( ( A2
= ( insert_a @ B4 @ B3 ) )
& ~ ( member_a @ B4 @ B3 )
& ( ( finite_card_a @ B3 )
= K2 )
& ( ( K2 = zero_zero_nat )
=> ( B3 = bot_bot_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_1231_card__Suc__eq,axiom,
! [A2: set_set_a,K2: nat] :
( ( ( finite_card_set_a @ A2 )
= ( suc @ K2 ) )
= ( ? [B6: set_a,B5: set_set_a] :
( ( A2
= ( insert_set_a @ B6 @ B5 ) )
& ~ ( member_set_a @ B6 @ B5 )
& ( ( finite_card_set_a @ B5 )
= K2 )
& ( ( K2 = zero_zero_nat )
=> ( B5 = bot_bot_set_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1232_card__Suc__eq,axiom,
! [A2: set_a,K2: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K2 ) )
= ( ? [B6: a,B5: set_a] :
( ( A2
= ( insert_a @ B6 @ B5 ) )
& ~ ( member_a @ B6 @ B5 )
& ( ( finite_card_a @ B5 )
= K2 )
& ( ( K2 = zero_zero_nat )
=> ( B5 = bot_bot_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1233_card__gt__0__iff,axiom,
! [A2: set_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ A2 ) )
= ( ( A2 != bot_bot_set_set_a )
& ( finite_finite_set_a @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_1234_card__gt__0__iff,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
= ( ( A2 != bot_bot_set_a )
& ( finite_finite_a @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_1235_card__le__Suc__iff,axiom,
! [N: nat,A2: set_set_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_set_a @ A2 ) )
= ( ? [A3: set_a,B5: set_set_a] :
( ( A2
= ( insert_set_a @ A3 @ B5 ) )
& ~ ( member_set_a @ A3 @ B5 )
& ( ord_less_eq_nat @ N @ ( finite_card_set_a @ B5 ) )
& ( finite_finite_set_a @ B5 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1236_card__le__Suc__iff,axiom,
! [N: nat,A2: set_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_a @ A2 ) )
= ( ? [A3: a,B5: set_a] :
( ( A2
= ( insert_a @ A3 @ B5 ) )
& ~ ( member_a @ A3 @ B5 )
& ( ord_less_eq_nat @ N @ ( finite_card_a @ B5 ) )
& ( finite_finite_a @ B5 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1237_card__Diff__subset,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1238_card__Diff__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1239_card__le__Suc0__iff__eq,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ! [Y5: set_a] :
( ( member_set_a @ Y5 @ A2 )
=> ( X2 = Y5 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1240_card__le__Suc0__iff__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y5: a] :
( ( member_a @ Y5 @ A2 )
=> ( X2 = Y5 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1241_card__Diff1__le,axiom,
! [A2: set_set_a,X: set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A2 ) ) ).
% card_Diff1_le
thf(fact_1242_card__Diff1__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ).
% card_Diff1_le
thf(fact_1243_diff__card__le__card__Diff,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1244_diff__card__le__card__Diff,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1245_card__psubset,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B ) )
=> ( ord_less_set_set_a @ A2 @ B ) ) ) ) ).
% card_psubset
thf(fact_1246_card__psubset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B ) )
=> ( ord_less_set_a @ A2 @ B ) ) ) ) ).
% card_psubset
thf(fact_1247_card__Diff__subset__Int,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_1248_card__Diff__subset__Int,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ ( inf_inf_set_a @ A2 @ B ) )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_1249_power__diff,axiom,
! [A: int,N: nat,M2: nat] :
( ( A != zero_zero_int )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N ) )
= ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_1250_power__diff,axiom,
! [A: nat,N: nat,M2: nat] :
( ( A != zero_zero_nat )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N ) )
= ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_1251_subgroup_Ofinite__imp__card__positive,axiom,
! [H: set_a,G: partia2175431115845679010xt_a_b] :
( ( subgro1816942748394427906xt_a_b @ H @ G )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ H ) ) ) ) ).
% subgroup.finite_imp_card_positive
thf(fact_1252_subgroup_Ofinite__imp__card__positive,axiom,
! [H: set_a,G: partia8223610829204095565t_unit] :
( ( subgro3222307229058429633t_unit @ H @ G )
=> ( ( finite_finite_a @ ( partia6735698275553448452t_unit @ G ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ H ) ) ) ) ).
% subgroup.finite_imp_card_positive
thf(fact_1253_card__Suc__Diff1,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( suc @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) )
= ( finite_card_set_a @ A2 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_1254_card__Suc__Diff1,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_1255_card_Oinsert__remove,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_card_set_a @ ( insert_set_a @ X @ A2 ) )
= ( suc @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_1256_card_Oinsert__remove,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ ( insert_a @ X @ A2 ) )
= ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_1257_card_Oremove,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ A2 )
= ( suc @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ) ) ) ).
% card.remove
thf(fact_1258_card_Oremove,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ A2 )
= ( suc @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ) ) ).
% card.remove
thf(fact_1259_card__Diff1__less,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1260_card__Diff1__less,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1261_card__Diff2__less,axiom,
! [A2: set_set_a,X: set_a,Y2: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( member_set_a @ Y2 @ A2 )
=> ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ ( insert_set_a @ Y2 @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1262_card__Diff2__less,axiom,
! [A2: set_a,X: a,Y2: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y2 @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( insert_a @ Y2 @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1263_card__Diff1__less__iff,axiom,
! [A2: set_set_a,X: set_a] :
( ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A2 ) )
= ( ( finite_finite_set_a @ A2 )
& ( member_set_a @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1264_card__Diff1__less__iff,axiom,
! [A2: set_a,X: a] :
( ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) )
= ( ( finite_finite_a @ A2 )
& ( member_a @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1265_card__Diff__singleton__if,axiom,
! [X: set_a,A2: set_set_a] :
( ( ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
= ( finite_card_set_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1266_card__Diff__singleton__if,axiom,
! [X: a,A2: set_a] :
( ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1267_card__Diff__singleton,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1268_a__card__cosets__equal,axiom,
! [C: set_a,H: set_a] :
( ( member_set_a @ C @ ( a_RCOSETS_a_b @ r @ H ) )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_card_a @ C )
= ( finite_card_a @ H ) ) ) ) ) ).
% a_card_cosets_equal
thf(fact_1269_mult__of_Olagrange__finite,axiom,
! [H: set_a] :
( ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( ( times_times_nat @ ( finite_card_set_a @ ( rCOSET407642731378740692t_unit @ ( ring_mult_of_a_b @ r ) @ H ) ) @ ( finite_card_a @ H ) )
= ( order_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.lagrange_finite
thf(fact_1270_nat__pow__pow,axiom,
! [X: a,N: nat,M2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ M2 )
= ( pow_a_1026414303147256608_b_nat @ r @ X @ ( times_times_nat @ N @ M2 ) ) ) ) ).
% nat_pow_pow
thf(fact_1271_telescopic__base,axiom,
! [K: set_a,F: set_a,N: nat,M2: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( subfield_a_b @ F @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ F )
=> ( ( embedd2795209813406577254on_a_b @ r @ M2 @ F @ E )
=> ( embedd2795209813406577254on_a_b @ r @ ( times_times_nat @ N @ M2 ) @ K @ E ) ) ) ) ) ).
% telescopic_base
thf(fact_1272_mult__of_Onat__pow__pow,axiom,
! [X: a,N: nat,M2: nat] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ N ) @ M2 )
= ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X @ ( times_times_nat @ N @ M2 ) ) ) ) ).
% mult_of.nat_pow_pow
thf(fact_1273_mult__of_Oabelian__ord__mul__divides,axiom,
! [X: a,Y2: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( dvd_dvd_nat @ ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ X @ Y2 ) ) @ ( times_times_nat @ ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ X ) @ ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ Y2 ) ) ) ) ) ).
% mult_of.abelian_ord_mul_divides
thf(fact_1274_mult__of_Oord__mul__divides,axiom,
! [X: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( dvd_dvd_nat @ ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ X @ Y2 ) ) @ ( times_times_nat @ ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ X ) @ ( multip1500854282228996350t_unit @ ( ring_mult_of_a_b @ r ) @ Y2 ) ) ) ) ) ) ).
% mult_of.ord_mul_divides
thf(fact_1275_mult__of_Olagrange,axiom,
! [H: set_a] :
( ( subgro3222307229058429633t_unit @ H @ ( ring_mult_of_a_b @ r ) )
=> ( ( times_times_nat @ ( finite_card_set_a @ ( rCOSET407642731378740692t_unit @ ( ring_mult_of_a_b @ r ) @ H ) ) @ ( finite_card_a @ H ) )
= ( order_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.lagrange
thf(fact_1276_one__le__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1277_mult__le__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
% Conjectures (1)
thf(conj_0,conjecture,
member_a @ a2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
%------------------------------------------------------------------------------