TPTP Problem File: SLH0901^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Interpolation_Polynomials_HOL_Algebra/0001_Lagrange_Interpolation/prob_00215_008214__17277802_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1465 ( 274 unt; 189 typ; 0 def)
% Number of atoms : 4808 (1194 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 19253 ( 295 ~; 29 |; 271 &;15634 @)
% ( 0 <=>;3024 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 9 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 763 ( 763 >; 0 *; 0 +; 0 <<)
% Number of symbols : 172 ( 171 usr; 9 con; 0-4 aty)
% Number of variables : 3920 ( 99 ^;3589 !; 232 ?;3920 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:37:44.562
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
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thf(ty_n_tf__a,type,
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% Explicit typings (171)
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thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Set__Oset_Itf__a_J,type,
cons_set_a: set_a > list_set_a > list_set_a ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
nil_list_list_a: list_list_list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
list_a3802133873445908231list_a: ( list_a > list_a > $o ) > list_list_a > list_list_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_Itf__a_J_001tf__a,type,
list_all2_list_a_a: ( list_a > a > $o ) > list_list_a > list_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__Nat__Onat_001t__Nat__Onat,type,
list_all2_nat_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
list_all2_nat_set_a: ( nat > set_a > $o ) > list_nat > list_set_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__Nat__Onat_001tf__a,type,
list_all2_nat_a: ( nat > a > $o ) > list_nat > list_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
list_all2_set_a_nat: ( set_a > nat > $o ) > list_set_a > list_nat > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
list_a5961261016436360967_set_a: ( set_a > set_a > $o ) > list_set_a > list_set_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__Set__Oset_Itf__a_J_001tf__a,type,
list_all2_set_a_a: ( set_a > a > $o ) > list_set_a > list_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001tf__a_001t__List__Olist_Itf__a_J,type,
list_all2_a_list_a: ( a > list_a > $o ) > list_a > list_list_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001tf__a_001t__Nat__Onat,type,
list_all2_a_nat: ( a > nat > $o ) > list_a > list_nat > $o ).
thf(sy_c_List_Olist_Olist__all2_001tf__a_001t__Set__Oset_Itf__a_J,type,
list_all2_a_set_a: ( a > set_a > $o ) > list_a > list_set_a > $o ).
thf(sy_c_List_Olist_Olist__all2_001tf__a_001tf__a,type,
list_all2_a_a: ( a > a > $o ) > list_a > list_a > $o ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_Itf__a_J,type,
set_set_a2: list_set_a > set_set_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_Multiplicative__Group_Omult__of_001tf__a_001tf__b,type,
multip3210463924028840165of_a_b: partia2175431115845679010xt_a_b > partia8223610829204095565t_unit ).
thf(sy_c_Multiset_Omset_001tf__a,type,
mset_a: list_a > multiset_a ).
thf(sy_c_Multiset_Osubseteq__mset_001t__Set__Oset_Itf__a_J,type,
subseteq_mset_set_a: multiset_set_a > multiset_set_a > $o ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
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__List__Olist_Itf__a_J_J,type,
ord_less_set_list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_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_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_M_062_It__List__Olist_It__List__Olist_Itf__a_J_J_M_Eo_J_J,type,
ord_le3776173323681337614st_a_o: ( list_list_a > list_list_a > $o ) > ( list_list_a > list_list_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_Itf__a_J_M_062_It__List__Olist_Itf__a_J_M_Eo_J_J,type,
ord_le5542992221119063950st_a_o: ( list_a > list_a > $o ) > ( list_a > list_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
ord_less_eq_a_a_o: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
ord_le7905258569527593284_set_a: multiset_set_a > multiset_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Oexp__base_001tf__a_001tf__b,type,
polyno2922411391617481336se_a_b: partia2175431115845679010xt_a_b > a > nat > list_a ).
thf(sy_c_Polynomial__Divisibility_Oring_Ois__root_001tf__a_001tf__b,type,
polyno4133073214067823460ot_a_b: partia2175431115845679010xt_a_b > list_a > a > $o ).
thf(sy_c_Polynomials_Oring_Oconst__term_001tf__a_001tf__b,type,
const_term_a_b: partia2175431115845679010xt_a_b > list_a > a ).
thf(sy_c_Polynomials_Oring_Oeval_001tf__a_001tf__b,type,
eval_a_b: partia2175431115845679010xt_a_b > list_a > a > a ).
thf(sy_c_Polynomials_Oring_Omonom_001tf__a_001tf__b,type,
monom_a_b: partia2175431115845679010xt_a_b > a > nat > list_a ).
thf(sy_c_Polynomials_Oring_Opoly__of__const_001tf__a_001tf__b,type,
poly_of_const_a_b: partia2175431115845679010xt_a_b > a > list_a ).
thf(sy_c_Polynomials_Ovar_001tf__a_001tf__b,type,
var_a_b: partia2175431115845679010xt_a_b > list_a ).
thf(sy_c_Ring_Oabelian__group_001tf__a_001tf__b,type,
abelian_group_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oabelian__monoid_001tf__a_001tf__b,type,
abelian_monoid_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Odomain_001tf__a_001tf__b,type,
domain_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Ofield_001tf__a_001tf__b,type,
field_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_001tf__a_001tf__b,type,
ring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
add_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
semiring_a_b: partia2175431115845679010xt_a_b > $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_001tf__a_001tf__b,type,
ring_mult_of_a_b: partia2175431115845679010xt_a_b > partia8223610829204095565t_unit ).
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_001tf__a_001tf__b,type,
ring_n3639167112692572309ng_a_b: partia2175431115845679010xt_a_b > $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_001tf__a_001tf__b,type,
ring_r999134135267193926le_a_b: partia2175431115845679010xt_a_b > 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_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
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_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_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_001_062_It__List__Olist_Itf__a_J_M_062_It__List__Olist_Itf__a_J_M_Eo_J_J,type,
member5128058179956651799st_a_o: ( list_a > list_a > $o ) > set_list_a_list_a_o > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
member_a_a_o: ( a > a > $o ) > set_a_a_o > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
member2747690772047059533_set_a: multiset_set_a > set_multiset_set_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_S,type,
s: set_a ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1275)
thf(fact_0_factorial__domain__axioms,axiom,
ring_f5272581269873410839in_a_b @ r ).
% factorial_domain_axioms
thf(fact_1_local_Ofield__axioms,axiom,
field_a_b @ r ).
% local.field_axioms
thf(fact_2_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_3_assms_I1_J,axiom,
finite_finite_a @ s ).
% assms(1)
thf(fact_4_ring_Olagrange__basis__polynomial_Ocong,axiom,
lagran2649660974587678107al_a_b = lagran2649660974587678107al_a_b ).
% ring.lagrange_basis_polynomial.cong
thf(fact_5_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_6_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_7_assms_I3_J,axiom,
member_a @ x @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ s ) ).
% assms(3)
thf(fact_8_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_9_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_10_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_11_assms_I2_J,axiom,
ord_less_eq_set_a @ s @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(2)
thf(fact_12_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_13_finprod__one__eqI,axiom,
! [A: set_nat,F: nat > a] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( F @ X )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_14_finprod__one__eqI,axiom,
! [A: set_set_a,F: set_a > a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ( F @ X )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro934595834566309783_set_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_15_finprod__one__eqI,axiom,
! [A: set_a,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( F @ X )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_16_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_17_is__abelian__group,axiom,
abelian_group_a_b @ r ).
% is_abelian_group
thf(fact_18_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_19_carrier__is__subfield,axiom,
subfield_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subfield
thf(fact_20_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_21_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_22_finprod__zero__iff,axiom,
! [A: set_nat,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ A )
=> ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro1280035270526425175_b_nat @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_23_finprod__zero__iff,axiom,
! [A: set_set_a,F: set_a > a] :
( ( finite_finite_set_a @ A )
=> ( ! [A2: set_a] :
( ( member_set_a @ A2 @ A )
=> ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro934595834566309783_set_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_24_finprod__zero__iff,axiom,
! [A: set_list_a,F: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ! [A2: list_a] :
( ( member_list_a @ A2 @ A )
=> ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro6052973074229812797list_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_25_finprod__zero__iff,axiom,
! [A: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ! [A2: a] :
( ( member_a @ A2 @ A )
=> ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_26_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_27_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_28_finprod__infinite,axiom,
! [A: set_nat,F: nat > a] :
( ~ ( finite_finite_nat @ A )
=> ( ( finpro1280035270526425175_b_nat @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_29_finprod__infinite,axiom,
! [A: set_set_a,F: set_a > a] :
( ~ ( finite_finite_set_a @ A )
=> ( ( finpro934595834566309783_set_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_30_finprod__infinite,axiom,
! [A: set_list_a,F: list_a > a] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finpro6052973074229812797list_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_31_finprod__infinite,axiom,
! [A: set_a,F: a > a] :
( ~ ( finite_finite_a @ A )
=> ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_32_domain_Ofinprod__zero__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_nat,F: nat > a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_nat @ A )
=> ( ! [A2: nat] :
( ( member_nat @ A2 @ A )
=> ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
=> ( ( ( finpro1280035270526425175_b_nat @ R @ F @ A )
= ( zero_a_b @ R ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_33_domain_Ofinprod__zero__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_set_a,F: set_a > a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_set_a @ A )
=> ( ! [A2: set_a] :
( ( member_set_a @ A2 @ A )
=> ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
=> ( ( ( finpro934595834566309783_set_a @ R @ F @ A )
= ( zero_a_b @ R ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_34_domain_Ofinprod__zero__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_list_a,F: list_a > a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_list_a @ A )
=> ( ! [A2: list_a] :
( ( member_list_a @ A2 @ A )
=> ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
=> ( ( ( finpro6052973074229812797list_a @ R @ F @ A )
= ( zero_a_b @ R ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_35_domain_Ofinprod__zero__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: set_a,F: a > a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_a @ A )
=> ( ! [A2: a] :
( ( member_a @ A2 @ A )
=> ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
=> ( ( ( finpro205304725090349623_a_b_a @ R @ F @ A )
= ( zero_a_b @ R ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ R ) ) ) ) ) ) ) ) ).
% domain.finprod_zero_iff
thf(fact_36_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_37_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_38_ring__irreducibleE_I1_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( R2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_39_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_40_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_41_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_42_finite__Diff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_43_finite__Diff,axiom,
! [A: set_list_a,B: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_44_finite__Diff,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) ) ) ).
% finite_Diff
thf(fact_45_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_46_finite__Diff2,axiom,
! [B: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( finite_finite_set_a @ A ) ) ) ).
% finite_Diff2
thf(fact_47_finite__Diff2,axiom,
! [B: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B ) )
= ( finite_finite_list_a @ A ) ) ) ).
% finite_Diff2
thf(fact_48_finite__Diff2,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A @ B ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_Diff2
thf(fact_49_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A2
!= ( zero_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A2 @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_50_subalgebra__in__carrier,axiom,
! [K: set_a,V: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_51_carrier__is__subalgebra,axiom,
! [K: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_52_a__l__coset__subset__G,axiom,
! [H: set_a,X4: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X4 @ H ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_53_ring_Osubring__props_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.subring_props(1)
thf(fact_54_m__assoc,axiom,
! [X4: a,Y: a,Z: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X4 @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X4 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_55_m__comm,axiom,
! [X4: a,Y: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X4 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X4 ) ) ) ) ).
% m_comm
thf(fact_56_m__lcomm,axiom,
! [X4: a,Y: a,Z: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X4 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X4 @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_57_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_58_local_Ointegral,axiom,
! [A3: a,B2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A3 @ B2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A3
= ( zero_a_b @ r ) )
| ( B2
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_59_integral__iff,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A3 @ B2 )
= ( zero_a_b @ r ) )
= ( ( A3
= ( zero_a_b @ r ) )
| ( B2
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_60_m__lcancel,axiom,
! [A3: a,B2: a,C: a] :
( ( A3
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A3 @ ( 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 @ A3 @ B2 )
= ( mult_a_ring_ext_a_b @ r @ A3 @ C ) )
= ( B2 = C ) ) ) ) ) ) ).
% m_lcancel
thf(fact_61_m__rcancel,axiom,
! [A3: a,B2: a,C: a] :
( ( A3
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A3 @ ( 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 @ A3 )
= ( mult_a_ring_ext_a_b @ r @ C @ A3 ) )
= ( B2 = C ) ) ) ) ) ) ).
% m_rcancel
thf(fact_62_mem__Collect__eq,axiom,
! [A3: nat,P2: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A3: set_a,P2: set_a > $o] :
( ( member_set_a @ A3 @ ( collect_set_a @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A3: a,P2: a > $o] :
( ( member_a @ A3 @ ( collect_a @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_65_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_68_Collect__cong,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collect_a @ P2 )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_69_inv__unique,axiom,
! [Y: a,X4: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y @ X4 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X4 @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% inv_unique
thf(fact_70_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X )
= X ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_71_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_72_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_73_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_74_m__closed,axiom,
! [X4: a,Y: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X4 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_75_l__null,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X4 )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_76_r__null,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X4 @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_77_l__one,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X4 )
= X4 ) ) ).
% l_one
thf(fact_78_r__one,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X4 @ ( one_a_ring_ext_a_b @ r ) )
= X4 ) ) ).
% r_one
thf(fact_79_subalgebra_Osmult__closed,axiom,
! [K: set_a,V: set_a,R: partia2175431115845679010xt_a_b,K2: a,V2: a] :
( ( embedd9027525575939734154ra_a_b @ K @ V @ R )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ V2 @ V )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K2 @ V2 ) @ V ) ) ) ) ).
% subalgebra.smult_closed
thf(fact_80_ring_Osubring__props_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,H1: a,H2: a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% ring.subring_props(6)
thf(fact_81_ring_Osubalgebra__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,V: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V @ R )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_82_ring_Ocarrier__is__subalgebra,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ R ) )
=> ( embedd9027525575939734154ra_a_b @ K @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_83_finite__has__minimal2,axiom,
! [A: set_a_a_o,A3: a > a > $o] :
( ( finite_finite_a_a_o @ A )
=> ( ( member_a_a_o @ A3 @ A )
=> ? [X: a > a > $o] :
( ( member_a_a_o @ X @ A )
& ( ord_less_eq_a_a_o @ X @ A3 )
& ! [Xa: a > a > $o] :
( ( member_a_a_o @ Xa @ A )
=> ( ( ord_less_eq_a_a_o @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_84_finite__has__minimal2,axiom,
! [A: set_list_a_list_a_o,A3: list_a > list_a > $o] :
( ( finite407736375021937175st_a_o @ A )
=> ( ( member5128058179956651799st_a_o @ A3 @ A )
=> ? [X: list_a > list_a > $o] :
( ( member5128058179956651799st_a_o @ X @ A )
& ( ord_le5542992221119063950st_a_o @ X @ A3 )
& ! [Xa: list_a > list_a > $o] :
( ( member5128058179956651799st_a_o @ Xa @ A )
=> ( ( ord_le5542992221119063950st_a_o @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_85_finite__has__minimal2,axiom,
! [A: set_multiset_set_a,A3: multiset_set_a] :
( ( finite2815193924343055693_set_a @ A )
=> ( ( member2747690772047059533_set_a @ A3 @ A )
=> ? [X: multiset_set_a] :
( ( member2747690772047059533_set_a @ X @ A )
& ( ord_le7905258569527593284_set_a @ X @ A3 )
& ! [Xa: multiset_set_a] :
( ( member2747690772047059533_set_a @ Xa @ A )
=> ( ( ord_le7905258569527593284_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_86_finite__has__minimal2,axiom,
! [A: set_set_a,A3: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A3 @ A )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( ord_less_eq_set_a @ X @ A3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_87_finite__has__minimal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ X @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_88_finite__has__maximal2,axiom,
! [A: set_a_a_o,A3: a > a > $o] :
( ( finite_finite_a_a_o @ A )
=> ( ( member_a_a_o @ A3 @ A )
=> ? [X: a > a > $o] :
( ( member_a_a_o @ X @ A )
& ( ord_less_eq_a_a_o @ A3 @ X )
& ! [Xa: a > a > $o] :
( ( member_a_a_o @ Xa @ A )
=> ( ( ord_less_eq_a_a_o @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_89_finite__has__maximal2,axiom,
! [A: set_list_a_list_a_o,A3: list_a > list_a > $o] :
( ( finite407736375021937175st_a_o @ A )
=> ( ( member5128058179956651799st_a_o @ A3 @ A )
=> ? [X: list_a > list_a > $o] :
( ( member5128058179956651799st_a_o @ X @ A )
& ( ord_le5542992221119063950st_a_o @ A3 @ X )
& ! [Xa: list_a > list_a > $o] :
( ( member5128058179956651799st_a_o @ Xa @ A )
=> ( ( ord_le5542992221119063950st_a_o @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_90_finite__has__maximal2,axiom,
! [A: set_multiset_set_a,A3: multiset_set_a] :
( ( finite2815193924343055693_set_a @ A )
=> ( ( member2747690772047059533_set_a @ A3 @ A )
=> ? [X: multiset_set_a] :
( ( member2747690772047059533_set_a @ X @ A )
& ( ord_le7905258569527593284_set_a @ A3 @ X )
& ! [Xa: multiset_set_a] :
( ( member2747690772047059533_set_a @ Xa @ A )
=> ( ( ord_le7905258569527593284_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_91_finite__has__maximal2,axiom,
! [A: set_set_a,A3: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A3 @ A )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( ord_less_eq_set_a @ A3 @ X )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_92_finite__has__maximal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ A3 @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_93_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_94_rev__finite__subset,axiom,
! [B: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( finite_finite_set_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_95_rev__finite__subset,axiom,
! [B: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( finite_finite_list_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_96_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_97_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_98_infinite__super,axiom,
! [S: set_set_a,T: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T ) ) ) ).
% infinite_super
thf(fact_99_infinite__super,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_super
thf(fact_100_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_101_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_102_finite__subset,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( finite_finite_set_a @ B )
=> ( finite_finite_set_a @ A ) ) ) ).
% finite_subset
thf(fact_103_finite__subset,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( finite_finite_list_a @ B )
=> ( finite_finite_list_a @ A ) ) ) ).
% finite_subset
thf(fact_104_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_105_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_106_Diff__infinite__finite,axiom,
! [T: set_set_a,S: set_set_a] :
( ( finite_finite_set_a @ T )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_107_Diff__infinite__finite,axiom,
! [T: set_list_a,S: set_list_a] :
( ( finite_finite_list_a @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_108_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_109_ring_Osubring__props_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( member_a @ ( zero_a_b @ R ) @ K ) ) ) ).
% ring.subring_props(2)
thf(fact_110_ring_Osubring__props_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ K ) ) ) ).
% ring.subring_props(3)
thf(fact_111_line__extension__smult__closed,axiom,
! [K: set_a,E: set_a,A3: a,K2: a,U: a] :
( ( subfield_a_b @ K @ r )
=> ( ! [K3: a,V3: a] :
( ( member_a @ K3 @ K )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ V3 ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A3 @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ U ) @ ( embedd971793762689825387on_a_b @ r @ K @ A3 @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_112_monoid__cancelI,axiom,
( ! [A2: a,B3: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A2 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B3 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B3 ) ) ) ) )
=> ( ! [A2: a,B3: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A2 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B3 @ C2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B3 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_113_abelian__group_Oa__lcos__mult__one,axiom,
! [G: partia2175431115845679010xt_a_b,M: set_a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_l_coset_a_b @ G @ ( zero_a_b @ G ) @ M )
= M ) ) ) ).
% abelian_group.a_lcos_mult_one
thf(fact_114_subalbegra__incl__imp__finite__dimension,axiom,
! [K: set_a,E: set_a,V: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ( ord_less_eq_set_a @ V @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K @ V ) ) ) ) ) ).
% subalbegra_incl_imp_finite_dimension
thf(fact_115_principal__domain_Oprimeness__condition,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ P )
= ( ring_ring_prime_a_b @ R @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_116_domain_Oring__primeE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P )
=> ( prime_a_ring_ext_a_b @ R @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_117_ring_Oring__primeI,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( ring_a_b @ R )
=> ( ( P
!= ( zero_a_b @ R ) )
=> ( ( prime_a_ring_ext_a_b @ R @ P )
=> ( ring_ring_prime_a_b @ R @ P ) ) ) ) ).
% ring.ring_primeI
thf(fact_118_domain_Oring__primeE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P )
=> ( P
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_119_abelian__group_Oa__l__coset__subset__G,axiom,
! [G: partia2175431115845679010xt_a_b,H: set_a,X4: a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ G @ X4 @ H ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_group.a_l_coset_subset_G
thf(fact_120_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X4 )
= X4 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_121_semiring_Or__null,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X4 @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.r_null
thf(fact_122_telescopic__base__dim_I1_J,axiom,
! [K: set_a,F2: set_a,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( subfield_a_b @ F2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ F2 )
=> ( ( embedd8708762675212832759on_a_b @ r @ F2 @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K @ E ) ) ) ) ) ).
% telescopic_base_dim(1)
thf(fact_123_line__extension__in__carrier,axiom,
! [K: set_a,A3: a,E: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K @ A3 @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_124_finite__dimension__imp__subalgebra,axiom,
! [K: set_a,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ( embedd9027525575939734154ra_a_b @ K @ E @ r ) ) ) ).
% finite_dimension_imp_subalgebra
thf(fact_125_ring_Ofinite__dimension_Ocong,axiom,
embedd8708762675212832759on_a_b = embedd8708762675212832759on_a_b ).
% ring.finite_dimension.cong
thf(fact_126_ring_Oline__extension_Ocong,axiom,
embedd971793762689825387on_a_b = embedd971793762689825387on_a_b ).
% ring.line_extension.cong
thf(fact_127_ring_Otelescopic__base__dim_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,F2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( subfield_a_b @ F2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K @ F2 )
=> ( ( embedd8708762675212832759on_a_b @ R @ F2 @ E )
=> ( embedd8708762675212832759on_a_b @ R @ K @ E ) ) ) ) ) ) ).
% ring.telescopic_base_dim(1)
thf(fact_128_ring_Oline__extension__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,A3: a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ R @ K @ A3 @ E ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_129_ring_Ofinite__dimension__imp__subalgebra,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K @ E )
=> ( embedd9027525575939734154ra_a_b @ K @ E @ R ) ) ) ) ).
% ring.finite_dimension_imp_subalgebra
thf(fact_130_ring_Osubalbegra__incl__imp__finite__dimension,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,E: set_a,V: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V @ R )
=> ( ( ord_less_eq_set_a @ V @ E )
=> ( embedd8708762675212832759on_a_b @ R @ K @ V ) ) ) ) ) ) ).
% ring.subalbegra_incl_imp_finite_dimension
thf(fact_131_ring_Oline__extension__smult__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,E: set_a,A3: a,K2: a,U: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ! [K3: a,V3: a] :
( ( member_a @ K3 @ K )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K3 @ V3 ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ R @ K @ A3 @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K2 @ U ) @ ( embedd971793762689825387on_a_b @ R @ K @ A3 @ E ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_132_field_Ois__ring,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ring_a_b @ R ) ) ).
% field.is_ring
thf(fact_133_ring_Ois__abelian__group,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( abelian_group_a_b @ R ) ) ).
% ring.is_abelian_group
thf(fact_134_field_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% field.axioms(1)
thf(fact_135_abelian__group_Oaxioms_I1_J,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ G )
=> ( abelian_monoid_a_b @ G ) ) ).
% abelian_group.axioms(1)
thf(fact_136_noetherian__ring_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_n3639167112692572309ng_a_b @ R )
=> ( ring_a_b @ R ) ) ).
% noetherian_ring.axioms(1)
thf(fact_137_semiring_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( abelian_monoid_a_b @ R ) ) ).
% semiring.axioms(1)
thf(fact_138_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_139_noetherian__domain_Oaxioms_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_n4045954140777738665in_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% noetherian_domain.axioms(2)
thf(fact_140_factorial__domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_f5272581269873410839in_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% factorial_domain.axioms(1)
thf(fact_141_noetherian__domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_n4045954140777738665in_a_b @ R )
=> ( ring_n3639167112692572309ng_a_b @ R ) ) ).
% noetherian_domain.axioms(1)
thf(fact_142_ring_Oring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.ring_simprules(2)
thf(fact_143_ring_Oring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X4 @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R @ X4 @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_144_ring_Oring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X4 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_145_ring_Oring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.ring_simprules(6)
thf(fact_146_domain_Oone__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% domain.one_not_zero
thf(fact_147_domain_Ozero__not__one,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( zero_a_b @ R )
!= ( one_a_ring_ext_a_b @ R ) ) ) ).
% domain.zero_not_one
thf(fact_148_abelian__groupE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% abelian_groupE(2)
thf(fact_149_abelian__monoidE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_150_abelian__monoid_Ozero__closed,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ G )
=> ( member_a @ ( zero_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_151_Ring_Oone__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_152_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_153_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X4 @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R @ X4 @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_154_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X4 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_155_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_156_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( prime_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_157_ring__prime__def,axiom,
( ring_ring_prime_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,A4: a] :
( ( A4
!= ( zero_a_b @ R3 ) )
& ( prime_a_ring_ext_a_b @ R3 @ A4 ) ) ) ) ).
% ring_prime_def
thf(fact_158_noetherian__domain_Ointro,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_n3639167112692572309ng_a_b @ R )
=> ( ( domain_a_b @ R )
=> ( ring_n4045954140777738665in_a_b @ R ) ) ) ).
% noetherian_domain.intro
thf(fact_159_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_160_ring_Oring__simprules_I25_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X4 @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_161_ring_Oring__simprules_I24_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X4 )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_162_domain_Ointegral,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( domain_a_b @ R )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A3 @ B2 )
= ( zero_a_b @ R ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A3
= ( zero_a_b @ R ) )
| ( B2
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_163_domain_Om__lcancel,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a,B2: a,C: a] :
( ( domain_a_b @ R )
=> ( ( A3
!= ( zero_a_b @ R ) )
=> ( ( member_a @ A3 @ ( 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 @ A3 @ B2 )
= ( mult_a_ring_ext_a_b @ R @ A3 @ C ) )
= ( B2 = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_164_domain_Om__rcancel,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a,B2: a,C: a] :
( ( domain_a_b @ R )
=> ( ( A3
!= ( zero_a_b @ R ) )
=> ( ( member_a @ A3 @ ( 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 @ A3 )
= ( mult_a_ring_ext_a_b @ R @ C @ A3 ) )
= ( B2 = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_165_domain_Ointegral__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A3 @ B2 )
= ( zero_a_b @ R ) )
= ( ( A3
= ( zero_a_b @ R ) )
| ( B2
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_166_ring_Oring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X4 )
= X4 ) ) ) ).
% ring.ring_simprules(12)
thf(fact_167_Ring_Ointegral,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( field_a_b @ R )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A3 @ B2 )
= ( zero_a_b @ R ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A3
= ( zero_a_b @ R ) )
| ( B2
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_168_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ( R2
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_169_semiring_Ol__null,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X4 )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.l_null
thf(fact_170_eval__var,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X4 )
= X4 ) ) ).
% eval_var
thf(fact_171_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_172_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_173_a__lcos__m__assoc,axiom,
! [M: set_a,G2: a,H3: a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G2 @ ( a_l_coset_a_b @ r @ H3 @ M ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G2 @ H3 ) @ M ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_174_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_175_eval__in__carrier,axiom,
! [P: list_a,X4: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X4 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier
thf(fact_176_eval__poly__of__const,axiom,
! [X4: a,Y: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_of_const_a_b @ r @ X4 ) @ Y )
= X4 ) ) ).
% eval_poly_of_const
thf(fact_177_ring__irreducibleE_I2_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( irredu6211895646901577903xt_a_b @ r @ R2 ) ) ) ).
% ring_irreducibleE(2)
thf(fact_178_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_179_add_Ol__cancel,axiom,
! [C: a,A3: a,B2: a] :
( ( ( add_a_b @ r @ C @ A3 )
= ( add_a_b @ r @ C @ B2 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B2 ) ) ) ) ) ).
% add.l_cancel
thf(fact_180_add_Or__cancel,axiom,
! [A3: a,C: a,B2: a] :
( ( ( add_a_b @ r @ A3 @ C )
= ( add_a_b @ r @ B2 @ C ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B2 ) ) ) ) ) ).
% add.r_cancel
thf(fact_181_a__assoc,axiom,
! [X4: a,Y: a,Z: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X4 @ Y ) @ Z )
= ( add_a_b @ r @ X4 @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_182_a__comm,axiom,
! [X4: a,Y: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X4 @ Y )
= ( add_a_b @ r @ Y @ X4 ) ) ) ) ).
% a_comm
thf(fact_183_a__lcomm,axiom,
! [X4: a,Y: a,Z: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X4 @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X4 @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_184_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_185_cgenideal__self,axiom,
! [I2: a] :
( ( member_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I2 @ ( cgenid547466209912283029xt_a_b @ r @ I2 ) ) ) ).
% cgenideal_self
thf(fact_186_add_Oinv__comm,axiom,
! [X4: a,Y: a] :
( ( ( add_a_b @ r @ X4 @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X4 )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_187_add_Ol__inv__ex,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X4 )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_188_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X )
= X ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_189_add_Or__inv__ex,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X4 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_190_local_Ominus__unique,axiom,
! [Y: a,X4: a,Y2: a] :
( ( ( add_a_b @ r @ Y @ X4 )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X4 @ Y2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_191_l__distr,axiom,
! [X4: a,Y: a,Z: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X4 @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X4 @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_192_r__distr,axiom,
! [X4: a,Y: a,Z: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X4 @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X4 ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_193_line__extension__mem__iff,axiom,
! [U: a,K: set_a,A3: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A3 @ E ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ K )
& ? [Y3: a] :
( ( member_a @ Y3 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ A3 ) @ Y3 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_194_local_Oadd_Oright__cancel,axiom,
! [X4: a,Y: a,Z: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X4 )
= ( add_a_b @ r @ Z @ X4 ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_195_a__closed,axiom,
! [X4: a,Y: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X4 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_196_add_Ol__cancel__one,axiom,
! [X4: a,A3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X4 @ A3 )
= X4 )
= ( A3
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_197_add_Ol__cancel__one_H,axiom,
! [X4: a,A3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X4
= ( add_a_b @ r @ X4 @ A3 ) )
= ( A3
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_198_add_Or__cancel__one,axiom,
! [X4: a,A3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A3 @ X4 )
= X4 )
= ( A3
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_199_add_Or__cancel__one_H,axiom,
! [X4: a,A3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X4
= ( add_a_b @ r @ A3 @ X4 ) )
= ( A3
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_200_l__zero,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X4 )
= X4 ) ) ).
% l_zero
thf(fact_201_r__zero,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X4 @ ( zero_a_b @ r ) )
= X4 ) ) ).
% r_zero
thf(fact_202_ring_Oring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X4 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_203_ring_Oring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X4 @ Y ) @ Z )
= ( add_a_b @ R @ X4 @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_204_ring_Oring__simprules_I10_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X4 @ Y )
= ( add_a_b @ R @ Y @ X4 ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_205_ring_Oring__simprules_I22_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X4 @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X4 @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_206_ring_Osubring__props_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,H1: a,H2: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H2 @ K )
=> ( member_a @ ( add_a_b @ R @ H1 @ H2 ) @ K ) ) ) ) ) ).
% ring.subring_props(7)
thf(fact_207_abelian__groupE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X4 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_208_abelian__groupE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a,Z: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X4 @ Y ) @ Z )
= ( add_a_b @ R @ X4 @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_209_abelian__groupE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X4 @ Y )
= ( add_a_b @ R @ Y @ X4 ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_210_abelian__monoid_Oa__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X4: a,Y: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( add_a_b @ G @ X4 @ Y ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_211_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2175431115845679010xt_a_b,X4: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X4 @ ( add_a_b @ G @ Y @ Z ) )
= ( add_a_b @ G @ Y @ ( add_a_b @ G @ X4 @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_212_abelian__monoid_Oa__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,X4: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( add_a_b @ G @ X4 @ Y ) @ Z )
= ( add_a_b @ G @ X4 @ ( add_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_213_abelian__monoid_Oa__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X4: a,Y: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X4 @ Y )
= ( add_a_b @ G @ Y @ X4 ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_214_abelian__monoidE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X4 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_215_abelian__monoidE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X4 @ Y ) @ Z )
= ( add_a_b @ R @ X4 @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_216_abelian__monoidE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X4 @ Y )
= ( add_a_b @ R @ Y @ X4 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_217_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X4 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_218_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X4 @ Y ) @ Z )
= ( add_a_b @ R @ X4 @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_219_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X4 @ Y )
= ( add_a_b @ R @ Y @ X4 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_220_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X4 @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X4 @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_221_ring__irreducible__def,axiom,
( ring_r999134135267193926le_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,A4: a] :
( ( A4
!= ( zero_a_b @ R3 ) )
& ( irredu6211895646901577903xt_a_b @ R3 @ A4 ) ) ) ) ).
% ring_irreducible_def
thf(fact_222_ring_Oring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X4 )
= X4 ) ) ) ).
% ring.ring_simprules(8)
thf(fact_223_ring_Oring__simprules_I15_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X4 @ ( zero_a_b @ R ) )
= X4 ) ) ) ).
% ring.ring_simprules(15)
thf(fact_224_ring_Oring__simprules_I13_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X4 @ Y ) @ Z )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X4 @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_225_ring_Oring__simprules_I23_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X4 @ Y ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X4 ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_226_abelian__groupI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y4 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Z2: a] :
( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y4 ) @ Z2 )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y4 @ Z2 ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y4 )
= ( add_a_b @ R @ Y4 @ X ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( add_a_b @ R @ Xa @ X )
= ( zero_a_b @ R ) ) ) )
=> ( abelian_group_a_b @ R ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_227_abelian__groupE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X4 )
= X4 ) ) ) ).
% abelian_groupE(5)
thf(fact_228_abelian__groupE_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( add_a_b @ R @ X @ X4 )
= ( zero_a_b @ R ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_229_abelian__monoidI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X: a,Y4: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y4 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ! [X: a,Y4: a,Z2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y4 ) @ Z2 )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y4 @ Z2 ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) )
=> ( ! [X: a,Y4: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y4 )
= ( add_a_b @ R @ Y4 @ X ) ) ) )
=> ( abelian_monoid_a_b @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_230_abelian__monoid_Ominus__unique,axiom,
! [G: partia2175431115845679010xt_a_b,Y: a,X4: a,Y2: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( ( add_a_b @ G @ Y @ X4 )
= ( zero_a_b @ G ) )
=> ( ( ( add_a_b @ G @ X4 @ Y2 )
= ( zero_a_b @ G ) )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_231_abelian__monoid_Or__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X4: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X4 @ ( zero_a_b @ G ) )
= X4 ) ) ) ).
% abelian_monoid.r_zero
thf(fact_232_abelian__monoid_Ol__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X4: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( zero_a_b @ G ) @ X4 )
= X4 ) ) ) ).
% abelian_monoid.l_zero
thf(fact_233_abelian__monoidE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X4 )
= X4 ) ) ) ).
% abelian_monoidE(4)
thf(fact_234_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X4 )
= X4 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_235_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X4 @ ( zero_a_b @ R ) )
= X4 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_236_ring_Oline__extension__mem__iff,axiom,
! [R: partia2175431115845679010xt_a_b,U: a,K: set_a,A3: a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ R @ K @ A3 @ E ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ K )
& ? [Y3: a] :
( ( member_a @ Y3 @ E )
& ( U
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X2 @ A3 ) @ Y3 ) ) ) ) ) ) ) ).
% ring.line_extension_mem_iff
thf(fact_237_semiring_Or__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X4 @ Y ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X4 ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_238_semiring_Ol__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X4 @ Y ) @ Z )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X4 @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_239_domain_Ozero__is__irreducible__iff__field,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( irredu6211895646901577903xt_a_b @ R @ ( zero_a_b @ R ) )
= ( field_a_b @ R ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_240_domain_Oring__irreducibleE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ( irredu6211895646901577903xt_a_b @ R @ R2 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_241_abelian__group_Oa__lcos__m__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,M: set_a,G2: a,H3: a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ G2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ H3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_l_coset_a_b @ G @ G2 @ ( a_l_coset_a_b @ G @ H3 @ M ) )
= ( a_l_coset_a_b @ G @ ( add_a_b @ G @ G2 @ H3 ) @ M ) ) ) ) ) ) ).
% abelian_group.a_lcos_m_assoc
thf(fact_242_ring_Oeval__poly__of__const,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( poly_of_const_a_b @ R @ X4 ) @ Y )
= X4 ) ) ) ).
% ring.eval_poly_of_const
thf(fact_243_exp__base__closed,axiom,
! [X4: a,N: nat] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X4 @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% exp_base_closed
thf(fact_244_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_245_ee__trans,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
=> ( ( essent8953798148185448568xt_a_b @ r @ Bs @ Cs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( essent8953798148185448568xt_a_b @ r @ As @ Cs ) ) ) ) ) ) ).
% ee_trans
thf(fact_246_ee__sym,axiom,
! [As: list_a,Bs: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( essent8953798148185448568xt_a_b @ r @ Bs @ As ) ) ) ) ).
% ee_sym
thf(fact_247_ring_Oeval__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,X4: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( eval_a_b @ R @ P @ X4 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.eval_in_carrier
thf(fact_248_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_249_List_Ofinite__set,axiom,
! [Xs: list_set_a] : ( finite_finite_set_a @ ( set_set_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_250_List_Ofinite__set,axiom,
! [Xs: list_list_a] : ( finite_finite_list_a @ ( set_list_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_251_List_Ofinite__set,axiom,
! [Xs: list_a] : ( finite_finite_a @ ( set_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_252_noetherian__ringI,axiom,
( ! [I3: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ? [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( finite_finite_a @ A5 )
& ( I3
= ( genideal_a_b @ r @ A5 ) ) ) )
=> ( ring_n3639167112692572309ng_a_b @ r ) ) ).
% noetherian_ringI
thf(fact_253_factors__closed,axiom,
! [Fs: list_a,A3: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fs @ A3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% factors_closed
thf(fact_254_Span__is__subalgebra,axiom,
! [K: set_a,Us: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K @ ( embedded_Span_a_b @ r @ K @ Us ) @ r ) ) ) ).
% Span_is_subalgebra
thf(fact_255_oneideal,axiom,
ideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% oneideal
thf(fact_256_exists__gen,axiom,
! [I: set_a] :
( ( ideal_a_b @ I @ r )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( I
= ( cgenid547466209912283029xt_a_b @ r @ X ) ) ) ) ).
% exists_gen
thf(fact_257_cgenideal__ideal,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ A3 ) @ r ) ) ).
% cgenideal_ideal
thf(fact_258_cgenideal__minimal,axiom,
! [J: set_a,A3: a] :
( ( ideal_a_b @ J @ r )
=> ( ( member_a @ A3 @ J )
=> ( ord_less_eq_set_a @ ( cgenid547466209912283029xt_a_b @ r @ A3 ) @ J ) ) ) ).
% cgenideal_minimal
thf(fact_259_genideal__minimal,axiom,
! [I: set_a,S: set_a] :
( ( ideal_a_b @ I @ r )
=> ( ( ord_less_eq_set_a @ S @ I )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ S ) @ I ) ) ) ).
% genideal_minimal
thf(fact_260_Span__in__carrier,axiom,
! [K: set_a,Us: list_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% Span_in_carrier
thf(fact_261_Idl__subset__ideal,axiom,
! [I: set_a,H: set_a] :
( ( ideal_a_b @ I @ r )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H ) @ I )
= ( ord_less_eq_set_a @ H @ I ) ) ) ) ).
% Idl_subset_ideal
thf(fact_262_genideal__ideal,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ideal_a_b @ ( genideal_a_b @ r @ S ) @ r ) ) ).
% genideal_ideal
thf(fact_263_ideal__is__subalgebra,axiom,
! [K: set_a,I: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ideal_a_b @ I @ r )
=> ( embedd9027525575939734154ra_a_b @ K @ I @ r ) ) ) ).
% ideal_is_subalgebra
thf(fact_264_Span__subgroup__props_I1_J,axiom,
! [K: set_a,Us: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% Span_subgroup_props(1)
thf(fact_265_Span__base__incl,axiom,
! [K: set_a,Us: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( embedded_Span_a_b @ r @ K @ Us ) ) ) ) ).
% Span_base_incl
thf(fact_266_Span__same__set,axiom,
! [K: set_a,Us: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( set_a2 @ Us )
= ( set_a2 @ Vs ) )
=> ( ( embedded_Span_a_b @ r @ K @ Us )
= ( embedded_Span_a_b @ r @ K @ Vs ) ) ) ) ) ).
% Span_same_set
thf(fact_267_mono__Span__sublist,axiom,
! [K: set_a,Us: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( set_a2 @ Vs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us ) @ ( embedded_Span_a_b @ r @ K @ Vs ) ) ) ) ) ).
% mono_Span_sublist
thf(fact_268_mono__Span__subset,axiom,
! [K: set_a,Us: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( embedded_Span_a_b @ r @ K @ Vs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us ) @ ( embedded_Span_a_b @ r @ K @ Vs ) ) ) ) ) ).
% mono_Span_subset
thf(fact_269_Span__subalgebraI,axiom,
! [K: set_a,E: set_a,Us: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K @ E @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ E )
=> ( ! [V4: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V4 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ V4 )
=> ( ord_less_eq_set_a @ E @ V4 ) ) )
=> ( E
= ( embedded_Span_a_b @ r @ K @ Us ) ) ) ) ) ) ).
% Span_subalgebraI
thf(fact_270_subalgebra__Span__incl,axiom,
! [K: set_a,V: set_a,Us: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ V )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us ) @ V ) ) ) ) ).
% subalgebra_Span_incl
thf(fact_271_finetely__gen,axiom,
! [I: set_a] :
( ( ideal_a_b @ I @ r )
=> ? [A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
& ( finite_finite_a @ A6 )
& ( I
= ( genideal_a_b @ r @ A6 ) ) ) ) ).
% finetely_gen
thf(fact_272_Span__subgroup__props_I2_J,axiom,
! [K: set_a,Us: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( zero_a_b @ r ) @ ( embedded_Span_a_b @ r @ K @ Us ) ) ) ) ).
% Span_subgroup_props(2)
thf(fact_273_Span__subgroup__props_I3_J,axiom,
! [K: set_a,Us: list_a,V1: a,V22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ V1 @ ( embedded_Span_a_b @ r @ K @ Us ) )
=> ( ( member_a @ V22 @ ( embedded_Span_a_b @ r @ K @ Us ) )
=> ( member_a @ ( add_a_b @ r @ V1 @ V22 ) @ ( embedded_Span_a_b @ r @ K @ Us ) ) ) ) ) ) ).
% Span_subgroup_props(3)
thf(fact_274_Span__smult__closed,axiom,
! [K: set_a,Us: list_a,K2: a,V2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ V2 @ ( embedded_Span_a_b @ r @ K @ Us ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ V2 ) @ ( embedded_Span_a_b @ r @ K @ Us ) ) ) ) ) ) ).
% Span_smult_closed
thf(fact_275_Span__finite__dimension,axiom,
! [K: set_a,Us: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd8708762675212832759on_a_b @ r @ K @ ( embedded_Span_a_b @ r @ K @ Us ) ) ) ) ).
% Span_finite_dimension
thf(fact_276_ee__refl,axiom,
! [As: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( essent8953798148185448568xt_a_b @ r @ As @ As ) ) ).
% ee_refl
thf(fact_277_ring_OSpan_Ocong,axiom,
embedded_Span_a_b = embedded_Span_a_b ).
% ring.Span.cong
thf(fact_278_principal__domain_Oexists__gen,axiom,
! [R: partia2175431115845679010xt_a_b,I: set_a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( ideal_a_b @ I @ R )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
& ( I
= ( cgenid547466209912283029xt_a_b @ R @ X ) ) ) ) ) ).
% principal_domain.exists_gen
thf(fact_279_ring_Oeval_Ocong,axiom,
eval_a_b = eval_a_b ).
% ring.eval.cong
thf(fact_280_ring_OSpan__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K @ Us ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.Span_in_carrier
thf(fact_281_ring_Oideal__is__subalgebra,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,I: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ideal_a_b @ I @ R )
=> ( embedd9027525575939734154ra_a_b @ K @ I @ R ) ) ) ) ).
% ring.ideal_is_subalgebra
thf(fact_282_ring_Opoly__of__const_Ocong,axiom,
poly_of_const_a_b = poly_of_const_a_b ).
% ring.poly_of_const.cong
thf(fact_283_ring_OSpan__subgroup__props_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K @ Us ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.Span_subgroup_props(1)
thf(fact_284_ring_OSpan__same__set,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a,Vs: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( set_a2 @ Us )
= ( set_a2 @ Vs ) )
=> ( ( embedded_Span_a_b @ R @ K @ Us )
= ( embedded_Span_a_b @ R @ K @ Vs ) ) ) ) ) ) ).
% ring.Span_same_set
thf(fact_285_ring_OSpan__base__incl,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( embedded_Span_a_b @ R @ K @ Us ) ) ) ) ) ).
% ring.Span_base_incl
thf(fact_286_ring_Omono__Span__subset,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a,Vs: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( embedded_Span_a_b @ R @ K @ Vs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K @ Us ) @ ( embedded_Span_a_b @ R @ K @ Vs ) ) ) ) ) ) ).
% ring.mono_Span_subset
thf(fact_287_ring_Omono__Span__sublist,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a,Vs: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( set_a2 @ Vs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K @ Us ) @ ( embedded_Span_a_b @ R @ K @ Vs ) ) ) ) ) ) ).
% ring.mono_Span_sublist
thf(fact_288_ring_OSpan__subalgebraI,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,E: set_a,Us: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( embedd9027525575939734154ra_a_b @ K @ E @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ E )
=> ( ! [V4: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V4 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ V4 )
=> ( ord_less_eq_set_a @ E @ V4 ) ) )
=> ( E
= ( embedded_Span_a_b @ R @ K @ Us ) ) ) ) ) ) ) ).
% ring.Span_subalgebraI
thf(fact_289_ring_Osubalgebra__Span__incl,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,V: set_a,Us: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ V )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K @ Us ) @ V ) ) ) ) ) ).
% ring.subalgebra_Span_incl
thf(fact_290_noetherian__ring_Ofinetely__gen,axiom,
! [R: partia2175431115845679010xt_a_b,I: set_a] :
( ( ring_n3639167112692572309ng_a_b @ R )
=> ( ( ideal_a_b @ I @ R )
=> ? [A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ ( partia707051561876973205xt_a_b @ R ) )
& ( finite_finite_a @ A6 )
& ( I
= ( genideal_a_b @ R @ A6 ) ) ) ) ) ).
% noetherian_ring.finetely_gen
thf(fact_291_ring_OSpan__subgroup__props_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( zero_a_b @ R ) @ ( embedded_Span_a_b @ R @ K @ Us ) ) ) ) ) ).
% ring.Span_subgroup_props(2)
thf(fact_292_ring_OSpan__subgroup__props_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a,V1: a,V22: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ V1 @ ( embedded_Span_a_b @ R @ K @ Us ) )
=> ( ( member_a @ V22 @ ( embedded_Span_a_b @ R @ K @ Us ) )
=> ( member_a @ ( add_a_b @ R @ V1 @ V22 ) @ ( embedded_Span_a_b @ R @ K @ Us ) ) ) ) ) ) ) ).
% ring.Span_subgroup_props(3)
thf(fact_293_ring_OSpan__smult__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a,K2: a,V2: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ V2 @ ( embedded_Span_a_b @ R @ K @ Us ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K2 @ V2 ) @ ( embedded_Span_a_b @ R @ K @ Us ) ) ) ) ) ) ) ).
% ring.Span_smult_closed
thf(fact_294_ring_OSpan__finite__dimension,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( embedd8708762675212832759on_a_b @ R @ K @ ( embedded_Span_a_b @ R @ K @ Us ) ) ) ) ) ).
% ring.Span_finite_dimension
thf(fact_295_ring_OSpan__is__subalgebra,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( embedd9027525575939734154ra_a_b @ K @ ( embedded_Span_a_b @ R @ K @ Us ) @ R ) ) ) ) ).
% ring.Span_is_subalgebra
thf(fact_296_ring_Onoetherian__ringI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ! [I3: set_a] :
( ( ideal_a_b @ I3 @ R )
=> ? [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ ( partia707051561876973205xt_a_b @ R ) )
& ( finite_finite_a @ A5 )
& ( I3
= ( genideal_a_b @ R @ A5 ) ) ) )
=> ( ring_n3639167112692572309ng_a_b @ R ) ) ) ).
% ring.noetherian_ringI
thf(fact_297_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_298_subset__code_I1_J,axiom,
! [Xs: list_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_299_subset__code_I1_J,axiom,
! [Xs: list_a,B: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( member_a @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_300_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_301_finite__list,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ? [Xs2: list_set_a] :
( ( set_set_a2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_302_finite__list,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ? [Xs2: list_list_a] :
( ( set_list_a2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_303_finite__list,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ? [Xs2: list_a] :
( ( set_a2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_304_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ P )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ R @ P ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_305_ring_Oeval__var,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( var_a_b @ R ) @ X4 )
= X4 ) ) ) ).
% ring.eval_var
thf(fact_306_Span__strict__incl,axiom,
! [K: set_a,Us: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_set_a @ ( embedded_Span_a_b @ r @ K @ Us ) @ ( embedded_Span_a_b @ r @ K @ Vs ) )
=> ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Vs ) )
& ~ ( member_a @ X @ ( embedded_Span_a_b @ r @ K @ Us ) ) ) ) ) ) ) ).
% Span_strict_incl
thf(fact_307_ring_Ogenideal__ideal,axiom,
! [R: partia2175431115845679010xt_a_b,S: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ideal_a_b @ ( genideal_a_b @ R @ S ) @ R ) ) ) ).
% ring.genideal_ideal
thf(fact_308_ring_OIdl__subset__ideal,axiom,
! [R: partia2175431115845679010xt_a_b,I: set_a,H: set_a] :
( ( ring_a_b @ R )
=> ( ( ideal_a_b @ I @ R )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ R @ H ) @ I )
= ( ord_less_eq_set_a @ H @ I ) ) ) ) ) ).
% ring.Idl_subset_ideal
thf(fact_309_ring_Oexp__base__closed,axiom,
! [R: partia2175431115845679010xt_a_b,X4: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ R @ X4 @ N ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.exp_base_closed
thf(fact_310_mono__Span,axiom,
! [K: set_a,Us: list_a,U: a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us ) @ ( embedded_Span_a_b @ r @ K @ ( cons_a @ U @ Us ) ) ) ) ) ) ).
% mono_Span
thf(fact_311_factors__mult,axiom,
! [Fa: list_a,A3: a,Fb: list_a,B2: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fa @ A3 )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) ) ) ) ) ) ).
% factors_mult
thf(fact_312_factorsI,axiom,
! [Fs: list_a,A3: a] :
( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Fs ) )
=> ( irredu6211895646901577903xt_a_b @ r @ X ) )
=> ( ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) )
= A3 )
=> ( factor5638265376665762323xt_a_b @ r @ Fs @ A3 ) ) ) ).
% factorsI
thf(fact_313_list_Oinject,axiom,
! [X21: list_a,X22: list_list_a,Y21: list_a,Y22: list_list_a] :
( ( ( cons_list_a @ X21 @ X22 )
= ( cons_list_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_314_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_315_append_Oassoc,axiom,
! [A3: list_a,B2: list_a,C: list_a] :
( ( append_a @ ( append_a @ A3 @ B2 ) @ C )
= ( append_a @ A3 @ ( append_a @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_316_append__assoc,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_317_append__same__eq,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_318_same__append__eq,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_319_factors__mult__single,axiom,
! [A3: a,Fb: list_a,B2: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A3 )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( cons_a @ A3 @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) ) ) ) ) ).
% factors_mult_single
thf(fact_320_mono__Span__append_I1_J,axiom,
! [K: set_a,Us: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us ) @ ( embedded_Span_a_b @ r @ K @ ( append_a @ Us @ Vs ) ) ) ) ) ) ).
% mono_Span_append(1)
thf(fact_321_mono__Span__append_I2_J,axiom,
! [K: set_a,Us: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us ) @ ( embedded_Span_a_b @ r @ K @ ( append_a @ Vs @ Us ) ) ) ) ) ) ).
% mono_Span_append(2)
thf(fact_322_foldr__append,axiom,
! [F: a > a > a,Xs: list_a,Ys: list_a,A3: a] :
( ( foldr_a_a @ F @ ( append_a @ Xs @ Ys ) @ A3 )
= ( foldr_a_a @ F @ Xs @ ( foldr_a_a @ F @ Ys @ A3 ) ) ) ).
% foldr_append
thf(fact_323_add_Omultlist__closed,axiom,
! [Fs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( foldr_a_a @ ( add_a_b @ r ) @ Fs @ ( zero_a_b @ r ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.multlist_closed
thf(fact_324_multlist__closed,axiom,
! [Fs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% multlist_closed
thf(fact_325_split__list__first__prop__iff,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ( ? [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys2: list_list_a,X2: list_a] :
( ? [Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs2 ) ) )
& ( P2 @ X2 )
& ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( set_list_a2 @ Ys2 ) )
=> ~ ( P2 @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_326_split__list__first__prop__iff,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P2 @ X2 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Ys2 ) )
=> ~ ( P2 @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_327_split__list__last__prop__iff,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ( ? [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys2: list_list_a,X2: list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs2 ) ) )
& ( P2 @ X2 )
& ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( set_list_a2 @ Zs2 ) )
=> ~ ( P2 @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_328_split__list__last__prop__iff,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( P2 @ X2 ) ) )
= ( ? [Ys2: list_a,X2: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P2 @ X2 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Zs2 ) )
=> ~ ( P2 @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_329_in__set__conv__decomp__first,axiom,
! [X4: nat,Xs: list_nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs2 ) ) )
& ~ ( member_nat @ X4 @ ( set_nat2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_330_in__set__conv__decomp__first,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys2: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X4 @ Zs2 ) ) )
& ~ ( member_set_a @ X4 @ ( set_set_a2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_331_in__set__conv__decomp__first,axiom,
! [X4: list_a,Xs: list_list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
= ( ? [Ys2: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X4 @ Zs2 ) ) )
& ~ ( member_list_a @ X4 @ ( set_list_a2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_332_in__set__conv__decomp__first,axiom,
! [X4: a,Xs: list_a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
= ( ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
& ~ ( member_a @ X4 @ ( set_a2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_333_in__set__conv__decomp__last,axiom,
! [X4: nat,Xs: list_nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs2 ) ) )
& ~ ( member_nat @ X4 @ ( set_nat2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_334_in__set__conv__decomp__last,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys2: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X4 @ Zs2 ) ) )
& ~ ( member_set_a @ X4 @ ( set_set_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_335_in__set__conv__decomp__last,axiom,
! [X4: list_a,Xs: list_list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
= ( ? [Ys2: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X4 @ Zs2 ) ) )
& ~ ( member_list_a @ X4 @ ( set_list_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_336_in__set__conv__decomp__last,axiom,
! [X4: a,Xs: list_a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
= ( ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
& ~ ( member_a @ X4 @ ( set_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_337_split__list__first__propE,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ? [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X3 ) )
=> ~ ! [Ys3: list_list_a,X: list_a] :
( ? [Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs3 ) ) )
=> ( ( P2 @ X )
=> ~ ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( set_list_a2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_338_split__list__first__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P2 @ X3 ) )
=> ~ ! [Ys3: list_a,X: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
=> ( ( P2 @ X )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_339_split__list__last__propE,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ? [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X3 ) )
=> ~ ! [Ys3: list_list_a,X: list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs3 ) ) )
=> ( ( P2 @ X )
=> ~ ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( set_list_a2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_340_split__list__last__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P2 @ X3 ) )
=> ~ ! [Ys3: list_a,X: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
=> ( ( P2 @ X )
=> ~ ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_341_split__list__first__prop,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ? [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X3 ) )
=> ? [Ys3: list_list_a,X: list_a] :
( ? [Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs3 ) ) )
& ( P2 @ X )
& ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( set_list_a2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_342_split__list__first__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P2 @ X3 ) )
=> ? [Ys3: list_a,X: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
& ( P2 @ X )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Ys3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_343_split__list__last__prop,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ? [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X3 ) )
=> ? [Ys3: list_list_a,X: list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs3 ) ) )
& ( P2 @ X )
& ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( set_list_a2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_344_split__list__last__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P2 @ X3 ) )
=> ? [Ys3: list_a,X: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
& ( P2 @ X )
& ! [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Zs3 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_345_in__set__conv__decomp,axiom,
! [X4: nat,Xs: list_nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_346_in__set__conv__decomp,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys2: list_set_a,Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X4 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_347_in__set__conv__decomp,axiom,
! [X4: list_a,Xs: list_list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
= ( ? [Ys2: list_list_a,Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X4 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_348_in__set__conv__decomp,axiom,
! [X4: a,Xs: list_a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
= ( ? [Ys2: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_349_append__Cons__eq__iff,axiom,
! [X4: nat,Xs: list_nat,Ys: list_nat,Xs3: list_nat,Ys4: list_nat] :
( ~ ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X4 @ Ys ) )
= ( append_nat @ Xs3 @ ( cons_nat @ X4 @ Ys4 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_350_append__Cons__eq__iff,axiom,
! [X4: set_a,Xs: list_set_a,Ys: list_set_a,Xs3: list_set_a,Ys4: list_set_a] :
( ~ ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
=> ( ~ ( member_set_a @ X4 @ ( set_set_a2 @ Ys ) )
=> ( ( ( append_set_a @ Xs @ ( cons_set_a @ X4 @ Ys ) )
= ( append_set_a @ Xs3 @ ( cons_set_a @ X4 @ Ys4 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_351_append__Cons__eq__iff,axiom,
! [X4: list_a,Xs: list_list_a,Ys: list_list_a,Xs3: list_list_a,Ys4: list_list_a] :
( ~ ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
=> ( ~ ( member_list_a @ X4 @ ( set_list_a2 @ Ys ) )
=> ( ( ( append_list_a @ Xs @ ( cons_list_a @ X4 @ Ys ) )
= ( append_list_a @ Xs3 @ ( cons_list_a @ X4 @ Ys4 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_352_append__Cons__eq__iff,axiom,
! [X4: a,Xs: list_a,Ys: list_a,Xs3: list_a,Ys4: list_a] :
( ~ ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X4 @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X4 @ Ys ) )
= ( append_a @ Xs3 @ ( cons_a @ X4 @ Ys4 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_353_split__list__propE,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ? [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X3 ) )
=> ~ ! [Ys3: list_list_a,X: list_a] :
( ? [Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs3 ) ) )
=> ~ ( P2 @ X ) ) ) ).
% split_list_propE
thf(fact_354_split__list__propE,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P2 @ X3 ) )
=> ~ ! [Ys3: list_a,X: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
=> ~ ( P2 @ X ) ) ) ).
% split_list_propE
thf(fact_355_split__list__first,axiom,
! [X4: nat,Xs: list_nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) )
& ~ ( member_nat @ X4 @ ( set_nat2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_356_split__list__first,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
=> ? [Ys3: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X4 @ Zs3 ) ) )
& ~ ( member_set_a @ X4 @ ( set_set_a2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_357_split__list__first,axiom,
! [X4: list_a,Xs: list_list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
=> ? [Ys3: list_list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X4 @ Zs3 ) ) )
& ~ ( member_list_a @ X4 @ ( set_list_a2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_358_split__list__first,axiom,
! [X4: a,Xs: list_a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs3 ) ) )
& ~ ( member_a @ X4 @ ( set_a2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_359_split__list__prop,axiom,
! [Xs: list_list_a,P2: list_a > $o] :
( ? [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
& ( P2 @ X3 ) )
=> ? [Ys3: list_list_a,X: list_a] :
( ? [Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs3 ) ) )
& ( P2 @ X ) ) ) ).
% split_list_prop
thf(fact_360_split__list__prop,axiom,
! [Xs: list_a,P2: a > $o] :
( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P2 @ X3 ) )
=> ? [Ys3: list_a,X: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
& ( P2 @ X ) ) ) ).
% split_list_prop
thf(fact_361_split__list__last,axiom,
! [X4: nat,Xs: list_nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) )
& ~ ( member_nat @ X4 @ ( set_nat2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_362_split__list__last,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
=> ? [Ys3: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X4 @ Zs3 ) ) )
& ~ ( member_set_a @ X4 @ ( set_set_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_363_split__list__last,axiom,
! [X4: list_a,Xs: list_list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
=> ? [Ys3: list_list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X4 @ Zs3 ) ) )
& ~ ( member_list_a @ X4 @ ( set_list_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_364_split__list__last,axiom,
! [X4: a,Xs: list_a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs3 ) ) )
& ~ ( member_a @ X4 @ ( set_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_365_split__list,axiom,
! [X4: nat,Xs: list_nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_366_split__list,axiom,
! [X4: set_a,Xs: list_set_a] :
( ( member_set_a @ X4 @ ( set_set_a2 @ Xs ) )
=> ? [Ys3: list_set_a,Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys3 @ ( cons_set_a @ X4 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_367_split__list,axiom,
! [X4: list_a,Xs: list_list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
=> ? [Ys3: list_list_a,Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys3 @ ( cons_list_a @ X4 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_368_split__list,axiom,
! [X4: a,Xs: list_a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ? [Ys3: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_369_append__Cons,axiom,
! [X4: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( append_list_a @ ( cons_list_a @ X4 @ Xs ) @ Ys )
= ( cons_list_a @ X4 @ ( append_list_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_370_append__Cons,axiom,
! [X4: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X4 @ Xs ) @ Ys )
= ( cons_a @ X4 @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_371_not__Cons__self2,axiom,
! [X4: list_a,Xs: list_list_a] :
( ( cons_list_a @ X4 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_372_not__Cons__self2,axiom,
! [X4: a,Xs: list_a] :
( ( cons_a @ X4 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_373_Cons__eq__appendI,axiom,
! [X4: list_a,Xs1: list_list_a,Ys: list_list_a,Xs: list_list_a,Zs: list_list_a] :
( ( ( cons_list_a @ X4 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_a @ Xs1 @ Zs ) )
=> ( ( cons_list_a @ X4 @ Xs )
= ( append_list_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_374_Cons__eq__appendI,axiom,
! [X4: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X4 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X4 @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_375_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us2: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us2 ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_376_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us3: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us3 ) )
& ( ( append_a @ Us3 @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us3 )
= Zs )
& ( Ys
= ( append_a @ Us3 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_377_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_378_list_Oset__intros_I2_J,axiom,
! [Y: set_a,X22: list_set_a,X21: set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ X22 ) )
=> ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_379_list_Oset__intros_I2_J,axiom,
! [Y: list_a,X22: list_list_a,X21: list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ X22 ) )
=> ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_380_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X22 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_381_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_382_list_Oset__intros_I1_J,axiom,
! [X21: set_a,X22: list_set_a] : ( member_set_a @ X21 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_383_list_Oset__intros_I1_J,axiom,
! [X21: list_a,X22: list_list_a] : ( member_list_a @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_384_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_385_list_Oset__cases,axiom,
! [E2: nat,A3: list_nat] :
( ( member_nat @ E2 @ ( set_nat2 @ A3 ) )
=> ( ! [Z22: list_nat] :
( A3
!= ( cons_nat @ E2 @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A3
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E2 @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_386_list_Oset__cases,axiom,
! [E2: set_a,A3: list_set_a] :
( ( member_set_a @ E2 @ ( set_set_a2 @ A3 ) )
=> ( ! [Z22: list_set_a] :
( A3
!= ( cons_set_a @ E2 @ Z22 ) )
=> ~ ! [Z1: set_a,Z22: list_set_a] :
( ( A3
= ( cons_set_a @ Z1 @ Z22 ) )
=> ~ ( member_set_a @ E2 @ ( set_set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_387_list_Oset__cases,axiom,
! [E2: list_a,A3: list_list_a] :
( ( member_list_a @ E2 @ ( set_list_a2 @ A3 ) )
=> ( ! [Z22: list_list_a] :
( A3
!= ( cons_list_a @ E2 @ Z22 ) )
=> ~ ! [Z1: list_a,Z22: list_list_a] :
( ( A3
= ( cons_list_a @ Z1 @ Z22 ) )
=> ~ ( member_list_a @ E2 @ ( set_list_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_388_list_Oset__cases,axiom,
! [E2: a,A3: list_a] :
( ( member_a @ E2 @ ( set_a2 @ A3 ) )
=> ( ! [Z22: list_a] :
( A3
!= ( cons_a @ E2 @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A3
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E2 @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_389_set__ConsD,axiom,
! [Y: nat,X4: nat,Xs: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X4 @ Xs ) ) )
=> ( ( Y = X4 )
| ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_390_set__ConsD,axiom,
! [Y: set_a,X4: set_a,Xs: list_set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X4 @ Xs ) ) )
=> ( ( Y = X4 )
| ( member_set_a @ Y @ ( set_set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_391_set__ConsD,axiom,
! [Y: list_a,X4: list_a,Xs: list_list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X4 @ Xs ) ) )
=> ( ( Y = X4 )
| ( member_list_a @ Y @ ( set_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_392_set__ConsD,axiom,
! [Y: a,X4: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X4 @ Xs ) ) )
=> ( ( Y = X4 )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_393_finite__psubset__induct,axiom,
! [A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B4: set_nat] :
( ( ord_less_set_nat @ B4 @ A6 )
=> ( P2 @ B4 ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A ) ) ) ).
% finite_psubset_induct
thf(fact_394_finite__psubset__induct,axiom,
! [A: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ A )
=> ( ! [A6: set_set_a] :
( ( finite_finite_set_a @ A6 )
=> ( ! [B4: set_set_a] :
( ( ord_less_set_set_a @ B4 @ A6 )
=> ( P2 @ B4 ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A ) ) ) ).
% finite_psubset_induct
thf(fact_395_finite__psubset__induct,axiom,
! [A: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ A )
=> ( ! [A6: set_list_a] :
( ( finite_finite_list_a @ A6 )
=> ( ! [B4: set_list_a] :
( ( ord_less_set_list_a @ B4 @ A6 )
=> ( P2 @ B4 ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A ) ) ) ).
% finite_psubset_induct
thf(fact_396_finite__psubset__induct,axiom,
! [A: set_a,P2: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ! [B4: set_a] :
( ( ord_less_set_a @ B4 @ A6 )
=> ( P2 @ B4 ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A ) ) ) ).
% finite_psubset_induct
thf(fact_397_foldr__cong,axiom,
! [A3: a,B2: a,L: list_a,K2: list_a,F: a > a > a,G2: a > a > a] :
( ( A3 = B2 )
=> ( ( L = K2 )
=> ( ! [A2: a,X: a] :
( ( member_a @ X @ ( set_a2 @ L ) )
=> ( ( F @ X @ A2 )
= ( G2 @ X @ A2 ) ) )
=> ( ( foldr_a_a @ F @ L @ A3 )
= ( foldr_a_a @ G2 @ K2 @ B2 ) ) ) ) ) ).
% foldr_cong
thf(fact_398_set__subset__Cons,axiom,
! [Xs: list_list_a,X4: list_a] : ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ ( cons_list_a @ X4 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_399_set__subset__Cons,axiom,
! [Xs: list_a,X4: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X4 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_400_ideal_Ois__ideal,axiom,
! [I: set_a,R: partia2175431115845679010xt_a_b] :
( ( ideal_a_b @ I @ R )
=> ( ideal_a_b @ I @ R ) ) ).
% ideal.is_ideal
thf(fact_401_ring_Oexp__base_Ocong,axiom,
polyno2922411391617481336se_a_b = polyno2922411391617481336se_a_b ).
% ring.exp_base.cong
thf(fact_402_maximalideal_Ois__maximalideal,axiom,
! [I: set_a,R: partia2175431115845679010xt_a_b] :
( ( maximalideal_a_b @ I @ R )
=> ( maximalideal_a_b @ I @ R ) ) ).
% maximalideal.is_maximalideal
thf(fact_403_principalideal_Ois__principalideal,axiom,
! [I: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I @ R )
=> ( principalideal_a_b @ I @ R ) ) ).
% principalideal.is_principalideal
thf(fact_404_ring_Omono__Span__append_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a,Vs: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K @ Us ) @ ( embedded_Span_a_b @ R @ K @ ( append_a @ Us @ Vs ) ) ) ) ) ) ) ).
% ring.mono_Span_append(1)
thf(fact_405_ring_Omono__Span__append_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a,Vs: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K @ Us ) @ ( embedded_Span_a_b @ R @ K @ ( append_a @ Vs @ Us ) ) ) ) ) ) ) ).
% ring.mono_Span_append(2)
thf(fact_406_ring_Omono__Span,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a,U: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K @ Us ) @ ( embedded_Span_a_b @ R @ K @ ( cons_a @ U @ Us ) ) ) ) ) ) ) ).
% ring.mono_Span
thf(fact_407_ring_OSpan__strict__incl,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a,Vs: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_set_a @ ( embedded_Span_a_b @ R @ K @ Us ) @ ( embedded_Span_a_b @ R @ K @ Vs ) )
=> ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Vs ) )
& ~ ( member_a @ X @ ( embedded_Span_a_b @ R @ K @ Us ) ) ) ) ) ) ) ) ).
% ring.Span_strict_incl
thf(fact_408_ideal_OIcarr,axiom,
! [I: set_a,R: partia2175431115845679010xt_a_b,I2: a] :
( ( ideal_a_b @ I @ R )
=> ( ( member_a @ I2 @ I )
=> ( member_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ideal.Icarr
thf(fact_409_ideal_Oaxioms_I2_J,axiom,
! [I: set_a,R: partia2175431115845679010xt_a_b] :
( ( ideal_a_b @ I @ R )
=> ( ring_a_b @ R ) ) ).
% ideal.axioms(2)
thf(fact_410_maximalideal_OI__notcarr,axiom,
! [I: set_a,R: partia2175431115845679010xt_a_b] :
( ( maximalideal_a_b @ I @ R )
=> ( ( partia707051561876973205xt_a_b @ R )
!= I ) ) ).
% maximalideal.I_notcarr
thf(fact_411_maximalideal_Oaxioms_I1_J,axiom,
! [I: set_a,R: partia2175431115845679010xt_a_b] :
( ( maximalideal_a_b @ I @ R )
=> ( ideal_a_b @ I @ R ) ) ).
% maximalideal.axioms(1)
thf(fact_412_principalideal_Oaxioms_I1_J,axiom,
! [I: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I @ R )
=> ( ideal_a_b @ I @ R ) ) ).
% principalideal.axioms(1)
thf(fact_413_ideal_OI__l__closed,axiom,
! [I: set_a,R: partia2175431115845679010xt_a_b,A3: a,X4: a] :
( ( ideal_a_b @ I @ R )
=> ( ( member_a @ A3 @ I )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X4 @ A3 ) @ I ) ) ) ) ).
% ideal.I_l_closed
thf(fact_414_ideal_OI__r__closed,axiom,
! [I: set_a,R: partia2175431115845679010xt_a_b,A3: a,X4: a] :
( ( ideal_a_b @ I @ R )
=> ( ( member_a @ A3 @ I )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ A3 @ X4 ) @ I ) ) ) ) ).
% ideal.I_r_closed
thf(fact_415_ideal_Ohelper__I__closed,axiom,
! [I: set_a,R: partia2175431115845679010xt_a_b,A3: a,X4: a,Y: a] :
( ( ideal_a_b @ I @ R )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ R @ A3 @ X4 ) @ I )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ A3 @ ( mult_a_ring_ext_a_b @ R @ X4 @ Y ) ) @ I ) ) ) ) ) ) ).
% ideal.helper_I_closed
thf(fact_416_ideal_Oone__imp__carrier,axiom,
! [I: set_a,R: partia2175431115845679010xt_a_b] :
( ( ideal_a_b @ I @ R )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ R ) @ I )
=> ( I
= ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ideal.one_imp_carrier
thf(fact_417_ring_Ooneideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ideal_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% ring.oneideal
thf(fact_418_ring_Ocgenideal__self,axiom,
! [R: partia2175431115845679010xt_a_b,I2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ I2 @ ( cgenid547466209912283029xt_a_b @ R @ I2 ) ) ) ) ).
% ring.cgenideal_self
thf(fact_419_ring_Oonepideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( principalideal_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% ring.onepideal
thf(fact_420_ring_Osubset__Idl__subset,axiom,
! [R: partia2175431115845679010xt_a_b,I: set_a,H: set_a] :
( ( ring_a_b @ R )
=> ( ( 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 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_421_ring_Ogenideal__self,axiom,
! [R: partia2175431115845679010xt_a_b,S: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ R @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_422_ring_Ocgenideal__minimal,axiom,
! [R: partia2175431115845679010xt_a_b,J: set_a,A3: a] :
( ( ring_a_b @ R )
=> ( ( ideal_a_b @ J @ R )
=> ( ( member_a @ A3 @ J )
=> ( ord_less_eq_set_a @ ( cgenid547466209912283029xt_a_b @ R @ A3 ) @ J ) ) ) ) ).
% ring.cgenideal_minimal
thf(fact_423_ring_Ogenideal__minimal,axiom,
! [R: partia2175431115845679010xt_a_b,I: set_a,S: set_a] :
( ( ring_a_b @ R )
=> ( ( ideal_a_b @ I @ R )
=> ( ( ord_less_eq_set_a @ S @ I )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ R @ S ) @ I ) ) ) ) ).
% ring.genideal_minimal
thf(fact_424_maximalidealI,axiom,
! [I: set_a,R: partia2175431115845679010xt_a_b] :
( ( ideal_a_b @ I @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
!= I )
=> ( ! [J2: set_a] :
( ( ideal_a_b @ J2 @ R )
=> ( ( ord_less_eq_set_a @ I @ J2 )
=> ( ( ord_less_eq_set_a @ J2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( J2 = I )
| ( J2
= ( partia707051561876973205xt_a_b @ R ) ) ) ) ) )
=> ( maximalideal_a_b @ I @ R ) ) ) ) ).
% maximalidealI
thf(fact_425_maximalideal_OI__maximal,axiom,
! [I: set_a,R: partia2175431115845679010xt_a_b,J: set_a] :
( ( maximalideal_a_b @ I @ R )
=> ( ( ideal_a_b @ J @ R )
=> ( ( ord_less_eq_set_a @ I @ J )
=> ( ( ord_less_eq_set_a @ J @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( J = I )
| ( J
= ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ).
% maximalideal.I_maximal
thf(fact_426_eval__append__aux,axiom,
! [P: list_a,B2: a,A3: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P @ ( cons_a @ B2 @ nil_a ) ) @ A3 )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A3 ) @ A3 ) @ B2 ) ) ) ) ) ).
% eval_append_aux
thf(fact_427_factors__def,axiom,
( factor5638265376665762323xt_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,Fs2: list_a,A4: a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Fs2 ) )
=> ( irredu6211895646901577903xt_a_b @ G3 @ X2 ) )
& ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ G3 ) @ Fs2 @ ( one_a_ring_ext_a_b @ G3 ) )
= A4 ) ) ) ) ).
% factors_def
thf(fact_428_factors__def,axiom,
( factor4979495158039764464t_unit
= ( ^ [G3: partia8223610829204095565t_unit,Fs2: list_a,A4: a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Fs2 ) )
=> ( irredu4023057619401689684t_unit @ G3 @ X2 ) )
& ( ( foldr_a_a @ ( mult_a_Product_unit @ G3 ) @ Fs2 @ ( one_a_Product_unit @ G3 ) )
= A4 ) ) ) ) ).
% factors_def
thf(fact_429_factorsE,axiom,
! [G: partia2175431115845679010xt_a_b,Fs: list_a,A3: a] :
( ( factor5638265376665762323xt_a_b @ G @ Fs @ A3 )
=> ~ ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Fs ) )
=> ( irredu6211895646901577903xt_a_b @ G @ X3 ) )
=> ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ G ) @ Fs @ ( one_a_ring_ext_a_b @ G ) )
!= A3 ) ) ) ).
% factorsE
thf(fact_430_factorsE,axiom,
! [G: partia8223610829204095565t_unit,Fs: list_a,A3: a] :
( ( factor4979495158039764464t_unit @ G @ Fs @ A3 )
=> ~ ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Fs ) )
=> ( irredu4023057619401689684t_unit @ G @ X3 ) )
=> ( ( foldr_a_a @ ( mult_a_Product_unit @ G ) @ Fs @ ( one_a_Product_unit @ G ) )
!= A3 ) ) ) ).
% factorsE
thf(fact_431_Span__append__eq__set__add,axiom,
! [K: set_a,Us: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_Span_a_b @ r @ K @ ( append_a @ Us @ Vs ) )
= ( set_add_a_b @ r @ ( embedded_Span_a_b @ r @ K @ Us ) @ ( embedded_Span_a_b @ r @ K @ Vs ) ) ) ) ) ) ).
% Span_append_eq_set_add
thf(fact_432_multlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ As @ ( one_a_ring_ext_a_b @ r ) )
= ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Bs @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% multlist_perm_cong
thf(fact_433_add_Omultlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( foldr_a_a @ ( add_a_b @ r ) @ As @ ( zero_a_b @ r ) )
= ( foldr_a_a @ ( add_a_b @ r ) @ Bs @ ( zero_a_b @ r ) ) ) ) ) ).
% add.multlist_perm_cong
thf(fact_434_psubsetI,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% psubsetI
thf(fact_435_normalize_Ocases,axiom,
! [X4: list_a] :
( ( X4 != nil_a )
=> ~ ! [V3: a,Va: list_a] :
( X4
!= ( cons_a @ V3 @ Va ) ) ) ).
% normalize.cases
thf(fact_436_add__ideals,axiom,
! [I: set_a,J: set_a] :
( ( ideal_a_b @ I @ r )
=> ( ( ideal_a_b @ J @ r )
=> ( ideal_a_b @ ( set_add_a_b @ r @ I @ J ) @ r ) ) ) ).
% add_ideals
thf(fact_437_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_438_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( member_set_a @ X @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_439_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ X @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_440_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_441_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_442_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_443_set__add__comm,axiom,
! [I: set_a,J: set_a] :
( ( ord_less_eq_set_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ J @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ I @ J )
= ( set_add_a_b @ r @ J @ I ) ) ) ) ).
% set_add_comm
thf(fact_444_set__add__closed,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A @ B ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_add_closed
thf(fact_445_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_446_DiffI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_447_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_448_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_449_Diff__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ~ ( member_set_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_450_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_451_Diff__idemp,axiom,
! [A: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ).
% Diff_idemp
thf(fact_452_irrlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ As ) )
=> ( irredu6211895646901577903xt_a_b @ r @ X ) )
=> ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Bs ) )
=> ( irredu6211895646901577903xt_a_b @ r @ X3 ) ) ) ) ).
% irrlist_perm_cong
thf(fact_453_sum__space__dim_I1_J,axiom,
! [K: set_a,E: set_a,F2: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ F2 )
=> ( embedd8708762675212832759on_a_b @ r @ K @ ( set_add_a_b @ r @ E @ F2 ) ) ) ) ) ).
% sum_space_dim(1)
thf(fact_454_perm__closed,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% perm_closed
thf(fact_455_append__is__Nil__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= nil_list_a )
= ( ( Xs = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% append_is_Nil_conv
thf(fact_456_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_457_Nil__is__append__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( nil_list_a
= ( append_list_a @ Xs @ Ys ) )
= ( ( Xs = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% Nil_is_append_conv
thf(fact_458_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_459_self__append__conv2,axiom,
! [Y: list_list_a,Xs: list_list_a] :
( ( Y
= ( append_list_a @ Xs @ Y ) )
= ( Xs = nil_list_a ) ) ).
% self_append_conv2
thf(fact_460_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_461_append__self__conv2,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_a ) ) ).
% append_self_conv2
thf(fact_462_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_463_self__append__conv,axiom,
! [Y: list_list_a,Ys: list_list_a] :
( ( Y
= ( append_list_a @ Y @ Ys ) )
= ( Ys = nil_list_a ) ) ).
% self_append_conv
thf(fact_464_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_465_append__self__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_a ) ) ).
% append_self_conv
thf(fact_466_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_467_append__Nil2,axiom,
! [Xs: list_list_a] :
( ( append_list_a @ Xs @ nil_list_a )
= Xs ) ).
% append_Nil2
thf(fact_468_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_469_append_Oright__neutral,axiom,
! [A3: list_list_a] :
( ( append_list_a @ A3 @ nil_list_a )
= A3 ) ).
% append.right_neutral
thf(fact_470_append_Oright__neutral,axiom,
! [A3: list_a] :
( ( append_a @ A3 @ nil_a )
= A3 ) ).
% append.right_neutral
thf(fact_471_append1__eq__conv,axiom,
! [Xs: list_list_a,X4: list_a,Ys: list_list_a,Y: list_a] :
( ( ( append_list_a @ Xs @ ( cons_list_a @ X4 @ nil_list_a ) )
= ( append_list_a @ Ys @ ( cons_list_a @ Y @ nil_list_a ) ) )
= ( ( Xs = Ys )
& ( X4 = Y ) ) ) ).
% append1_eq_conv
thf(fact_472_append1__eq__conv,axiom,
! [Xs: list_a,X4: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X4 @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X4 = Y ) ) ) ).
% append1_eq_conv
thf(fact_473_transpose_Ocases,axiom,
! [X4: list_list_list_a] :
( ( X4 != nil_list_list_a )
=> ( ! [Xss: list_list_list_a] :
( X4
!= ( cons_list_list_a @ nil_list_a @ Xss ) )
=> ~ ! [X: list_a,Xs2: list_list_a,Xss: list_list_list_a] :
( X4
!= ( cons_list_list_a @ ( cons_list_a @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_474_transpose_Ocases,axiom,
! [X4: list_list_a] :
( ( X4 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X4
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X: a,Xs2: list_a,Xss: list_list_a] :
( X4
!= ( cons_list_a @ ( cons_a @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_475_perm__setP,axiom,
! [As: list_a,Bs: list_a,P2: set_a > $o] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( P2 @ ( set_a2 @ As ) )
=> ( P2 @ ( set_a2 @ Bs ) ) ) ) ).
% perm_setP
thf(fact_476_list__nonempty__induct,axiom,
! [Xs: list_list_a,P2: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X: list_a] : ( P2 @ ( cons_list_a @ X @ nil_list_a ) )
=> ( ! [X: list_a,Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_477_list__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X: a] : ( P2 @ ( cons_a @ X @ nil_a ) )
=> ( ! [X: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_478_list__induct2_H,axiom,
! [P2: list_a > list_list_a > $o,Xs: list_a,Ys: list_list_a] :
( ( P2 @ nil_a @ nil_list_a )
=> ( ! [X: a,Xs2: list_a] : ( P2 @ ( cons_a @ X @ Xs2 ) @ nil_list_a )
=> ( ! [Y4: list_a,Ys3: list_list_a] : ( P2 @ nil_a @ ( cons_list_a @ Y4 @ Ys3 ) )
=> ( ! [X: a,Xs2: list_a,Y4: list_a,Ys3: list_list_a] :
( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_479_list__induct2_H,axiom,
! [P2: list_list_a > list_a > $o,Xs: list_list_a,Ys: list_a] :
( ( P2 @ nil_list_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a] : ( P2 @ ( cons_list_a @ X @ Xs2 ) @ nil_a )
=> ( ! [Y4: a,Ys3: list_a] : ( P2 @ nil_list_a @ ( cons_a @ Y4 @ Ys3 ) )
=> ( ! [X: list_a,Xs2: list_list_a,Y4: a,Ys3: list_a] :
( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_480_list__induct2_H,axiom,
! [P2: list_list_a > list_list_a > $o,Xs: list_list_a,Ys: list_list_a] :
( ( P2 @ nil_list_a @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a] : ( P2 @ ( cons_list_a @ X @ Xs2 ) @ nil_list_a )
=> ( ! [Y4: list_a,Ys3: list_list_a] : ( P2 @ nil_list_a @ ( cons_list_a @ Y4 @ Ys3 ) )
=> ( ! [X: list_a,Xs2: list_list_a,Y4: list_a,Ys3: list_list_a] :
( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_481_list__induct2_H,axiom,
! [P2: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P2 @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a] : ( P2 @ ( cons_a @ X @ Xs2 ) @ nil_a )
=> ( ! [Y4: a,Ys3: list_a] : ( P2 @ nil_a @ ( cons_a @ Y4 @ Ys3 ) )
=> ( ! [X: a,Xs2: list_a,Y4: a,Ys3: list_a] :
( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_482_neq__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
= ( ? [Y3: list_a,Ys2: list_list_a] :
( Xs
= ( cons_list_a @ Y3 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_483_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y3: a,Ys2: list_a] :
( Xs
= ( cons_a @ Y3 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_484_remdups__adj_Ocases,axiom,
! [X4: list_list_a] :
( ( X4 != nil_list_a )
=> ( ! [X: list_a] :
( X4
!= ( cons_list_a @ X @ nil_list_a ) )
=> ~ ! [X: list_a,Y4: list_a,Xs2: list_list_a] :
( X4
!= ( cons_list_a @ X @ ( cons_list_a @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_485_remdups__adj_Ocases,axiom,
! [X4: list_a] :
( ( X4 != nil_a )
=> ( ! [X: a] :
( X4
!= ( cons_a @ X @ nil_a ) )
=> ~ ! [X: a,Y4: a,Xs2: list_a] :
( X4
!= ( cons_a @ X @ ( cons_a @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_486_list_Oexhaust,axiom,
! [Y: list_list_a] :
( ( Y != nil_list_a )
=> ~ ! [X212: list_a,X222: list_list_a] :
( Y
!= ( cons_list_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_487_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_488_list_OdiscI,axiom,
! [List: list_list_a,X21: list_a,X22: list_list_a] :
( ( List
= ( cons_list_a @ X21 @ X22 ) )
=> ( List != nil_list_a ) ) ).
% list.discI
thf(fact_489_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_490_list_Odistinct_I1_J,axiom,
! [X21: list_a,X22: list_list_a] :
( nil_list_a
!= ( cons_list_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_491_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_492_eq__Nil__appendI,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_a @ nil_list_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_493_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_494_append_Oleft__neutral,axiom,
! [A3: list_list_a] :
( ( append_list_a @ nil_list_a @ A3 )
= A3 ) ).
% append.left_neutral
thf(fact_495_append_Oleft__neutral,axiom,
! [A3: list_a] :
( ( append_a @ nil_a @ A3 )
= A3 ) ).
% append.left_neutral
thf(fact_496_append__Nil,axiom,
! [Ys: list_list_a] :
( ( append_list_a @ nil_list_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_497_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_498_rev__nonempty__induct,axiom,
! [Xs: list_list_a,P2: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X: list_a] : ( P2 @ ( cons_list_a @ X @ nil_list_a ) )
=> ( ! [X: list_a,Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append_list_a @ Xs2 @ ( cons_list_a @ X @ nil_list_a ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_499_rev__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X: a] : ( P2 @ ( cons_a @ X @ nil_a ) )
=> ( ! [X: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_500_append__eq__Cons__conv,axiom,
! [Ys: list_list_a,Zs: list_list_a,X4: list_a,Xs: list_list_a] :
( ( ( append_list_a @ Ys @ Zs )
= ( cons_list_a @ X4 @ Xs ) )
= ( ( ( Ys = nil_list_a )
& ( Zs
= ( cons_list_a @ X4 @ Xs ) ) )
| ? [Ys5: list_list_a] :
( ( Ys
= ( cons_list_a @ X4 @ Ys5 ) )
& ( ( append_list_a @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_501_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X4: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X4 @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X4 @ Xs ) ) )
| ? [Ys5: list_a] :
( ( Ys
= ( cons_a @ X4 @ Ys5 ) )
& ( ( append_a @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_502_Cons__eq__append__conv,axiom,
! [X4: list_a,Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( cons_list_a @ X4 @ Xs )
= ( append_list_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_a )
& ( ( cons_list_a @ X4 @ Xs )
= Zs ) )
| ? [Ys5: list_list_a] :
( ( ( cons_list_a @ X4 @ Ys5 )
= Ys )
& ( Xs
= ( append_list_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_503_Cons__eq__append__conv,axiom,
! [X4: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X4 @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X4 @ Xs )
= Zs ) )
| ? [Ys5: list_a] :
( ( ( cons_a @ X4 @ Ys5 )
= Ys )
& ( Xs
= ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_504_rev__exhaust,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ~ ! [Ys3: list_list_a,Y4: list_a] :
( Xs
!= ( append_list_a @ Ys3 @ ( cons_list_a @ Y4 @ nil_list_a ) ) ) ) ).
% rev_exhaust
thf(fact_505_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys3: list_a,Y4: a] :
( Xs
!= ( append_a @ Ys3 @ ( cons_a @ Y4 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_506_rev__induct,axiom,
! [P2: list_list_a > $o,Xs: list_list_a] :
( ( P2 @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append_list_a @ Xs2 @ ( cons_list_a @ X @ nil_list_a ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_507_rev__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ( P2 @ nil_a )
=> ( ! [X: a,Xs2: list_a] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_508_ring_Odense__repr_Ocases,axiom,
! [R: partia2175431115845679010xt_a_b,X4: list_a] :
( ( ring_a_b @ R )
=> ( ( X4 != nil_a )
=> ~ ! [V3: a,Va: list_a] :
( X4
!= ( cons_a @ V3 @ Va ) ) ) ) ).
% ring.dense_repr.cases
thf(fact_509_ring_Oadd__ideals,axiom,
! [R: partia2175431115845679010xt_a_b,I: set_a,J: set_a] :
( ( ring_a_b @ R )
=> ( ( ideal_a_b @ I @ R )
=> ( ( ideal_a_b @ J @ R )
=> ( ideal_a_b @ ( set_add_a_b @ R @ I @ J ) @ R ) ) ) ) ).
% ring.add_ideals
thf(fact_510_in__mono,axiom,
! [A: set_nat,B: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B ) ) ) ).
% in_mono
thf(fact_511_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X4: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X4 @ A )
=> ( member_set_a @ X4 @ B ) ) ) ).
% in_mono
thf(fact_512_in__mono,axiom,
! [A: set_a,B: set_a,X4: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X4 @ A )
=> ( member_a @ X4 @ B ) ) ) ).
% in_mono
thf(fact_513_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_514_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_515_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_516_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_517_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A7: set_nat,B5: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A7 )
=> ( member_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_518_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A7: set_set_a,B5: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A7 )
=> ( member_set_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_519_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A7 )
=> ( member_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_520_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_521_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_522_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A7: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A7 )
=> ( member_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_523_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A7: set_set_a,B5: set_set_a] :
! [T2: set_a] :
( ( member_set_a @ T2 @ A7 )
=> ( member_set_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_524_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B5: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A7 )
=> ( member_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_525_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_526_Collect__mono,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_527_subset__trans,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C3 )
=> ( ord_less_eq_set_a @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_528_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A7: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A7 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_529_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_530_ring_Oeval_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( eval_a_b @ R @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ R ) ) ) ) ).
% ring.eval.simps(1)
thf(fact_531_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_532_DiffE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% DiffE
thf(fact_533_DiffE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_534_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_535_DiffD1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ( member_set_a @ C @ A ) ) ).
% DiffD1
thf(fact_536_DiffD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_537_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_538_DiffD2,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B ) )
=> ~ ( member_set_a @ C @ B ) ) ).
% DiffD2
thf(fact_539_DiffD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_540_ring_Oset__add__comm,axiom,
! [R: partia2175431115845679010xt_a_b,I: set_a,J: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ J @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( set_add_a_b @ R @ I @ J )
= ( set_add_a_b @ R @ J @ I ) ) ) ) ) ).
% ring.set_add_comm
thf(fact_541_abelian__group_Osetadd__subset__G,axiom,
! [G: partia2175431115845679010xt_a_b,H: set_a,K: set_a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ G @ H @ K ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_group.setadd_subset_G
thf(fact_542_abelian__monoid_Oset__add__closed,axiom,
! [G: partia2175431115845679010xt_a_b,A: set_a,B: set_a] :
( ( abelian_monoid_a_b @ G )
=> ( ( ord_less_eq_set_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ G @ A @ B ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_monoid.set_add_closed
thf(fact_543_ring_Osum__space__dim_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,E: set_a,F2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K @ E )
=> ( ( embedd8708762675212832759on_a_b @ R @ K @ F2 )
=> ( embedd8708762675212832759on_a_b @ R @ K @ ( set_add_a_b @ R @ E @ F2 ) ) ) ) ) ) ).
% ring.sum_space_dim(1)
thf(fact_544_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_545_monoid__cancel_Ois__monoid__cancel,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid1999574367301118026t_unit @ G )
=> ( monoid1999574367301118026t_unit @ G ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_546_var__def,axiom,
( var_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b] : ( cons_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( cons_a @ ( zero_a_b @ R3 ) @ nil_a ) ) ) ) ).
% var_def
thf(fact_547_Diff__mono,axiom,
! [A: set_a,C3: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ ( minus_minus_set_a @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_548_Diff__subset,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_549_double__diff,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C3 )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C3 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_550_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B5: set_a] :
( ( ord_less_set_a @ A7 @ B5 )
| ( A7 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_551_subset__psubset__trans,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C3 )
=> ( ord_less_set_a @ A @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_552_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A7: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A7 @ B5 )
& ~ ( ord_less_eq_set_a @ B5 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_553_psubset__subset__trans,axiom,
! [A: set_a,B: set_a,C3: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C3 )
=> ( ord_less_set_a @ A @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_554_psubset__imp__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_555_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A7: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A7 @ B5 )
& ( A7 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_556_psubsetE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ B @ A ) ) ) ).
% psubsetE
thf(fact_557_psubset__imp__ex__mem,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_558_psubset__imp__ex__mem,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_less_set_set_a @ A @ B )
=> ? [B3: set_a] : ( member_set_a @ B3 @ ( minus_5736297505244876581_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_559_psubset__imp__ex__mem,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ? [B3: a] : ( member_a @ B3 @ ( minus_minus_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_560_ring_OSpan__append__eq__set__add,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a,Vs: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( embedded_Span_a_b @ R @ K @ ( append_a @ Us @ Vs ) )
= ( set_add_a_b @ R @ ( embedded_Span_a_b @ R @ K @ Us ) @ ( embedded_Span_a_b @ R @ K @ Vs ) ) ) ) ) ) ) ).
% ring.Span_append_eq_set_add
thf(fact_561_ring_Oeval__append__aux,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,B2: a,A3: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( append_a @ P @ ( cons_a @ B2 @ nil_a ) ) @ A3 )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ ( eval_a_b @ R @ P @ A3 ) @ A3 ) @ B2 ) ) ) ) ) ) ).
% ring.eval_append_aux
thf(fact_562_monoid__cancel_Or__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,C: a,B2: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ A3 @ C )
= ( mult_a_ring_ext_a_b @ G @ B2 @ C ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A3 = B2 ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_563_monoid__cancel_Or__cancel,axiom,
! [G: partia8223610829204095565t_unit,A3: a,C: a,B2: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ A3 @ C )
= ( mult_a_Product_unit @ G @ B2 @ C ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( A3 = B2 ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_564_monoid__cancel_Ol__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,C: a,A3: a,B2: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ C @ A3 )
= ( mult_a_ring_ext_a_b @ G @ C @ B2 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A3 = B2 ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_565_monoid__cancel_Ol__cancel,axiom,
! [G: partia8223610829204095565t_unit,C: a,A3: a,B2: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ C @ A3 )
= ( mult_a_Product_unit @ G @ C @ B2 ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( A3 = B2 ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_566_ideal__sum__iff__gcd,axiom,
! [A3: a,B2: a,D2: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ D2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( cgenid547466209912283029xt_a_b @ r @ D2 )
= ( set_add_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A3 ) @ ( cgenid547466209912283029xt_a_b @ r @ B2 ) ) )
= ( isgcd_a_ring_ext_a_b @ r @ D2 @ A3 @ B2 ) ) ) ) ) ).
% ideal_sum_iff_gcd
thf(fact_567_bezout__identity,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A3 ) @ ( cgenid547466209912283029xt_a_b @ r @ B2 ) )
= ( cgenid547466209912283029xt_a_b @ r @ ( somegc1600592057159103747xt_a_b @ r @ A3 @ B2 ) ) ) ) ) ).
% bezout_identity
thf(fact_568_is__root__def,axiom,
! [P: list_a,X4: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P @ X4 )
= ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P @ X4 )
= ( zero_a_b @ r ) )
& ( P != nil_a ) ) ) ).
% is_root_def
thf(fact_569_const__term__eq__last,axiom,
! [P: list_a,A3: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( append_a @ P @ ( cons_a @ A3 @ nil_a ) ) )
= A3 ) ) ) ).
% const_term_eq_last
thf(fact_570_const__term__explicit,axiom,
! [P: list_a,A3: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= A3 )
=> ~ ! [P3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( P
!= ( append_a @ P3 @ ( cons_a @ A3 @ nil_a ) ) ) ) ) ) ) ).
% const_term_explicit
thf(fact_571_combine__append__zero,axiom,
! [Us: list_a,Ks: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( append_a @ Ks @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Us )
= ( embedded_combine_a_b @ r @ Ks @ Us ) ) ) ).
% combine_append_zero
thf(fact_572_combine_Osimps_I3_J,axiom,
! [Ks: list_a] :
( ( embedded_combine_a_b @ r @ Ks @ nil_a )
= ( zero_a_b @ r ) ) ).
% combine.simps(3)
thf(fact_573_combine_Osimps_I2_J,axiom,
! [Us: list_a] :
( ( embedded_combine_a_b @ r @ nil_a @ Us )
= ( zero_a_b @ r ) ) ).
% combine.simps(2)
thf(fact_574_const__term__not__zero,axiom,
! [P: list_a] :
( ( ( const_term_a_b @ r @ P )
!= ( zero_a_b @ r ) )
=> ( P != nil_a ) ) ).
% const_term_not_zero
thf(fact_575_const__term__def,axiom,
! [P: list_a] :
( ( const_term_a_b @ r @ P )
= ( eval_a_b @ r @ P @ ( zero_a_b @ r ) ) ) ).
% const_term_def
thf(fact_576_const__term__simprules_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( const_term_a_b @ r @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% const_term_simprules(1)
thf(fact_577_combine_Osimps_I1_J,axiom,
! [K2: a,Ks: list_a,U: a,Us: list_a] :
( ( embedded_combine_a_b @ r @ ( cons_a @ K2 @ Ks ) @ ( cons_a @ U @ Us ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K2 @ U ) @ ( embedded_combine_a_b @ r @ Ks @ Us ) ) ) ).
% combine.simps(1)
thf(fact_578_combine_Oelims,axiom,
! [X4: list_a,Xa2: list_a,Y: a] :
( ( ( embedded_combine_a_b @ r @ X4 @ Xa2 )
= Y )
=> ( ! [K3: a,Ks2: list_a] :
( ( X4
= ( cons_a @ K3 @ Ks2 ) )
=> ! [U2: a,Us4: list_a] :
( ( Xa2
= ( cons_a @ U2 @ Us4 ) )
=> ( Y
!= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K3 @ U2 ) @ ( embedded_combine_a_b @ r @ Ks2 @ Us4 ) ) ) ) )
=> ( ( ( X4 = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) )
=> ~ ( ( Xa2 = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% combine.elims
thf(fact_579_combine__in__carrier,axiom,
! [Ks: list_a,Us: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( embedded_combine_a_b @ r @ Ks @ Us ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% combine_in_carrier
thf(fact_580_ring_Oconst__term_Ocong,axiom,
const_term_a_b = const_term_a_b ).
% ring.const_term.cong
thf(fact_581_ring_Ocombine_Ocong,axiom,
embedded_combine_a_b = embedded_combine_a_b ).
% ring.combine.cong
thf(fact_582_ring_Ocombine_Osimps_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,Us: list_a] :
( ( ring_a_b @ R )
=> ( ( embedded_combine_a_b @ R @ nil_a @ Us )
= ( zero_a_b @ R ) ) ) ).
% ring.combine.simps(2)
thf(fact_583_ring_Ocombine_Osimps_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,Ks: list_a] :
( ( ring_a_b @ R )
=> ( ( embedded_combine_a_b @ R @ Ks @ nil_a )
= ( zero_a_b @ R ) ) ) ).
% ring.combine.simps(3)
thf(fact_584_ring_Oconst__term__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ( const_term_a_b @ R @ P )
!= ( zero_a_b @ R ) )
=> ( P != nil_a ) ) ) ).
% ring.const_term_not_zero
thf(fact_585_ring_Oconst__term__def,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( const_term_a_b @ R @ P )
= ( eval_a_b @ R @ P @ ( zero_a_b @ R ) ) ) ) ).
% ring.const_term_def
thf(fact_586_ring_Ocombine__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,Ks: list_a,Us: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( embedded_combine_a_b @ R @ Ks @ Us ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.combine_in_carrier
thf(fact_587_ring_Ocombine_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: a,Ks: list_a,U: a,Us: list_a] :
( ( ring_a_b @ R )
=> ( ( embedded_combine_a_b @ R @ ( cons_a @ K2 @ Ks ) @ ( cons_a @ U @ Us ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ K2 @ U ) @ ( embedded_combine_a_b @ R @ Ks @ Us ) ) ) ) ).
% ring.combine.simps(1)
thf(fact_588_principal__domain_Obezout__identity,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( set_add_a_b @ R @ ( cgenid547466209912283029xt_a_b @ R @ A3 ) @ ( cgenid547466209912283029xt_a_b @ R @ B2 ) )
= ( cgenid547466209912283029xt_a_b @ R @ ( somegc1600592057159103747xt_a_b @ R @ A3 @ B2 ) ) ) ) ) ) ).
% principal_domain.bezout_identity
thf(fact_589_ring_Ocombine_Oelims,axiom,
! [R: partia2175431115845679010xt_a_b,X4: list_a,Xa2: list_a,Y: a] :
( ( ring_a_b @ R )
=> ( ( ( embedded_combine_a_b @ R @ X4 @ Xa2 )
= Y )
=> ( ! [K3: a,Ks2: list_a] :
( ( X4
= ( cons_a @ K3 @ Ks2 ) )
=> ! [U2: a,Us4: list_a] :
( ( Xa2
= ( cons_a @ U2 @ Us4 ) )
=> ( Y
!= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ K3 @ U2 ) @ ( embedded_combine_a_b @ R @ Ks2 @ Us4 ) ) ) ) )
=> ( ( ( X4 = nil_a )
=> ( Y
!= ( zero_a_b @ R ) ) )
=> ~ ( ( Xa2 = nil_a )
=> ( Y
!= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% ring.combine.elims
thf(fact_590_principal__domain_Oideal__sum__iff__gcd,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a,B2: a,D2: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ D2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( cgenid547466209912283029xt_a_b @ R @ D2 )
= ( set_add_a_b @ R @ ( cgenid547466209912283029xt_a_b @ R @ A3 ) @ ( cgenid547466209912283029xt_a_b @ R @ B2 ) ) )
= ( isgcd_a_ring_ext_a_b @ R @ D2 @ A3 @ B2 ) ) ) ) ) ) ).
% principal_domain.ideal_sum_iff_gcd
thf(fact_591_ring_Ois__root__def,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,X4: a] :
( ( ring_a_b @ R )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P @ X4 )
= ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( eval_a_b @ R @ P @ X4 )
= ( zero_a_b @ R ) )
& ( P != nil_a ) ) ) ) ).
% ring.is_root_def
thf(fact_592_ring_Ocombine__append__zero,axiom,
! [R: partia2175431115845679010xt_a_b,Us: list_a,Ks: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( embedded_combine_a_b @ R @ ( append_a @ Ks @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) @ Us )
= ( embedded_combine_a_b @ R @ Ks @ Us ) ) ) ) ).
% ring.combine_append_zero
thf(fact_593_ring_Oconst__term__eq__last,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,A3: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( const_term_a_b @ R @ ( append_a @ P @ ( cons_a @ A3 @ nil_a ) ) )
= A3 ) ) ) ) ).
% ring.const_term_eq_last
thf(fact_594_ring_Oconst__term__explicit,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,A3: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ R @ P )
= A3 )
=> ~ ! [P3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( P
!= ( append_a @ P3 @ ( cons_a @ A3 @ nil_a ) ) ) ) ) ) ) ) ).
% ring.const_term_explicit
thf(fact_595_mult__of_Omultlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ As @ ( one_a_ring_ext_a_b @ r ) )
= ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Bs @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.multlist_perm_cong
thf(fact_596_Span__mem__iff__length__version,axiom,
! [K: set_a,Us: list_a,A3: a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( embedded_Span_a_b @ r @ K @ Us ) )
= ( ? [Ks3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ K )
& ( ( size_size_list_a @ Ks3 )
= ( size_size_list_a @ Us ) )
& ( A3
= ( embedded_combine_a_b @ r @ Ks3 @ Us ) ) ) ) ) ) ) ).
% Span_mem_iff_length_version
thf(fact_597_combine__append,axiom,
! [Ks: list_a,Us: list_a,Ks4: list_a,Vs: list_a] :
( ( ( size_size_list_a @ Ks )
= ( size_size_list_a @ Us ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( embedded_combine_a_b @ r @ Ks @ Us ) @ ( embedded_combine_a_b @ r @ Ks4 @ Vs ) )
= ( embedded_combine_a_b @ r @ ( append_a @ Ks @ Ks4 ) @ ( append_a @ Us @ Vs ) ) ) ) ) ) ) ) ).
% combine_append
thf(fact_598_multlist__dividesI,axiom,
! [F: a,Fs: list_a] :
( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ F @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% multlist_dividesI
thf(fact_599_Span__incl,axiom,
! [K: set_a,Us: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_mu8047982887099575916xt_a_b @ r @ K @ ( set_a2 @ Us ) ) @ ( embedded_Span_a_b @ r @ K @ Us ) ) ) ) ).
% Span_incl
thf(fact_600_mult__of_Omonoid__cancel__axioms,axiom,
monoid1999574367301118026t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.monoid_cancel_axioms
thf(fact_601_zero__is__irreducible__mult,axiom,
irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( zero_a_b @ r ) ).
% zero_is_irreducible_mult
thf(fact_602_divides__trans,axiom,
! [A3: a,B2: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A3 @ B2 )
=> ( ( factor8216151070175719842xt_a_b @ r @ B2 @ C )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A3 @ C ) ) ) ) ).
% divides_trans
thf(fact_603_zero__divides,axiom,
! [A3: a] :
( ( factor8216151070175719842xt_a_b @ r @ ( zero_a_b @ r ) @ A3 )
= ( A3
= ( zero_a_b @ r ) ) ) ).
% zero_divides
thf(fact_604_ee__length,axiom,
! [As: list_a,Bs: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
=> ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) ) ) ).
% ee_length
thf(fact_605_mult__of_Or__cancel,axiom,
! [A3: a,C: a,B2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A3 @ C )
= ( mult_a_ring_ext_a_b @ r @ B2 @ C ) )
=> ( ( member_a @ A3 @ ( 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 ) ) )
=> ( A3 = B2 ) ) ) ) ) ).
% mult_of.r_cancel
thf(fact_606_mult__of_Om__lcomm,axiom,
! [X4: a,Y: a,Z: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X4 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X4 @ Z ) ) ) ) ) ) ).
% mult_of.m_lcomm
thf(fact_607_mult__of_Om__comm,axiom,
! [X4: a,Y: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X4 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X4 ) ) ) ) ).
% mult_of.m_comm
thf(fact_608_mult__of_Om__assoc,axiom,
! [X4: a,Y: a,Z: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X4 @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X4 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% mult_of.m_assoc
thf(fact_609_mult__of_Ol__cancel,axiom,
! [C: a,A3: a,B2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C @ A3 )
= ( mult_a_ring_ext_a_b @ r @ C @ B2 ) )
=> ( ( member_a @ A3 @ ( 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 ) ) )
=> ( A3 = B2 ) ) ) ) ) ).
% mult_of.l_cancel
thf(fact_610_mult__of_Omonoid__cancelI,axiom,
( ! [A2: a,B3: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A2 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B3 ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A2 = B3 ) ) ) ) )
=> ( ! [A2: a,B3: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A2 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B3 @ C2 ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A2 = B3 ) ) ) ) )
=> ( monoid1999574367301118026t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.monoid_cancelI
thf(fact_611_divides__zero,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A3 @ ( zero_a_b @ r ) ) ) ).
% divides_zero
thf(fact_612_mult__of_Oirrlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ As ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ X ) )
=> ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Bs ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ X3 ) ) ) ) ).
% mult_of.irrlist_perm_cong
thf(fact_613_divides__prod__r,axiom,
! [A3: a,B2: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A3 @ ( mult_a_ring_ext_a_b @ r @ B2 @ C ) ) ) ) ) ).
% divides_prod_r
thf(fact_614_divides__prod__l,axiom,
! [A3: a,B2: a,C: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A3 @ B2 )
=> ( factor8216151070175719842xt_a_b @ r @ A3 @ ( mult_a_ring_ext_a_b @ r @ C @ B2 ) ) ) ) ) ) ).
% divides_prod_l
thf(fact_615_local_Odivides__mult,axiom,
! [A3: a,C: a,B2: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A3 @ B2 )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A3 ) @ ( mult_a_ring_ext_a_b @ r @ C @ B2 ) ) ) ) ) ).
% local.divides_mult
thf(fact_616_one__divides,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ A3 ) ) ).
% one_divides
thf(fact_617_irreducible__imp__irreducible__mult,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( irredu6211895646901577903xt_a_b @ r @ A3 )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A3 ) ) ) ).
% irreducible_imp_irreducible_mult
thf(fact_618_ring__irreducibleE_I3_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ R2 ) ) ) ).
% ring_irreducibleE(3)
thf(fact_619_set__mult__closed,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_mu8047982887099575916xt_a_b @ r @ H @ K ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_mult_closed
thf(fact_620_isgcd__divides__r,axiom,
! [B2: a,A3: a] :
( ( factor8216151070175719842xt_a_b @ r @ B2 @ A3 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ B2 @ A3 @ B2 ) ) ) ) ).
% isgcd_divides_r
thf(fact_621_isgcd__divides__l,axiom,
! [A3: a,B2: a] :
( ( factor8216151070175719842xt_a_b @ r @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ A3 @ A3 @ B2 ) ) ) ) ).
% isgcd_divides_l
thf(fact_622_mult__of_Ofactors__closed,axiom,
! [Fs: list_a,A3: a] :
( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.factors_closed
thf(fact_623_mult__of_Ofactors__mult__single,axiom,
! [A3: a,Fb: list_a,B2: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A3 )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fb @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ ( cons_a @ A3 @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) ) ) ) ) ).
% mult_of.factors_mult_single
thf(fact_624_mult__of_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X )
= X ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.one_unique
thf(fact_625_mult__of_Oinv__unique,axiom,
! [Y: a,X4: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y @ X4 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X4 @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% mult_of.inv_unique
thf(fact_626_to__contain__is__to__divide,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( 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 @ A3 ) )
= ( factor8216151070175719842xt_a_b @ r @ A3 @ B2 ) ) ) ) ).
% to_contain_is_to_divide
thf(fact_627_cgenideal__prod,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_mu8047982887099575916xt_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A3 ) @ ( cgenid547466209912283029xt_a_b @ r @ B2 ) )
= ( cgenid547466209912283029xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) ) ) ) ) ).
% cgenideal_prod
thf(fact_628_combine__eq__eval,axiom,
! [Ks: list_a,X4: a] :
( ( embedded_combine_a_b @ r @ Ks @ ( polyno2922411391617481336se_a_b @ r @ X4 @ ( size_size_list_a @ Ks ) ) )
= ( eval_a_b @ r @ Ks @ X4 ) ) ).
% combine_eq_eval
thf(fact_629_mult__of_Operm__closed,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.perm_closed
thf(fact_630_mult__of_OfactorsI,axiom,
! [Fs: list_a,A3: a] :
( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Fs ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ X ) )
=> ( ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) )
= A3 )
=> ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A3 ) ) ) ).
% mult_of.factorsI
thf(fact_631_factors__dividesI,axiom,
! [Fs: list_a,A3: a,F: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fs @ A3 )
=> ( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ F @ A3 ) ) ) ) ).
% factors_dividesI
thf(fact_632_mult__of_Ofactors__mult,axiom,
! [Fa: list_a,A3: a,Fb: list_a,B2: a] :
( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fa @ A3 )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fb @ B2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) ) ) ) ) ) ).
% mult_of.factors_mult
thf(fact_633_Ring__Divisibility_Omult__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( mult_a_Product_unit @ ( ring_mult_of_a_b @ R ) )
= ( mult_a_ring_ext_a_b @ R ) ) ).
% Ring_Divisibility.mult_mult_of
thf(fact_634_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us2: list_a,Vs2: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us2 )
= ( size_size_list_a @ Vs2 ) ) )
=> ( ( ( append_a @ Xs @ Us2 )
= ( append_a @ Ys @ Vs2 ) )
= ( ( Xs = Ys )
& ( Us2 = Vs2 ) ) ) ) ).
% append_eq_append_conv
thf(fact_635_Ring__Divisibility_Oone__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( one_a_Product_unit @ ( ring_mult_of_a_b @ R ) )
= ( one_a_ring_ext_a_b @ R ) ) ).
% Ring_Divisibility.one_mult_of
thf(fact_636_dividesI_H,axiom,
! [B2: a,G: partia2175431115845679010xt_a_b,A3: a,C: a] :
( ( B2
= ( mult_a_ring_ext_a_b @ G @ A3 @ C ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( factor8216151070175719842xt_a_b @ G @ A3 @ B2 ) ) ) ).
% dividesI'
thf(fact_637_dividesI_H,axiom,
! [B2: a,G: partia8223610829204095565t_unit,A3: a,C: a] :
( ( B2
= ( mult_a_Product_unit @ G @ A3 @ C ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ A3 @ B2 ) ) ) ).
% dividesI'
thf(fact_638_mult__of_Olcmof__exists,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( islcm_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ C2 @ A3 @ B2 ) ) ) ) ).
% mult_of.lcmof_exists
thf(fact_639_divides__refl,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A3 @ A3 ) ) ).
% divides_refl
thf(fact_640_mult__of_Om__closed,axiom,
! [X4: a,Y: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X4 @ Y ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.m_closed
thf(fact_641_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_642_divides__mult__rI,axiom,
! [A3: a,B2: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( 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 @ A3 @ C ) @ ( mult_a_ring_ext_a_b @ r @ B2 @ C ) ) ) ) ) ) ).
% divides_mult_rI
thf(fact_643_divides__mult__lI,axiom,
! [A3: a,B2: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A3 ) @ ( mult_a_ring_ext_a_b @ r @ C @ B2 ) ) ) ) ) ).
% divides_mult_lI
thf(fact_644_mult__of_Or__one,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X4 @ ( one_a_ring_ext_a_b @ r ) )
= X4 ) ) ).
% mult_of.r_one
thf(fact_645_mult__of_Ol__one,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X4 )
= X4 ) ) ).
% mult_of.l_one
thf(fact_646_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_647_mult__of_Omultlist__closed,axiom,
! [Fs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.multlist_closed
thf(fact_648_mult__of_Oprimeness__condition__monoid__axioms,axiom,
primen965786292471834261t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.primeness_condition_monoid_axioms
thf(fact_649_impossible__Cons,axiom,
! [Xs: list_list_a,Ys: list_list_a,X4: list_a] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs
!= ( cons_list_a @ X4 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_650_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X4: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X4 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_651_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_652_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_653_factor__def,axiom,
( factor8216151070175719842xt_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,A4: a,B6: a] :
? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G3 ) )
& ( B6
= ( mult_a_ring_ext_a_b @ G3 @ A4 @ X2 ) ) ) ) ) ).
% factor_def
thf(fact_654_factor__def,axiom,
( factor3040189038382604065t_unit
= ( ^ [G3: partia8223610829204095565t_unit,A4: a,B6: a] :
? [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ G3 ) )
& ( B6
= ( mult_a_Product_unit @ G3 @ A4 @ X2 ) ) ) ) ) ).
% factor_def
thf(fact_655_dividesI,axiom,
! [C: a,G: partia2175431115845679010xt_a_b,B2: a,A3: a] :
( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( B2
= ( mult_a_ring_ext_a_b @ G @ A3 @ C ) )
=> ( factor8216151070175719842xt_a_b @ G @ A3 @ B2 ) ) ) ).
% dividesI
thf(fact_656_dividesI,axiom,
! [C: a,G: partia8223610829204095565t_unit,B2: a,A3: a] :
( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( B2
= ( mult_a_Product_unit @ G @ A3 @ C ) )
=> ( factor3040189038382604065t_unit @ G @ A3 @ B2 ) ) ) ).
% dividesI
thf(fact_657_dividesE,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( factor8216151070175719842xt_a_b @ G @ A3 @ B2 )
=> ~ ! [C2: a] :
( ( B2
= ( mult_a_ring_ext_a_b @ G @ A3 @ C2 ) )
=> ~ ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% dividesE
thf(fact_658_dividesE,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a] :
( ( factor3040189038382604065t_unit @ G @ A3 @ B2 )
=> ~ ! [C2: a] :
( ( B2
= ( mult_a_Product_unit @ G @ A3 @ C2 ) )
=> ~ ( member_a @ C2 @ ( partia6735698275553448452t_unit @ G ) ) ) ) ).
% dividesE
thf(fact_659_dividesD,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( factor8216151070175719842xt_a_b @ G @ A3 @ B2 )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
& ( B2
= ( mult_a_ring_ext_a_b @ G @ A3 @ X ) ) ) ) ).
% dividesD
thf(fact_660_dividesD,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a] :
( ( factor3040189038382604065t_unit @ G @ A3 @ B2 )
=> ? [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
& ( B2
= ( mult_a_Product_unit @ G @ A3 @ X ) ) ) ) ).
% dividesD
thf(fact_661_ring_Ozero__divides,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a] :
( ( ring_a_b @ R )
=> ( ( factor8216151070175719842xt_a_b @ R @ ( zero_a_b @ R ) @ A3 )
= ( A3
= ( zero_a_b @ R ) ) ) ) ).
% ring.zero_divides
thf(fact_662_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_list_a,P2: list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P2 @ nil_list_a @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y4: list_a,Ys3: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_663_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_a,P2: list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_list_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y4: a,Ys3: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_664_list__induct2,axiom,
! [Xs: list_a,Ys: list_list_a,P2: list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_list_a )
=> ( ! [X: a,Xs2: list_a,Y4: list_a,Ys3: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_665_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P2: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y4: a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_666_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a,P2: list_list_a > list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_list_a @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y4: list_a,Ys3: list_list_a,Z2: list_a,Zs3: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs3 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) @ ( cons_list_a @ Z2 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_667_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_a,P2: list_list_a > list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_list_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y4: list_a,Ys3: list_list_a,Z2: a,Zs3: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) @ ( cons_a @ Z2 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_668_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_list_a,P2: list_list_a > list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y4: a,Ys3: list_a,Z2: list_a,Zs3: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs3 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_list_a @ Z2 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_669_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,P2: list_list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y4: a,Ys3: list_a,Z2: a,Zs3: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z2 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_670_list__induct3,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_list_a,P2: list_a > list_list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_list_a )
=> ( ! [X: a,Xs2: list_a,Y4: list_a,Ys3: list_list_a,Z2: list_a,Zs3: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs3 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) @ ( cons_list_a @ Z2 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_671_list__induct3,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,P2: list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y4: list_a,Ys3: list_list_a,Z2: a,Zs3: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) @ ( cons_a @ Z2 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_672_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,P2: list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X: a,Xs2: list_a,Y4: a,Ys3: list_a,Z2: list_a,Zs3: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs3 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_list_a @ Z2 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_673_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P2: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y4: a,Ys3: list_a,Z2: a,Zs3: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z2 @ Zs3 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_674_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P2: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y4: a,Ys3: list_a,Z2: a,Zs3: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z2 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_675_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,Ws: list_a,P2: list_list_a > list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y4: a,Ys3: list_a,Z2: a,Zs3: list_a,W: a,Ws2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z2 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_676_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,Ws: list_a,P2: list_a > list_list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y4: list_a,Ys3: list_list_a,Z2: a,Zs3: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) @ ( cons_a @ Z2 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_677_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,Ws: list_a,P2: list_a > list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y4: a,Ys3: list_a,Z2: list_a,Zs3: list_list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs3 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_list_a @ Z2 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_678_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_list_a,P2: list_a > list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X: a,Xs2: list_a,Y4: a,Ys3: list_a,Z2: a,Zs3: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z2 @ Zs3 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_679_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_a,Ws: list_a,P2: list_list_a > list_list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_list_a @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y4: list_a,Ys3: list_list_a,Z2: a,Zs3: list_a,W: a,Ws2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) @ ( cons_a @ Z2 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_680_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_list_a,Ws: list_a,P2: list_list_a > list_a > list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y4: a,Ys3: list_a,Z2: list_a,Zs3: list_list_a,W: a,Ws2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs3 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_list_a @ Z2 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_681_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,Ws: list_list_a,P2: list_list_a > list_a > list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y4: a,Ys3: list_a,Z2: a,Zs3: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z2 @ Zs3 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_682_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_list_a,Ws: list_a,P2: list_a > list_list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_list_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y4: list_a,Ys3: list_list_a,Z2: list_a,Zs3: list_list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs3 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) @ ( cons_list_a @ Z2 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_683_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,Ws: list_list_a,P2: list_a > list_list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_a @ nil_list_a )
=> ( ! [X: a,Xs2: list_a,Y4: list_a,Ys3: list_list_a,Z2: a,Zs3: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) @ ( cons_a @ Z2 @ Zs3 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_684_isgcd__def,axiom,
( isgcd_a_ring_ext_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,X2: a,A4: a,B6: a] :
( ( factor8216151070175719842xt_a_b @ G3 @ X2 @ A4 )
& ( factor8216151070175719842xt_a_b @ G3 @ X2 @ B6 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( ( ( factor8216151070175719842xt_a_b @ G3 @ Y3 @ A4 )
& ( factor8216151070175719842xt_a_b @ G3 @ Y3 @ B6 ) )
=> ( factor8216151070175719842xt_a_b @ G3 @ Y3 @ X2 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_685_isgcd__def,axiom,
( isgcd_a_Product_unit
= ( ^ [G3: partia8223610829204095565t_unit,X2: a,A4: a,B6: a] :
( ( factor3040189038382604065t_unit @ G3 @ X2 @ A4 )
& ( factor3040189038382604065t_unit @ G3 @ X2 @ B6 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G3 ) )
=> ( ( ( factor3040189038382604065t_unit @ G3 @ Y3 @ A4 )
& ( factor3040189038382604065t_unit @ G3 @ Y3 @ B6 ) )
=> ( factor3040189038382604065t_unit @ G3 @ Y3 @ X2 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_686_domain_Ozero__is__irreducible__mult,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R ) @ ( zero_a_b @ R ) ) ) ).
% domain.zero_is_irreducible_mult
thf(fact_687_ring_Odivides__zero,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( factor8216151070175719842xt_a_b @ R @ A3 @ ( zero_a_b @ R ) ) ) ) ).
% ring.divides_zero
thf(fact_688_ring_Odivides__mult,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a,C: a,B2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( factor8216151070175719842xt_a_b @ R @ A3 @ B2 )
=> ( factor8216151070175719842xt_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ C @ A3 ) @ ( mult_a_ring_ext_a_b @ R @ C @ B2 ) ) ) ) ) ) ).
% ring.divides_mult
thf(fact_689_ring_Oone__divides,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( factor8216151070175719842xt_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ A3 ) ) ) ).
% ring.one_divides
thf(fact_690_monoid__cancel_Odivides__mult__l,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A3 @ ( 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 @ A3 ) @ ( mult_a_ring_ext_a_b @ G @ C @ B2 ) )
= ( factor8216151070175719842xt_a_b @ G @ A3 @ B2 ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_691_monoid__cancel_Odivides__mult__l,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a,C: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ ( mult_a_Product_unit @ G @ C @ A3 ) @ ( mult_a_Product_unit @ G @ C @ B2 ) )
= ( factor3040189038382604065t_unit @ G @ A3 @ B2 ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_692_same__length__different,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs != Ys )
=> ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ? [Pre: list_list_a,X: list_a,Xs4: list_list_a,Y4: list_a,Ys6: list_list_a] :
( ( X != Y4 )
& ( Xs
= ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ X @ nil_list_a ) @ Xs4 ) ) )
& ( Ys
= ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ Y4 @ nil_list_a ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_693_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X: a,Xs4: list_a,Y4: a,Ys6: list_a] :
( ( X != Y4 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X @ nil_a ) @ Xs4 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y4 @ nil_a ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_694_domain_Oring__irreducibleE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R ) @ R2 ) ) ) ) ).
% domain.ring_irreducibleE(3)
thf(fact_695_domain_Oirreducible__imp__irreducible__mult,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( irredu6211895646901577903xt_a_b @ R @ A3 )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R ) @ A3 ) ) ) ) ).
% domain.irreducible_imp_irreducible_mult
thf(fact_696_ring_Ocombine__eq__eval,axiom,
! [R: partia2175431115845679010xt_a_b,Ks: list_a,X4: a] :
( ( ring_a_b @ R )
=> ( ( embedded_combine_a_b @ R @ Ks @ ( polyno2922411391617481336se_a_b @ R @ X4 @ ( size_size_list_a @ Ks ) ) )
= ( eval_a_b @ R @ Ks @ X4 ) ) ) ).
% ring.combine_eq_eval
thf(fact_697_ring_OSpan__incl,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_mu8047982887099575916xt_a_b @ R @ K @ ( set_a2 @ Us ) ) @ ( embedded_Span_a_b @ R @ K @ Us ) ) ) ) ) ).
% ring.Span_incl
thf(fact_698_ring_Ocombine__append,axiom,
! [R: partia2175431115845679010xt_a_b,Ks: list_a,Us: list_a,Ks4: list_a,Vs: list_a] :
( ( ring_a_b @ R )
=> ( ( ( size_size_list_a @ Ks )
= ( size_size_list_a @ Us ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( embedded_combine_a_b @ R @ Ks @ Us ) @ ( embedded_combine_a_b @ R @ Ks4 @ Vs ) )
= ( embedded_combine_a_b @ R @ ( append_a @ Ks @ Ks4 ) @ ( append_a @ Us @ Vs ) ) ) ) ) ) ) ) ) ).
% ring.combine_append
thf(fact_699_ring_OSpan__mem__iff__length__version,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Us: list_a,A3: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A3 @ ( embedded_Span_a_b @ R @ K @ Us ) )
= ( ? [Ks3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ K )
& ( ( size_size_list_a @ Ks3 )
= ( size_size_list_a @ Us ) )
& ( A3
= ( embedded_combine_a_b @ R @ Ks3 @ Us ) ) ) ) ) ) ) ) ).
% ring.Span_mem_iff_length_version
thf(fact_700_mult__of_Ogcdof__exists,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ C2 @ A3 @ B2 ) ) ) ) ).
% mult_of.gcdof_exists
thf(fact_701_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_702_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_703_eval__append,axiom,
! [P: list_a,Q2: list_a,A3: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P @ Q2 ) @ A3 )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A3 ) @ ( pow_a_1026414303147256608_b_nat @ r @ A3 @ ( size_size_list_a @ Q2 ) ) ) @ ( eval_a_b @ r @ Q2 @ A3 ) ) ) ) ) ) ).
% eval_append
thf(fact_704_mult__of_Odivides__trans,axiom,
! [A3: a,B2: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ B2 @ C )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ C ) ) ) ) ).
% mult_of.divides_trans
thf(fact_705_divides__mult__zero,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ ( zero_a_b @ r ) )
=> ( A3
= ( zero_a_b @ r ) ) ) ) ).
% divides_mult_zero
thf(fact_706_mult__of_Odivides__prod__r,axiom,
! [A3: a,B2: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( 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 ) @ A3 @ ( mult_a_ring_ext_a_b @ r @ B2 @ C ) ) ) ) ) ).
% mult_of.divides_prod_r
thf(fact_707_mult__of_Odivides__prod__l,axiom,
! [A3: a,B2: a,C: a] :
( ( member_a @ A3 @ ( 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 ) @ A3 @ B2 )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ ( mult_a_ring_ext_a_b @ r @ C @ B2 ) ) ) ) ) ) ).
% mult_of.divides_prod_l
thf(fact_708_group__commutes__pow,axiom,
! [X4: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X4 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X4 ) )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X4 @ N ) @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( pow_a_1026414303147256608_b_nat @ r @ X4 @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_709_nat__pow__comm,axiom,
! [X4: a,N: nat,M2: nat] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X4 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X4 @ M2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X4 @ M2 ) @ ( pow_a_1026414303147256608_b_nat @ r @ X4 @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_710_nat__pow__distrib,axiom,
! [X4: a,Y: a,N: nat] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X4 @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X4 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ).
% nat_pow_distrib
thf(fact_711_pow__mult__distrib,axiom,
! [X4: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X4 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X4 ) )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X4 @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X4 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_712_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_713_mult__of_Oisgcd__divides__l,axiom,
! [A3: a,B2: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( 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 ) @ A3 @ A3 @ B2 ) ) ) ) ).
% mult_of.isgcd_divides_l
thf(fact_714_mult__of_Oisgcd__divides__r,axiom,
! [B2: a,A3: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ B2 @ A3 )
=> ( ( member_a @ A3 @ ( 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 @ A3 @ B2 ) ) ) ) ).
% mult_of.isgcd_divides_r
thf(fact_715_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_716_mult__of_Oprime__divides,axiom,
! [A3: a,B2: a,P: a] :
( ( member_a @ A3 @ ( 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 @ A3 @ B2 ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ A3 )
| ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ B2 ) ) ) ) ) ) ).
% mult_of.prime_divides
thf(fact_717_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_718_mult__of_Ofactors__dividesI,axiom,
! [Fs: list_a,A3: a,F: a] :
( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A3 )
=> ( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ F @ A3 ) ) ) ) ).
% mult_of.factors_dividesI
thf(fact_719_mult__of_Omultlist__dividesI,axiom,
! [F: a,Fs: list_a] :
( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ F @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.multlist_dividesI
thf(fact_720_mult__of_Odivides__refl,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ A3 ) ) ).
% mult_of.divides_refl
thf(fact_721_nat__pow__closed,axiom,
! [X4: a,N: nat] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X4 @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% nat_pow_closed
thf(fact_722_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_723_mult__of_Odivides__mult__l,axiom,
! [A3: a,B2: a,C: a] :
( ( member_a @ A3 @ ( 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 @ A3 ) @ ( mult_a_ring_ext_a_b @ r @ C @ B2 ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) ) ) ) ) ).
% mult_of.divides_mult_l
thf(fact_724_mult__of_Odivides__mult__lI,axiom,
! [A3: a,B2: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( 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 @ A3 ) @ ( mult_a_ring_ext_a_b @ r @ C @ B2 ) ) ) ) ) ).
% mult_of.divides_mult_lI
thf(fact_725_mult__of_Odivides__mult__r,axiom,
! [A3: a,B2: a,C: a] :
( ( member_a @ A3 @ ( 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 @ A3 @ C ) @ ( mult_a_ring_ext_a_b @ r @ B2 @ C ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) ) ) ) ) ).
% mult_of.divides_mult_r
thf(fact_726_mult__of_Odivides__mult__rI,axiom,
! [A3: a,B2: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( 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 @ A3 @ C ) @ ( mult_a_ring_ext_a_b @ r @ B2 @ C ) ) ) ) ) ) ).
% mult_of.divides_mult_rI
thf(fact_727_divides__mult__imp__divides,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ R ) @ A3 @ B2 )
=> ( factor8216151070175719842xt_a_b @ R @ A3 @ B2 ) ) ).
% divides_mult_imp_divides
thf(fact_728_domain_Odivides__mult__zero,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ R ) @ A3 @ ( zero_a_b @ R ) )
=> ( A3
= ( zero_a_b @ R ) ) ) ) ) ).
% domain.divides_mult_zero
thf(fact_729_domain_Ozero__is__prime_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ R ) @ ( zero_a_b @ R ) ) ) ).
% domain.zero_is_prime(2)
thf(fact_730_islcm__def,axiom,
( islcm_a_ring_ext_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,X2: a,A4: a,B6: a] :
( ( factor8216151070175719842xt_a_b @ G3 @ A4 @ X2 )
& ( factor8216151070175719842xt_a_b @ G3 @ B6 @ X2 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( ( ( factor8216151070175719842xt_a_b @ G3 @ A4 @ Y3 )
& ( factor8216151070175719842xt_a_b @ G3 @ B6 @ Y3 ) )
=> ( factor8216151070175719842xt_a_b @ G3 @ X2 @ Y3 ) ) ) ) ) ) ).
% islcm_def
thf(fact_731_islcm__def,axiom,
( islcm_a_Product_unit
= ( ^ [G3: partia8223610829204095565t_unit,X2: a,A4: a,B6: a] :
( ( factor3040189038382604065t_unit @ G3 @ A4 @ X2 )
& ( factor3040189038382604065t_unit @ G3 @ B6 @ X2 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G3 ) )
=> ( ( ( factor3040189038382604065t_unit @ G3 @ A4 @ Y3 )
& ( factor3040189038382604065t_unit @ G3 @ B6 @ Y3 ) )
=> ( factor3040189038382604065t_unit @ G3 @ X2 @ Y3 ) ) ) ) ) ) ).
% islcm_def
thf(fact_732_primeness__condition__monoid_Oirreducible__prime,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a] :
( ( primen9005823089519874350xt_a_b @ G )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( irredu6211895646901577903xt_a_b @ G @ A3 )
=> ( prime_a_ring_ext_a_b @ G @ A3 ) ) ) ) ).
% primeness_condition_monoid.irreducible_prime
thf(fact_733_primeness__condition__monoid_Oirreducible__prime,axiom,
! [G: partia8223610829204095565t_unit,A3: a] :
( ( primen965786292471834261t_unit @ G )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( irredu4023057619401689684t_unit @ G @ A3 )
=> ( prime_a_Product_unit @ G @ A3 ) ) ) ) ).
% primeness_condition_monoid.irreducible_prime
thf(fact_734_domain_Oring__primeE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% domain.ring_primeE(2)
thf(fact_735_domain_Oprime__eq__prime__mult,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R )
=> ( ( 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 ) ) ) ) ).
% domain.prime_eq_prime_mult
thf(fact_736_ring_Oeval__append,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q2: list_a,A3: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( append_a @ P @ Q2 ) @ A3 )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ ( eval_a_b @ R @ P @ A3 ) @ ( pow_a_1026414303147256608_b_nat @ R @ A3 @ ( size_size_list_a @ Q2 ) ) ) @ ( eval_a_b @ R @ Q2 @ A3 ) ) ) ) ) ) ) ).
% ring.eval_append
thf(fact_737_mult__of_Odivides__fcount,axiom,
! [A3: a,B2: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( 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 ) @ A3 ) @ ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ B2 ) ) ) ) ) ).
% mult_of.divides_fcount
thf(fact_738_eval__monom,axiom,
! [B2: a,A3: a,N: nat] :
( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( monom_a_b @ r @ B2 @ N ) @ A3 )
= ( mult_a_ring_ext_a_b @ r @ B2 @ ( pow_a_1026414303147256608_b_nat @ r @ A3 @ N ) ) ) ) ) ).
% eval_monom
thf(fact_739_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_740_mult__of_Owfactors__mult__single,axiom,
! [A3: a,Fb: list_a,B2: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fb @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ ( cons_a @ A3 @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) ) ) ) ) ) ) ).
% mult_of.wfactors_mult_single
thf(fact_741_mult__of_Ofactorcount__exists,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [C2: nat] :
! [As2: list_a] :
( ( ( ord_less_eq_set_a @ ( set_a2 @ As2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As2 @ A3 ) )
=> ( C2
= ( size_size_list_a @ As2 ) ) ) ) ).
% mult_of.factorcount_exists
thf(fact_742_mult__of_Owfactors__dividesI,axiom,
! [Fs: list_a,A3: a,F: a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ F @ A3 ) ) ) ) ) ).
% mult_of.wfactors_dividesI
thf(fact_743_mult__of_Owfactors__perm__cong__l,axiom,
! [Fs: list_a,A3: a,Fs3: list_a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A3 )
=> ( ( ( mset_a @ Fs )
= ( mset_a @ Fs3 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs3 @ A3 ) ) ) ) ).
% mult_of.wfactors_perm_cong_l
thf(fact_744_mult__of_Ofactors__wfactors,axiom,
! [As: list_a,A3: a] :
( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 ) ) ) ).
% mult_of.factors_wfactors
thf(fact_745_mult__of_Ofactorcount__unique,axiom,
! [As: list_a,A3: a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ A3 )
= ( size_size_list_a @ As ) ) ) ) ) ).
% mult_of.factorcount_unique
thf(fact_746_monom__in__carrier,axiom,
! [A3: a,N: nat] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( monom_a_b @ r @ A3 @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% monom_in_carrier
thf(fact_747_mult__of_Owfactors__exist,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [Fs4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs4 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs4 @ A3 ) ) ) ).
% mult_of.wfactors_exist
thf(fact_748_mult__of_Owfactors__prod__exists,axiom,
! [As: list_a] :
( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ As ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ X ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [A2: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A2 ) ) ) ) ).
% mult_of.wfactors_prod_exists
thf(fact_749_mult__of_Owfactors__mult,axiom,
! [As: list_a,A3: a,Bs: list_a,B2: a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ ( append_a @ As @ Bs ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% mult_of.wfactors_mult
thf(fact_750_ring_Omonom_Ocong,axiom,
monom_a_b = monom_a_b ).
% ring.monom.cong
thf(fact_751_divisor__chain__condition__monoid_Owfactors__exist,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a] :
( ( diviso8226693441832491329xt_a_b @ G )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [As3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As3 ) @ ( partia707051561876973205xt_a_b @ G ) )
& ( wfacto3557276942076956612xt_a_b @ G @ As3 @ A3 ) ) ) ) ).
% divisor_chain_condition_monoid.wfactors_exist
thf(fact_752_divisor__chain__condition__monoid_Owfactors__exist,axiom,
! [G: partia8223610829204095565t_unit,A3: a] :
( ( diviso6259607970152342594t_unit @ G )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ? [As3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As3 ) @ ( partia6735698275553448452t_unit @ G ) )
& ( wfacto3536202916627062655t_unit @ G @ As3 @ A3 ) ) ) ) ).
% divisor_chain_condition_monoid.wfactors_exist
thf(fact_753_primeness__condition__monoid_Owfactors__unique,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,A3: a,As4: list_a] :
( ( primen9005823089519874350xt_a_b @ G )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ As @ A3 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ As4 @ A3 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As4 ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( essent8953798148185448568xt_a_b @ G @ As @ As4 ) ) ) ) ) ) ) ).
% primeness_condition_monoid.wfactors_unique
thf(fact_754_primeness__condition__monoid_Owfactors__unique,axiom,
! [G: partia8223610829204095565t_unit,As: list_a,A3: a,As4: list_a] :
( ( primen965786292471834261t_unit @ G )
=> ( ( wfacto3536202916627062655t_unit @ G @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ G @ As4 @ A3 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As4 ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( essent9005414202370111435t_unit @ G @ As @ As4 ) ) ) ) ) ) ) ).
% primeness_condition_monoid.wfactors_unique
thf(fact_755_ring_Omonom__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( monom_a_b @ R @ A3 @ N ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.monom_in_carrier
thf(fact_756_monoid__cancel_Owfactors__mult__single,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,Fb: list_a,B2: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( irredu6211895646901577903xt_a_b @ G @ A3 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ Fb @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( wfacto3557276942076956612xt_a_b @ G @ ( cons_a @ A3 @ Fb ) @ ( mult_a_ring_ext_a_b @ G @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% monoid_cancel.wfactors_mult_single
thf(fact_757_monoid__cancel_Owfactors__mult__single,axiom,
! [G: partia8223610829204095565t_unit,A3: a,Fb: list_a,B2: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( irredu4023057619401689684t_unit @ G @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ G @ Fb @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( wfacto3536202916627062655t_unit @ G @ ( cons_a @ A3 @ Fb ) @ ( mult_a_Product_unit @ G @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% monoid_cancel.wfactors_mult_single
thf(fact_758_ring_Oeval__monom,axiom,
! [R: partia2175431115845679010xt_a_b,B2: a,A3: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( monom_a_b @ R @ B2 @ N ) @ A3 )
= ( mult_a_ring_ext_a_b @ R @ B2 @ ( pow_a_1026414303147256608_b_nat @ R @ A3 @ N ) ) ) ) ) ) ).
% ring.eval_monom
thf(fact_759_mult__of__is__Units,axiom,
( ( multip3210463924028840165of_a_b @ r )
= ( units_8174867845824275201xt_a_b @ r ) ) ).
% mult_of_is_Units
thf(fact_760_Multiplicative__Group_Oone__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( one_a_Product_unit @ ( multip3210463924028840165of_a_b @ R ) )
= ( one_a_ring_ext_a_b @ R ) ) ).
% Multiplicative_Group.one_mult_of
thf(fact_761_Multiplicative__Group_Omult__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( mult_a_Product_unit @ ( multip3210463924028840165of_a_b @ R ) )
= ( mult_a_ring_ext_a_b @ R ) ) ).
% Multiplicative_Group.mult_mult_of
thf(fact_762_mult__of_Oee__length,axiom,
! [As: list_a,Bs: list_a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) ) ) ).
% mult_of.ee_length
thf(fact_763_mult__of_Oee__trans,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ Cs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Cs ) ) ) ) ) ) ).
% mult_of.ee_trans
thf(fact_764_mult__of_Oee__sym,axiom,
! [As: list_a,Bs: list_a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ As ) ) ) ) ).
% mult_of.ee_sym
thf(fact_765_wfactors__perm__cong__l,axiom,
! [Fs: list_a,A3: a,Fs3: list_a] :
( ( wfacto3557276942076956612xt_a_b @ r @ Fs @ A3 )
=> ( ( ( mset_a @ Fs )
= ( mset_a @ Fs3 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( wfacto3557276942076956612xt_a_b @ r @ Fs3 @ A3 ) ) ) ) ).
% wfactors_perm_cong_l
thf(fact_766_wfactors__dividesI,axiom,
! [Fs: list_a,A3: a,F: a] :
( ( wfacto3557276942076956612xt_a_b @ r @ Fs @ A3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( factor8216151070175719842xt_a_b @ r @ F @ A3 ) ) ) ) ) ).
% wfactors_dividesI
thf(fact_767_factors__wfactors,axiom,
! [As: list_a,A3: a] :
( ( factor5638265376665762323xt_a_b @ r @ As @ A3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( wfacto3557276942076956612xt_a_b @ r @ As @ A3 ) ) ) ).
% factors_wfactors
thf(fact_768_mult__of_Owfactors__unique,axiom,
! [Fs: list_a,A3: a,Fs3: list_a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs3 @ A3 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs3 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ Fs3 ) ) ) ) ) ) ).
% mult_of.wfactors_unique
thf(fact_769_mult__of_Owfactors__ee__cong__l,axiom,
! [As: list_a,Bs: list_a,B2: a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ B2 ) ) ) ) ) ) ).
% mult_of.wfactors_ee_cong_l
thf(fact_770_mult__of_Oee__refl,axiom,
! [As: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ As ) ) ).
% mult_of.ee_refl
thf(fact_771_wfactors__prod__exists,axiom,
! [As: list_a] :
( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ As ) )
=> ( irredu6211895646901577903xt_a_b @ r @ X ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( wfacto3557276942076956612xt_a_b @ r @ As @ A2 ) ) ) ) ).
% wfactors_prod_exists
thf(fact_772_field_Omult__of__is__Units,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( multip3210463924028840165of_a_b @ R )
= ( units_8174867845824275201xt_a_b @ R ) ) ) ).
% field.mult_of_is_Units
thf(fact_773_field_Ofinite__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ R ) )
=> ( finite_finite_a @ ( partia6735698275553448452t_unit @ ( multip3210463924028840165of_a_b @ R ) ) ) ) ) ).
% field.finite_mult_of
thf(fact_774_mult__of_Omultlist__ee__cong,axiom,
! [Fs: list_a,Fs3: list_a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ Fs3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs3 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs3 @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% mult_of.multlist_ee_cong
thf(fact_775_mult__of_Owfactors__factors,axiom,
! [As: list_a,A3: a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [A8: a] :
( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A8 )
& ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A8 @ A3 ) ) ) ) ).
% mult_of.wfactors_factors
thf(fact_776_mult__of_Operm__wfactorsD,axiom,
! [As: list_a,Bs: list_a,A3: a,B2: a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) ) ) ) ) ) ) ).
% mult_of.perm_wfactorsD
thf(fact_777_mult__of_Oassociated__sym,axiom,
! [A3: a,B2: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B2 @ A3 ) ) ).
% mult_of.associated_sym
thf(fact_778_mult__of_Oassociated__trans,axiom,
! [A3: a,B2: a,C: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B2 @ C )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ C ) ) ) ) ) ).
% mult_of.associated_trans
thf(fact_779_mult__of_Oassoc__subst,axiom,
! [A3: a,B2: a,F: a > a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ! [A2: a,B3: a] :
( ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( member_a @ B3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B3 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( member_a @ ( F @ B3 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( F @ A2 ) @ ( F @ B3 ) ) ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( F @ A3 ) @ ( F @ B2 ) ) ) ) ) ) ).
% mult_of.assoc_subst
thf(fact_780_mult__of_Oassoc__l__cancel,axiom,
! [A3: a,B2: a,B7: a] :
( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B7 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ B7 ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B2 @ B7 ) ) ) ) ) ).
% mult_of.assoc_l_cancel
thf(fact_781_mult__of_Oassoc__r__cancel,axiom,
! [A3: a,B2: a,A9: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) @ ( mult_a_ring_ext_a_b @ r @ A9 @ B2 ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A9 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ A9 ) ) ) ) ) ).
% mult_of.assoc_r_cancel
thf(fact_782_mult__of_Omult__cong__l,axiom,
! [A3: a,A9: a,B2: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ A9 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A9 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) @ ( mult_a_ring_ext_a_b @ r @ A9 @ B2 ) ) ) ) ) ) ).
% mult_of.mult_cong_l
thf(fact_783_mult__of_Omult__cong__r,axiom,
! [B2: a,B7: a,A3: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B2 @ B7 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B7 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ B7 ) ) ) ) ) ) ).
% mult_of.mult_cong_r
thf(fact_784_mult__of_Odivides__cong__r,axiom,
! [X4: a,Y: a,Y2: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X4 @ Y )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ Y @ Y2 )
=> ( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X4 @ Y2 ) ) ) ) ).
% mult_of.divides_cong_r
thf(fact_785_mult__of_Odivides__cong__l,axiom,
! [X4: a,X5: a,Y: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ X4 @ X5 )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X5 @ Y )
=> ( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X4 @ Y ) ) ) ) ).
% mult_of.divides_cong_l
thf(fact_786_mult__of_Oirreducible__cong,axiom,
! [A3: a,A9: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A3 )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ A9 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A9 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A9 ) ) ) ) ) ).
% mult_of.irreducible_cong
thf(fact_787_mult__of_Oprime__cong,axiom,
! [P: a,P4: a] :
( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ P @ P4 )
=> ( ( member_a @ P @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ P4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P4 ) ) ) ) ) ).
% mult_of.prime_cong
thf(fact_788_mult__of_Oassociated__fcount,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ A3 )
= ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ B2 ) ) ) ) ) ).
% mult_of.associated_fcount
thf(fact_789_mult__of_Ogcdof__cong__l,axiom,
! [A9: a,A3: a,B2: a,C: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A9 @ A3 )
=> ( ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 @ C )
=> ( ( member_a @ A9 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A3 @ ( 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 ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A9 @ B2 @ C ) ) ) ) ) ) ) ).
% mult_of.gcdof_cong_l
thf(fact_790_mult__of_Owfactors__cong__r,axiom,
! [Fs: list_a,A3: a,A9: a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A3 )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ A9 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A9 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A9 ) ) ) ) ) ) ).
% mult_of.wfactors_cong_r
thf(fact_791_mult__of_Oee__wfactorsI,axiom,
! [A3: a,B2: a,As: list_a,Bs: list_a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs ) ) ) ) ) ) ) ) ).
% mult_of.ee_wfactorsI
thf(fact_792_mult__of_Oee__wfactorsD,axiom,
! [As: list_a,Bs: list_a,A3: a,B2: a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% mult_of.ee_wfactorsD
thf(fact_793_mult__of_Oee__wfactors,axiom,
! [As: list_a,A3: a,Bs: list_a,B2: a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
= ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs ) ) ) ) ) ) ) ) ).
% mult_of.ee_wfactors
thf(fact_794_mult__of_Oee__factorsD,axiom,
! [As: list_a,Bs: list_a,A3: a,B2: a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) ) ) ) ) ) ).
% mult_of.ee_factorsD
thf(fact_795_mult__of_Oassociated__refl,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ A3 ) ) ).
% mult_of.associated_refl
thf(fact_796_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a,B7: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B7 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A3 @ B2 ) @ ( mult_a_ring_ext_a_b @ G @ A3 @ B7 ) )
=> ( associ5860276527279195403xt_a_b @ G @ B2 @ B7 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_797_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a,B7: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B7 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( associ6879500422977059064t_unit @ G @ ( mult_a_Product_unit @ G @ A3 @ B2 ) @ ( mult_a_Product_unit @ G @ A3 @ B7 ) )
=> ( associ6879500422977059064t_unit @ G @ B2 @ B7 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_798_monoid__cancel_Oirreducible__cong,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,A9: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( irredu6211895646901577903xt_a_b @ G @ A3 )
=> ( ( associ5860276527279195403xt_a_b @ G @ A3 @ A9 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ A9 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( irredu6211895646901577903xt_a_b @ G @ A9 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_799_monoid__cancel_Oirreducible__cong,axiom,
! [G: partia8223610829204095565t_unit,A3: a,A9: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( irredu4023057619401689684t_unit @ G @ A3 )
=> ( ( associ6879500422977059064t_unit @ G @ A3 @ A9 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A9 @ ( partia6735698275553448452t_unit @ G ) )
=> ( irredu4023057619401689684t_unit @ G @ A9 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_800_monoid__cancel_Oprime__cong,axiom,
! [G: partia2175431115845679010xt_a_b,P: a,P4: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( prime_a_ring_ext_a_b @ G @ P )
=> ( ( associ5860276527279195403xt_a_b @ G @ P @ P4 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ P4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( prime_a_ring_ext_a_b @ G @ P4 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_801_monoid__cancel_Oprime__cong,axiom,
! [G: partia8223610829204095565t_unit,P: a,P4: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( prime_a_Product_unit @ G @ P )
=> ( ( associ6879500422977059064t_unit @ G @ P @ P4 )
=> ( ( member_a @ P @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ P4 @ ( partia6735698275553448452t_unit @ G ) )
=> ( prime_a_Product_unit @ G @ P4 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_802_wfactorsE,axiom,
! [G: partia2175431115845679010xt_a_b,Fs: list_a,A3: a] :
( ( wfacto3557276942076956612xt_a_b @ G @ Fs @ A3 )
=> ~ ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Fs ) )
=> ( irredu6211895646901577903xt_a_b @ G @ X3 ) )
=> ~ ( associ5860276527279195403xt_a_b @ G @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ G ) @ Fs @ ( one_a_ring_ext_a_b @ G ) ) @ A3 ) ) ) ).
% wfactorsE
thf(fact_803_wfactorsE,axiom,
! [G: partia8223610829204095565t_unit,Fs: list_a,A3: a] :
( ( wfacto3536202916627062655t_unit @ G @ Fs @ A3 )
=> ~ ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Fs ) )
=> ( irredu4023057619401689684t_unit @ G @ X3 ) )
=> ~ ( associ6879500422977059064t_unit @ G @ ( foldr_a_a @ ( mult_a_Product_unit @ G ) @ Fs @ ( one_a_Product_unit @ G ) ) @ A3 ) ) ) ).
% wfactorsE
thf(fact_804_wfactorsI,axiom,
! [Fs: list_a,G: partia2175431115845679010xt_a_b,A3: a] :
( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Fs ) )
=> ( irredu6211895646901577903xt_a_b @ G @ X ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ G ) @ Fs @ ( one_a_ring_ext_a_b @ G ) ) @ A3 )
=> ( wfacto3557276942076956612xt_a_b @ G @ Fs @ A3 ) ) ) ).
% wfactorsI
thf(fact_805_wfactorsI,axiom,
! [Fs: list_a,G: partia8223610829204095565t_unit,A3: a] :
( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Fs ) )
=> ( irredu4023057619401689684t_unit @ G @ X ) )
=> ( ( associ6879500422977059064t_unit @ G @ ( foldr_a_a @ ( mult_a_Product_unit @ G ) @ Fs @ ( one_a_Product_unit @ G ) ) @ A3 )
=> ( wfacto3536202916627062655t_unit @ G @ Fs @ A3 ) ) ) ).
% wfactorsI
thf(fact_806_wfactors__def,axiom,
( wfacto3557276942076956612xt_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,Fs2: list_a,A4: a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Fs2 ) )
=> ( irredu6211895646901577903xt_a_b @ G3 @ X2 ) )
& ( associ5860276527279195403xt_a_b @ G3 @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ G3 ) @ Fs2 @ ( one_a_ring_ext_a_b @ G3 ) ) @ A4 ) ) ) ) ).
% wfactors_def
thf(fact_807_wfactors__def,axiom,
( wfacto3536202916627062655t_unit
= ( ^ [G3: partia8223610829204095565t_unit,Fs2: list_a,A4: a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Fs2 ) )
=> ( irredu4023057619401689684t_unit @ G3 @ X2 ) )
& ( associ6879500422977059064t_unit @ G3 @ ( foldr_a_a @ ( mult_a_Product_unit @ G3 ) @ Fs2 @ ( one_a_Product_unit @ G3 ) ) @ A4 ) ) ) ) ).
% wfactors_def
thf(fact_808_mult__of_Orelprime__mult,axiom,
! [A3: a,B2: a,C: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) @ ( one_a_ring_ext_a_b @ r ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ C ) @ ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( 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 ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ ( mult_a_ring_ext_a_b @ r @ B2 @ C ) ) @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.relprime_mult
thf(fact_809_mult__of_OgcdI2,axiom,
! [A3: a,B2: a,C: a] :
( ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 @ C )
=> ( ( member_a @ A3 @ ( 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 ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ B2 @ C ) ) ) ) ) ) ).
% mult_of.gcdI2
thf(fact_810_mult__of_Omultlist__listassoc__cong,axiom,
! [Fs: list_a,Fs3: list_a] :
( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ Fs @ Fs3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs3 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs3 @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% mult_of.multlist_listassoc_cong
thf(fact_811_associated__sym,axiom,
! [A3: a,B2: a] :
( ( associ5860276527279195403xt_a_b @ r @ A3 @ B2 )
=> ( associ5860276527279195403xt_a_b @ r @ B2 @ A3 ) ) ).
% associated_sym
thf(fact_812_assoc__subst,axiom,
! [A3: a,B2: a,F: a > a] :
( ( associ5860276527279195403xt_a_b @ r @ A3 @ B2 )
=> ( ! [A2: a,B3: a] :
( ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ A2 @ B3 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ ( F @ B3 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ ( F @ A2 ) @ ( F @ B3 ) ) ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( F @ A3 ) @ ( F @ B2 ) ) ) ) ) ) ).
% assoc_subst
thf(fact_813_associated__trans,axiom,
! [A3: a,B2: a,C: a] :
( ( associ5860276527279195403xt_a_b @ r @ A3 @ B2 )
=> ( ( associ5860276527279195403xt_a_b @ r @ B2 @ C )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A3 @ C ) ) ) ) ) ).
% associated_trans
thf(fact_814_mult__cong__l,axiom,
! [A3: a,A9: a,B2: a] :
( ( associ5860276527279195403xt_a_b @ r @ A3 @ A9 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A9 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) @ ( mult_a_ring_ext_a_b @ r @ A9 @ B2 ) ) ) ) ) ) ).
% mult_cong_l
thf(fact_815_mult__cong__r,axiom,
! [B2: a,B7: a,A3: a] :
( ( associ5860276527279195403xt_a_b @ r @ B2 @ B7 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B7 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ B7 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_816_divides__cong__l,axiom,
! [X4: a,X5: a,Y: a] :
( ( associ5860276527279195403xt_a_b @ r @ X4 @ X5 )
=> ( ( factor8216151070175719842xt_a_b @ r @ X5 @ Y )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ X4 @ Y ) ) ) ) ).
% divides_cong_l
thf(fact_817_divides__cong__r,axiom,
! [X4: a,Y: a,Y2: a] :
( ( factor8216151070175719842xt_a_b @ r @ X4 @ Y )
=> ( ( associ5860276527279195403xt_a_b @ r @ Y @ Y2 )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ X4 @ Y2 ) ) ) ) ).
% divides_cong_r
thf(fact_818_associated__iff__same__ideal,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A3 @ B2 )
= ( ( cgenid547466209912283029xt_a_b @ r @ A3 )
= ( cgenid547466209912283029xt_a_b @ r @ B2 ) ) ) ) ) ).
% associated_iff_same_ideal
thf(fact_819_perm__assoc__switch__r,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Bs @ Cs )
=> ? [Bs2: list_a] :
( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Bs2 )
& ( ( mset_a @ Bs2 )
= ( mset_a @ Cs ) ) ) ) ) ).
% perm_assoc_switch_r
thf(fact_820_perm__assoc__switch,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Bs )
=> ( ( ( mset_a @ Bs )
= ( mset_a @ Cs ) )
=> ? [Bs2: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs2 ) )
& ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Bs2 @ Cs ) ) ) ) ).
% perm_assoc_switch
thf(fact_821_assoc__iff__assoc__mult,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A3 @ B2 )
= ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) ) ) ) ).
% assoc_iff_assoc_mult
thf(fact_822_mult__of_Ogcd__closed,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( 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 ) @ A3 @ B2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.gcd_closed
thf(fact_823_mult__of_Ogcd__exists,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( 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 ) @ A3 @ B2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.gcd_exists
thf(fact_824_essentially__equalI,axiom,
! [Fs1: list_a,Fs12: list_a,Fs22: list_a] :
( ( ( mset_a @ Fs1 )
= ( mset_a @ Fs12 ) )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Fs12 @ Fs22 )
=> ( essent8953798148185448568xt_a_b @ r @ Fs1 @ Fs22 ) ) ) ).
% essentially_equalI
thf(fact_825_essentially__equalE,axiom,
! [Fs1: list_a,Fs22: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ Fs1 @ Fs22 )
=> ~ ! [Fs13: list_a] :
( ( ( mset_a @ Fs1 )
= ( mset_a @ Fs13 ) )
=> ~ ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Fs13 @ Fs22 ) ) ) ).
% essentially_equalE
thf(fact_826_listassoc__trans,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Bs )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Bs @ Cs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Cs ) ) ) ) ) ) ).
% listassoc_trans
thf(fact_827_listassoc__sym,axiom,
! [As: list_a,Bs: list_a] :
( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ Bs @ As ) ) ) ) ).
% listassoc_sym
thf(fact_828_wfactors__cong__r,axiom,
! [Fs: list_a,A3: a,A9: a] :
( ( wfacto3557276942076956612xt_a_b @ r @ Fs @ A3 )
=> ( ( associ5860276527279195403xt_a_b @ r @ A3 @ A9 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A9 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( wfacto3557276942076956612xt_a_b @ r @ Fs @ A9 ) ) ) ) ) ) ).
% wfactors_cong_r
thf(fact_829_mult__of_Ogcd__assoc,axiom,
! [A3: a,B2: a,C: a] :
( ( member_a @ A3 @ ( 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 ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) @ C ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ B2 @ C ) ) ) ) ) ) ).
% mult_of.gcd_assoc
thf(fact_830_mult__of_Ogcd__cong__l,axiom,
! [X4: a,X5: a,Y: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ X4 @ X5 )
=> ( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ X5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X4 @ Y ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X5 @ Y ) ) ) ) ) ) ).
% mult_of.gcd_cong_l
thf(fact_831_mult__of_Ogcd__cong__r,axiom,
! [Y: a,Y2: a,X4: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ Y @ Y2 )
=> ( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X4 @ Y ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X4 @ Y2 ) ) ) ) ) ) ).
% mult_of.gcd_cong_r
thf(fact_832_mult__of_Operm__assoc__switch__r,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ Bs @ Cs )
=> ? [Bs2: list_a] :
( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ As @ Bs2 )
& ( ( mset_a @ Bs2 )
= ( mset_a @ Cs ) ) ) ) ) ).
% mult_of.perm_assoc_switch_r
thf(fact_833_mult__of_Operm__assoc__switch,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ As @ Bs )
=> ( ( ( mset_a @ Bs )
= ( mset_a @ Cs ) )
=> ? [Bs2: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs2 ) )
& ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ Bs2 @ Cs ) ) ) ) ).
% mult_of.perm_assoc_switch
thf(fact_834_mult__of_Ogcd__divides,axiom,
! [Z: a,X4: a,Y: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Z @ X4 )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Z @ Y )
=> ( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Z @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X4 @ Y ) ) ) ) ) ) ) ).
% mult_of.gcd_divides
thf(fact_835_mult__of_Ogcd__divides__l,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( 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 ) @ A3 @ B2 ) @ A3 ) ) ) ).
% mult_of.gcd_divides_l
thf(fact_836_mult__of_Ogcd__divides__r,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( 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 ) @ A3 @ B2 ) @ B2 ) ) ) ).
% mult_of.gcd_divides_r
thf(fact_837_mult__of_Ogcd__isgcd,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( 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 ) @ A3 @ B2 ) @ A3 @ B2 ) ) ) ).
% mult_of.gcd_isgcd
thf(fact_838_mult__of_Ogcd__mult,axiom,
! [A3: a,B2: a,C: a] :
( ( member_a @ A3 @ ( 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 ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ A3 ) @ ( mult_a_ring_ext_a_b @ r @ C @ B2 ) ) ) ) ) ) ).
% mult_of.gcd_mult
thf(fact_839_perm__wfactorsD,axiom,
! [As: list_a,Bs: list_a,A3: a,B2: a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( wfacto3557276942076956612xt_a_b @ r @ As @ A3 )
=> ( ( wfacto3557276942076956612xt_a_b @ r @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A3 @ B2 ) ) ) ) ) ) ) ).
% perm_wfactorsD
thf(fact_840_mult__of_OgcdI,axiom,
! [A3: a,B2: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ C )
=> ( ! [Y4: a] :
( ( member_a @ Y4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Y4 @ B2 )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Y4 @ C )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Y4 @ A3 ) ) ) )
=> ( ( member_a @ A3 @ ( 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 ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ B2 @ C ) ) ) ) ) ) ) ) ).
% mult_of.gcdI
thf(fact_841_mult__of_Oessentially__equalI,axiom,
! [Fs1: list_a,Fs12: list_a,Fs22: list_a] :
( ( ( mset_a @ Fs1 )
= ( mset_a @ Fs12 ) )
=> ( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ Fs12 @ Fs22 )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Fs1 @ Fs22 ) ) ) ).
% mult_of.essentially_equalI
thf(fact_842_mult__of_Oessentially__equalE,axiom,
! [Fs1: list_a,Fs22: list_a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Fs1 @ Fs22 )
=> ~ ! [Fs13: list_a] :
( ( ( mset_a @ Fs1 )
= ( mset_a @ Fs13 ) )
=> ~ ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ Fs13 @ Fs22 ) ) ) ).
% mult_of.essentially_equalE
thf(fact_843_wfactors__factors,axiom,
! [As: list_a,A3: a] :
( ( wfacto3557276942076956612xt_a_b @ r @ As @ A3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [A8: a] :
( ( factor5638265376665762323xt_a_b @ r @ As @ A8 )
& ( associ5860276527279195403xt_a_b @ r @ A8 @ A3 ) ) ) ) ).
% wfactors_factors
thf(fact_844_mult__of_Olistassoc__sym,axiom,
! [As: list_a,Bs: list_a] :
( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ Bs @ As ) ) ) ) ).
% mult_of.listassoc_sym
thf(fact_845_mult__of_Olistassoc__trans,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ As @ Bs )
=> ( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ Bs @ Cs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ As @ Cs ) ) ) ) ) ) ).
% mult_of.listassoc_trans
thf(fact_846_list__all2__Nil2,axiom,
! [P2: list_a > a > $o,Xs: list_list_a] :
( ( list_all2_list_a_a @ P2 @ Xs @ nil_a )
= ( Xs = nil_list_a ) ) ).
% list_all2_Nil2
thf(fact_847_list__all2__Nil2,axiom,
! [P2: a > list_a > $o,Xs: list_a] :
( ( list_all2_a_list_a @ P2 @ Xs @ nil_list_a )
= ( Xs = nil_a ) ) ).
% list_all2_Nil2
thf(fact_848_list__all2__Nil2,axiom,
! [P2: list_a > list_a > $o,Xs: list_list_a] :
( ( list_a3802133873445908231list_a @ P2 @ Xs @ nil_list_a )
= ( Xs = nil_list_a ) ) ).
% list_all2_Nil2
thf(fact_849_list__all2__Nil2,axiom,
! [P2: a > a > $o,Xs: list_a] :
( ( list_all2_a_a @ P2 @ Xs @ nil_a )
= ( Xs = nil_a ) ) ).
% list_all2_Nil2
thf(fact_850_list__all2__Nil,axiom,
! [P2: a > list_a > $o,Ys: list_list_a] :
( ( list_all2_a_list_a @ P2 @ nil_a @ Ys )
= ( Ys = nil_list_a ) ) ).
% list_all2_Nil
thf(fact_851_list__all2__Nil,axiom,
! [P2: list_a > a > $o,Ys: list_a] :
( ( list_all2_list_a_a @ P2 @ nil_list_a @ Ys )
= ( Ys = nil_a ) ) ).
% list_all2_Nil
thf(fact_852_list__all2__Nil,axiom,
! [P2: list_a > list_a > $o,Ys: list_list_a] :
( ( list_a3802133873445908231list_a @ P2 @ nil_list_a @ Ys )
= ( Ys = nil_list_a ) ) ).
% list_all2_Nil
thf(fact_853_list__all2__Nil,axiom,
! [P2: a > a > $o,Ys: list_a] :
( ( list_all2_a_a @ P2 @ nil_a @ Ys )
= ( Ys = nil_a ) ) ).
% list_all2_Nil
thf(fact_854_mult__of_Olistassoc__wfactorsD,axiom,
! [As: list_a,Bs: list_a,A3: a,B2: a] :
( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ As @ Bs )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% mult_of.listassoc_wfactorsD
thf(fact_855_mult__of_Owfactors__listassoc__cong__l,axiom,
! [Fs: list_a,A3: a,Fs3: list_a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A3 )
=> ( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ Fs @ Fs3 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs3 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs3 @ A3 ) ) ) ) ) ) ).
% mult_of.wfactors_listassoc_cong_l
thf(fact_856_mult__of_Oirrlist__listassoc__cong,axiom,
! [As: list_a,Bs: list_a] :
( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ As ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ X ) )
=> ( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Bs ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ X3 ) ) ) ) ) ) ).
% mult_of.irrlist_listassoc_cong
thf(fact_857_associated__refl,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A3 @ A3 ) ) ).
% associated_refl
thf(fact_858_listassoc__refl,axiom,
! [As: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ r ) @ As @ As ) ) ).
% listassoc_refl
thf(fact_859_mult__of_Olistassoc__refl,axiom,
! [As: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ As @ As ) ) ).
% mult_of.listassoc_refl
thf(fact_860_list__all2__lengthD,axiom,
! [P2: a > a > $o,Xs: list_a,Ys: list_a] :
( ( list_all2_a_a @ P2 @ Xs @ Ys )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_861_list__all2__same,axiom,
! [P2: a > a > $o,Xs: list_a] :
( ( list_all2_a_a @ P2 @ Xs @ Xs )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( P2 @ X2 @ X2 ) ) ) ) ).
% list_all2_same
thf(fact_862_list_Orel__refl__strong,axiom,
! [X4: list_nat,Ra: nat > nat > $o] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ X4 ) )
=> ( Ra @ Z2 @ Z2 ) )
=> ( list_all2_nat_nat @ Ra @ X4 @ X4 ) ) ).
% list.rel_refl_strong
thf(fact_863_list_Orel__refl__strong,axiom,
! [X4: list_set_a,Ra: set_a > set_a > $o] :
( ! [Z2: set_a] :
( ( member_set_a @ Z2 @ ( set_set_a2 @ X4 ) )
=> ( Ra @ Z2 @ Z2 ) )
=> ( list_a5961261016436360967_set_a @ Ra @ X4 @ X4 ) ) ).
% list.rel_refl_strong
thf(fact_864_list_Orel__refl__strong,axiom,
! [X4: list_a,Ra: a > a > $o] :
( ! [Z2: a] :
( ( member_a @ Z2 @ ( set_a2 @ X4 ) )
=> ( Ra @ Z2 @ Z2 ) )
=> ( list_all2_a_a @ Ra @ X4 @ X4 ) ) ).
% list.rel_refl_strong
thf(fact_865_list_Orel__mono__strong,axiom,
! [R: nat > nat > $o,X4: list_nat,Y: list_nat,Ra: nat > nat > $o] :
( ( list_all2_nat_nat @ R @ X4 @ Y )
=> ( ! [Z2: nat,Yb: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ X4 ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Y ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( list_all2_nat_nat @ Ra @ X4 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_866_list_Orel__mono__strong,axiom,
! [R: nat > set_a > $o,X4: list_nat,Y: list_set_a,Ra: nat > set_a > $o] :
( ( list_all2_nat_set_a @ R @ X4 @ Y )
=> ( ! [Z2: nat,Yb: set_a] :
( ( member_nat @ Z2 @ ( set_nat2 @ X4 ) )
=> ( ( member_set_a @ Yb @ ( set_set_a2 @ Y ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( list_all2_nat_set_a @ Ra @ X4 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_867_list_Orel__mono__strong,axiom,
! [R: set_a > nat > $o,X4: list_set_a,Y: list_nat,Ra: set_a > nat > $o] :
( ( list_all2_set_a_nat @ R @ X4 @ Y )
=> ( ! [Z2: set_a,Yb: nat] :
( ( member_set_a @ Z2 @ ( set_set_a2 @ X4 ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Y ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( list_all2_set_a_nat @ Ra @ X4 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_868_list_Orel__mono__strong,axiom,
! [R: set_a > set_a > $o,X4: list_set_a,Y: list_set_a,Ra: set_a > set_a > $o] :
( ( list_a5961261016436360967_set_a @ R @ X4 @ Y )
=> ( ! [Z2: set_a,Yb: set_a] :
( ( member_set_a @ Z2 @ ( set_set_a2 @ X4 ) )
=> ( ( member_set_a @ Yb @ ( set_set_a2 @ Y ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( list_a5961261016436360967_set_a @ Ra @ X4 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_869_list_Orel__mono__strong,axiom,
! [R: nat > a > $o,X4: list_nat,Y: list_a,Ra: nat > a > $o] :
( ( list_all2_nat_a @ R @ X4 @ Y )
=> ( ! [Z2: nat,Yb: a] :
( ( member_nat @ Z2 @ ( set_nat2 @ X4 ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Y ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( list_all2_nat_a @ Ra @ X4 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_870_list_Orel__mono__strong,axiom,
! [R: set_a > a > $o,X4: list_set_a,Y: list_a,Ra: set_a > a > $o] :
( ( list_all2_set_a_a @ R @ X4 @ Y )
=> ( ! [Z2: set_a,Yb: a] :
( ( member_set_a @ Z2 @ ( set_set_a2 @ X4 ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Y ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( list_all2_set_a_a @ Ra @ X4 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_871_list_Orel__mono__strong,axiom,
! [R: a > nat > $o,X4: list_a,Y: list_nat,Ra: a > nat > $o] :
( ( list_all2_a_nat @ R @ X4 @ Y )
=> ( ! [Z2: a,Yb: nat] :
( ( member_a @ Z2 @ ( set_a2 @ X4 ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Y ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( list_all2_a_nat @ Ra @ X4 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_872_list_Orel__mono__strong,axiom,
! [R: a > set_a > $o,X4: list_a,Y: list_set_a,Ra: a > set_a > $o] :
( ( list_all2_a_set_a @ R @ X4 @ Y )
=> ( ! [Z2: a,Yb: set_a] :
( ( member_a @ Z2 @ ( set_a2 @ X4 ) )
=> ( ( member_set_a @ Yb @ ( set_set_a2 @ Y ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( list_all2_a_set_a @ Ra @ X4 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_873_list_Orel__mono__strong,axiom,
! [R: a > a > $o,X4: list_a,Y: list_a,Ra: a > a > $o] :
( ( list_all2_a_a @ R @ X4 @ Y )
=> ( ! [Z2: a,Yb: a] :
( ( member_a @ Z2 @ ( set_a2 @ X4 ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Y ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( list_all2_a_a @ Ra @ X4 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_874_list_Orel__cong,axiom,
! [X4: list_nat,Ya: list_nat,Y: list_nat,Xa2: list_nat,R: nat > nat > $o,Ra: nat > nat > $o] :
( ( X4 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z2: nat,Yb: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ Ya ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Xa2 ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( list_all2_nat_nat @ R @ X4 @ Y )
= ( list_all2_nat_nat @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_875_list_Orel__cong,axiom,
! [X4: list_nat,Ya: list_nat,Y: list_set_a,Xa2: list_set_a,R: nat > set_a > $o,Ra: nat > set_a > $o] :
( ( X4 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z2: nat,Yb: set_a] :
( ( member_nat @ Z2 @ ( set_nat2 @ Ya ) )
=> ( ( member_set_a @ Yb @ ( set_set_a2 @ Xa2 ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( list_all2_nat_set_a @ R @ X4 @ Y )
= ( list_all2_nat_set_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_876_list_Orel__cong,axiom,
! [X4: list_set_a,Ya: list_set_a,Y: list_nat,Xa2: list_nat,R: set_a > nat > $o,Ra: set_a > nat > $o] :
( ( X4 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z2: set_a,Yb: nat] :
( ( member_set_a @ Z2 @ ( set_set_a2 @ Ya ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Xa2 ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( list_all2_set_a_nat @ R @ X4 @ Y )
= ( list_all2_set_a_nat @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_877_list_Orel__cong,axiom,
! [X4: list_set_a,Ya: list_set_a,Y: list_set_a,Xa2: list_set_a,R: set_a > set_a > $o,Ra: set_a > set_a > $o] :
( ( X4 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z2: set_a,Yb: set_a] :
( ( member_set_a @ Z2 @ ( set_set_a2 @ Ya ) )
=> ( ( member_set_a @ Yb @ ( set_set_a2 @ Xa2 ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( list_a5961261016436360967_set_a @ R @ X4 @ Y )
= ( list_a5961261016436360967_set_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_878_list_Orel__cong,axiom,
! [X4: list_nat,Ya: list_nat,Y: list_a,Xa2: list_a,R: nat > a > $o,Ra: nat > a > $o] :
( ( X4 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z2: nat,Yb: a] :
( ( member_nat @ Z2 @ ( set_nat2 @ Ya ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Xa2 ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( list_all2_nat_a @ R @ X4 @ Y )
= ( list_all2_nat_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_879_list_Orel__cong,axiom,
! [X4: list_set_a,Ya: list_set_a,Y: list_a,Xa2: list_a,R: set_a > a > $o,Ra: set_a > a > $o] :
( ( X4 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z2: set_a,Yb: a] :
( ( member_set_a @ Z2 @ ( set_set_a2 @ Ya ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Xa2 ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( list_all2_set_a_a @ R @ X4 @ Y )
= ( list_all2_set_a_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_880_list_Orel__cong,axiom,
! [X4: list_a,Ya: list_a,Y: list_nat,Xa2: list_nat,R: a > nat > $o,Ra: a > nat > $o] :
( ( X4 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z2: a,Yb: nat] :
( ( member_a @ Z2 @ ( set_a2 @ Ya ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Xa2 ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( list_all2_a_nat @ R @ X4 @ Y )
= ( list_all2_a_nat @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_881_list_Orel__cong,axiom,
! [X4: list_a,Ya: list_a,Y: list_set_a,Xa2: list_set_a,R: a > set_a > $o,Ra: a > set_a > $o] :
( ( X4 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z2: a,Yb: set_a] :
( ( member_a @ Z2 @ ( set_a2 @ Ya ) )
=> ( ( member_set_a @ Yb @ ( set_set_a2 @ Xa2 ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( list_all2_a_set_a @ R @ X4 @ Y )
= ( list_all2_a_set_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_882_list_Orel__cong,axiom,
! [X4: list_a,Ya: list_a,Y: list_a,Xa2: list_a,R: a > a > $o,Ra: a > a > $o] :
( ( X4 = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z2: a,Yb: a] :
( ( member_a @ Z2 @ ( set_a2 @ Ya ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Xa2 ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( list_all2_a_a @ R @ X4 @ Y )
= ( list_all2_a_a @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_883_list_Orel__eq,axiom,
( ( list_all2_a_a
@ ^ [Y5: a,Z3: a] : ( Y5 = Z3 ) )
= ( ^ [Y5: list_a,Z3: list_a] : ( Y5 = Z3 ) ) ) ).
% list.rel_eq
thf(fact_884_list_Orel__refl,axiom,
! [Ra: a > a > $o,X4: list_a] :
( ! [X: a] : ( Ra @ X @ X )
=> ( list_all2_a_a @ Ra @ X4 @ X4 ) ) ).
% list.rel_refl
thf(fact_885_list__all2__eq,axiom,
( ( ^ [Y5: list_a,Z3: list_a] : ( Y5 = Z3 ) )
= ( list_all2_a_a
@ ^ [Y5: a,Z3: a] : ( Y5 = Z3 ) ) ) ).
% list_all2_eq
thf(fact_886_list__all2__mono,axiom,
! [P2: a > a > $o,Xs: list_a,Ys: list_a,Q: a > a > $o] :
( ( list_all2_a_a @ P2 @ Xs @ Ys )
=> ( ! [Xs2: a,Ys3: a] :
( ( P2 @ Xs2 @ Ys3 )
=> ( Q @ Xs2 @ Ys3 ) )
=> ( list_all2_a_a @ Q @ Xs @ Ys ) ) ) ).
% list_all2_mono
thf(fact_887_list__all2__refl,axiom,
! [P2: a > a > $o,Xs: list_a] :
( ! [X: a] : ( P2 @ X @ X )
=> ( list_all2_a_a @ P2 @ Xs @ Xs ) ) ).
% list_all2_refl
thf(fact_888_list__all2__trans,axiom,
! [P1: a > a > $o,P22: a > a > $o,P32: a > a > $o,As: list_a,Bs: list_a,Cs: list_a] :
( ! [A2: a,B3: a,C2: a] :
( ( P1 @ A2 @ B3 )
=> ( ( P22 @ B3 @ C2 )
=> ( P32 @ A2 @ C2 ) ) )
=> ( ( list_all2_a_a @ P1 @ As @ Bs )
=> ( ( list_all2_a_a @ P22 @ Bs @ Cs )
=> ( list_all2_a_a @ P32 @ As @ Cs ) ) ) ) ).
% list_all2_trans
thf(fact_889_list__all2__antisym,axiom,
! [P2: a > a > $o,Q: a > a > $o,Xs: list_a,Ys: list_a] :
( ! [X: a,Y4: a] :
( ( P2 @ X @ Y4 )
=> ( ( Q @ Y4 @ X )
=> ( X = Y4 ) ) )
=> ( ( list_all2_a_a @ P2 @ Xs @ Ys )
=> ( ( list_all2_a_a @ Q @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ) ).
% list_all2_antisym
thf(fact_890_list__all2__appendI,axiom,
! [P2: a > a > $o,A3: list_a,B2: list_a,C: list_a,D2: list_a] :
( ( list_all2_a_a @ P2 @ A3 @ B2 )
=> ( ( list_all2_a_a @ P2 @ C @ D2 )
=> ( list_all2_a_a @ P2 @ ( append_a @ A3 @ C ) @ ( append_a @ B2 @ D2 ) ) ) ) ).
% list_all2_appendI
thf(fact_891_list_Orel__inject_I2_J,axiom,
! [R: a > list_a > $o,X21: a,X22: list_a,Y21: list_a,Y22: list_list_a] :
( ( list_all2_a_list_a @ R @ ( cons_a @ X21 @ X22 ) @ ( cons_list_a @ Y21 @ Y22 ) )
= ( ( R @ X21 @ Y21 )
& ( list_all2_a_list_a @ R @ X22 @ Y22 ) ) ) ).
% list.rel_inject(2)
thf(fact_892_list_Orel__inject_I2_J,axiom,
! [R: list_a > a > $o,X21: list_a,X22: list_list_a,Y21: a,Y22: list_a] :
( ( list_all2_list_a_a @ R @ ( cons_list_a @ X21 @ X22 ) @ ( cons_a @ Y21 @ Y22 ) )
= ( ( R @ X21 @ Y21 )
& ( list_all2_list_a_a @ R @ X22 @ Y22 ) ) ) ).
% list.rel_inject(2)
thf(fact_893_list_Orel__inject_I2_J,axiom,
! [R: list_a > list_a > $o,X21: list_a,X22: list_list_a,Y21: list_a,Y22: list_list_a] :
( ( list_a3802133873445908231list_a @ R @ ( cons_list_a @ X21 @ X22 ) @ ( cons_list_a @ Y21 @ Y22 ) )
= ( ( R @ X21 @ Y21 )
& ( list_a3802133873445908231list_a @ R @ X22 @ Y22 ) ) ) ).
% list.rel_inject(2)
thf(fact_894_list_Orel__inject_I2_J,axiom,
! [R: a > a > $o,X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( list_all2_a_a @ R @ ( cons_a @ X21 @ X22 ) @ ( cons_a @ Y21 @ Y22 ) )
= ( ( R @ X21 @ Y21 )
& ( list_all2_a_a @ R @ X22 @ Y22 ) ) ) ).
% list.rel_inject(2)
thf(fact_895_list_Orel__intros_I2_J,axiom,
! [R: a > list_a > $o,X21: a,Y21: list_a,X22: list_a,Y22: list_list_a] :
( ( R @ X21 @ Y21 )
=> ( ( list_all2_a_list_a @ R @ X22 @ Y22 )
=> ( list_all2_a_list_a @ R @ ( cons_a @ X21 @ X22 ) @ ( cons_list_a @ Y21 @ Y22 ) ) ) ) ).
% list.rel_intros(2)
thf(fact_896_list_Orel__intros_I2_J,axiom,
! [R: list_a > a > $o,X21: list_a,Y21: a,X22: list_list_a,Y22: list_a] :
( ( R @ X21 @ Y21 )
=> ( ( list_all2_list_a_a @ R @ X22 @ Y22 )
=> ( list_all2_list_a_a @ R @ ( cons_list_a @ X21 @ X22 ) @ ( cons_a @ Y21 @ Y22 ) ) ) ) ).
% list.rel_intros(2)
thf(fact_897_list_Orel__intros_I2_J,axiom,
! [R: list_a > list_a > $o,X21: list_a,Y21: list_a,X22: list_list_a,Y22: list_list_a] :
( ( R @ X21 @ Y21 )
=> ( ( list_a3802133873445908231list_a @ R @ X22 @ Y22 )
=> ( list_a3802133873445908231list_a @ R @ ( cons_list_a @ X21 @ X22 ) @ ( cons_list_a @ Y21 @ Y22 ) ) ) ) ).
% list.rel_intros(2)
thf(fact_898_list_Orel__intros_I2_J,axiom,
! [R: a > a > $o,X21: a,Y21: a,X22: list_a,Y22: list_a] :
( ( R @ X21 @ Y21 )
=> ( ( list_all2_a_a @ R @ X22 @ Y22 )
=> ( list_all2_a_a @ R @ ( cons_a @ X21 @ X22 ) @ ( cons_a @ Y21 @ Y22 ) ) ) ) ).
% list.rel_intros(2)
thf(fact_899_list__all2__Cons,axiom,
! [P2: a > list_a > $o,X4: a,Xs: list_a,Y: list_a,Ys: list_list_a] :
( ( list_all2_a_list_a @ P2 @ ( cons_a @ X4 @ Xs ) @ ( cons_list_a @ Y @ Ys ) )
= ( ( P2 @ X4 @ Y )
& ( list_all2_a_list_a @ P2 @ Xs @ Ys ) ) ) ).
% list_all2_Cons
thf(fact_900_list__all2__Cons,axiom,
! [P2: list_a > a > $o,X4: list_a,Xs: list_list_a,Y: a,Ys: list_a] :
( ( list_all2_list_a_a @ P2 @ ( cons_list_a @ X4 @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( ( P2 @ X4 @ Y )
& ( list_all2_list_a_a @ P2 @ Xs @ Ys ) ) ) ).
% list_all2_Cons
thf(fact_901_list__all2__Cons,axiom,
! [P2: list_a > list_a > $o,X4: list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a] :
( ( list_a3802133873445908231list_a @ P2 @ ( cons_list_a @ X4 @ Xs ) @ ( cons_list_a @ Y @ Ys ) )
= ( ( P2 @ X4 @ Y )
& ( list_a3802133873445908231list_a @ P2 @ Xs @ Ys ) ) ) ).
% list_all2_Cons
thf(fact_902_list__all2__Cons,axiom,
! [P2: a > a > $o,X4: a,Xs: list_a,Y: a,Ys: list_a] :
( ( list_all2_a_a @ P2 @ ( cons_a @ X4 @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( ( P2 @ X4 @ Y )
& ( list_all2_a_a @ P2 @ Xs @ Ys ) ) ) ).
% list_all2_Cons
thf(fact_903_list__all2__Cons1,axiom,
! [P2: a > list_a > $o,X4: a,Xs: list_a,Ys: list_list_a] :
( ( list_all2_a_list_a @ P2 @ ( cons_a @ X4 @ Xs ) @ Ys )
= ( ? [Z4: list_a,Zs2: list_list_a] :
( ( Ys
= ( cons_list_a @ Z4 @ Zs2 ) )
& ( P2 @ X4 @ Z4 )
& ( list_all2_a_list_a @ P2 @ Xs @ Zs2 ) ) ) ) ).
% list_all2_Cons1
thf(fact_904_list__all2__Cons1,axiom,
! [P2: list_a > a > $o,X4: list_a,Xs: list_list_a,Ys: list_a] :
( ( list_all2_list_a_a @ P2 @ ( cons_list_a @ X4 @ Xs ) @ Ys )
= ( ? [Z4: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z4 @ Zs2 ) )
& ( P2 @ X4 @ Z4 )
& ( list_all2_list_a_a @ P2 @ Xs @ Zs2 ) ) ) ) ).
% list_all2_Cons1
thf(fact_905_list__all2__Cons1,axiom,
! [P2: list_a > list_a > $o,X4: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( list_a3802133873445908231list_a @ P2 @ ( cons_list_a @ X4 @ Xs ) @ Ys )
= ( ? [Z4: list_a,Zs2: list_list_a] :
( ( Ys
= ( cons_list_a @ Z4 @ Zs2 ) )
& ( P2 @ X4 @ Z4 )
& ( list_a3802133873445908231list_a @ P2 @ Xs @ Zs2 ) ) ) ) ).
% list_all2_Cons1
thf(fact_906_list__all2__Cons1,axiom,
! [P2: a > a > $o,X4: a,Xs: list_a,Ys: list_a] :
( ( list_all2_a_a @ P2 @ ( cons_a @ X4 @ Xs ) @ Ys )
= ( ? [Z4: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z4 @ Zs2 ) )
& ( P2 @ X4 @ Z4 )
& ( list_all2_a_a @ P2 @ Xs @ Zs2 ) ) ) ) ).
% list_all2_Cons1
thf(fact_907_list__all2__Cons2,axiom,
! [P2: list_a > a > $o,Xs: list_list_a,Y: a,Ys: list_a] :
( ( list_all2_list_a_a @ P2 @ Xs @ ( cons_a @ Y @ Ys ) )
= ( ? [Z4: list_a,Zs2: list_list_a] :
( ( Xs
= ( cons_list_a @ Z4 @ Zs2 ) )
& ( P2 @ Z4 @ Y )
& ( list_all2_list_a_a @ P2 @ Zs2 @ Ys ) ) ) ) ).
% list_all2_Cons2
thf(fact_908_list__all2__Cons2,axiom,
! [P2: a > list_a > $o,Xs: list_a,Y: list_a,Ys: list_list_a] :
( ( list_all2_a_list_a @ P2 @ Xs @ ( cons_list_a @ Y @ Ys ) )
= ( ? [Z4: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z4 @ Zs2 ) )
& ( P2 @ Z4 @ Y )
& ( list_all2_a_list_a @ P2 @ Zs2 @ Ys ) ) ) ) ).
% list_all2_Cons2
thf(fact_909_list__all2__Cons2,axiom,
! [P2: list_a > list_a > $o,Xs: list_list_a,Y: list_a,Ys: list_list_a] :
( ( list_a3802133873445908231list_a @ P2 @ Xs @ ( cons_list_a @ Y @ Ys ) )
= ( ? [Z4: list_a,Zs2: list_list_a] :
( ( Xs
= ( cons_list_a @ Z4 @ Zs2 ) )
& ( P2 @ Z4 @ Y )
& ( list_a3802133873445908231list_a @ P2 @ Zs2 @ Ys ) ) ) ) ).
% list_all2_Cons2
thf(fact_910_list__all2__Cons2,axiom,
! [P2: a > a > $o,Xs: list_a,Y: a,Ys: list_a] :
( ( list_all2_a_a @ P2 @ Xs @ ( cons_a @ Y @ Ys ) )
= ( ? [Z4: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z4 @ Zs2 ) )
& ( P2 @ Z4 @ Y )
& ( list_all2_a_a @ P2 @ Zs2 @ Ys ) ) ) ) ).
% list_all2_Cons2
thf(fact_911_list_Octr__transfer_I1_J,axiom,
! [R: a > list_a > $o] : ( list_all2_a_list_a @ R @ nil_a @ nil_list_a ) ).
% list.ctr_transfer(1)
thf(fact_912_list_Octr__transfer_I1_J,axiom,
! [R: list_a > a > $o] : ( list_all2_list_a_a @ R @ nil_list_a @ nil_a ) ).
% list.ctr_transfer(1)
thf(fact_913_list_Octr__transfer_I1_J,axiom,
! [R: list_a > list_a > $o] : ( list_a3802133873445908231list_a @ R @ nil_list_a @ nil_list_a ) ).
% list.ctr_transfer(1)
thf(fact_914_list_Octr__transfer_I1_J,axiom,
! [R: a > a > $o] : ( list_all2_a_a @ R @ nil_a @ nil_a ) ).
% list.ctr_transfer(1)
thf(fact_915_list_Orel__mono,axiom,
! [R: list_a > list_a > $o,Ra: list_a > list_a > $o] :
( ( ord_le5542992221119063950st_a_o @ R @ Ra )
=> ( ord_le3776173323681337614st_a_o @ ( list_a3802133873445908231list_a @ R ) @ ( list_a3802133873445908231list_a @ Ra ) ) ) ).
% list.rel_mono
thf(fact_916_list_Orel__mono,axiom,
! [R: a > a > $o,Ra: a > a > $o] :
( ( ord_less_eq_a_a_o @ R @ Ra )
=> ( ord_le5542992221119063950st_a_o @ ( list_all2_a_a @ R ) @ ( list_all2_a_a @ Ra ) ) ) ).
% list.rel_mono
thf(fact_917_list__all2__induct,axiom,
! [P2: list_a > list_a > $o,Xs: list_list_a,Ys: list_list_a,R: list_list_a > list_list_a > $o] :
( ( list_a3802133873445908231list_a @ P2 @ Xs @ Ys )
=> ( ( R @ nil_list_a @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y4: list_a,Ys3: list_list_a] :
( ( P2 @ X @ Y4 )
=> ( ( list_a3802133873445908231list_a @ P2 @ Xs2 @ Ys3 )
=> ( ( R @ Xs2 @ Ys3 )
=> ( R @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) ) ) ) )
=> ( R @ Xs @ Ys ) ) ) ) ).
% list_all2_induct
thf(fact_918_list__all2__induct,axiom,
! [P2: a > a > $o,Xs: list_a,Ys: list_a,R: list_a > list_a > $o] :
( ( list_all2_a_a @ P2 @ Xs @ Ys )
=> ( ( R @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y4: a,Ys3: list_a] :
( ( P2 @ X @ Y4 )
=> ( ( list_all2_a_a @ P2 @ Xs2 @ Ys3 )
=> ( ( R @ Xs2 @ Ys3 )
=> ( R @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) ) ) ) )
=> ( R @ Xs @ Ys ) ) ) ) ).
% list_all2_induct
thf(fact_919_list_Orel__induct,axiom,
! [R: a > a > $o,X4: list_a,Y: list_a,Q: list_a > list_a > $o] :
( ( list_all2_a_a @ R @ X4 @ Y )
=> ( ( Q @ nil_a @ nil_a )
=> ( ! [A21: a,A22: list_a,B21: a,B22: list_a] :
( ( R @ A21 @ B21 )
=> ( ( Q @ A22 @ B22 )
=> ( Q @ ( cons_a @ A21 @ A22 ) @ ( cons_a @ B21 @ B22 ) ) ) )
=> ( Q @ X4 @ Y ) ) ) ) ).
% list.rel_induct
thf(fact_920_list_Orel__cases,axiom,
! [R: a > a > $o,A3: list_a,B2: list_a] :
( ( list_all2_a_a @ R @ A3 @ B2 )
=> ( ( ( A3 = nil_a )
=> ( B2 != nil_a ) )
=> ~ ! [X1: a,X23: list_a] :
( ( A3
= ( cons_a @ X1 @ X23 ) )
=> ! [Y1: a,Y23: list_a] :
( ( B2
= ( cons_a @ Y1 @ Y23 ) )
=> ( ( R @ X1 @ Y1 )
=> ~ ( list_all2_a_a @ R @ X23 @ Y23 ) ) ) ) ) ) ).
% list.rel_cases
thf(fact_921_list_Orel__distinct_I1_J,axiom,
! [R: a > a > $o,Y21: a,Y22: list_a] :
~ ( list_all2_a_a @ R @ nil_a @ ( cons_a @ Y21 @ Y22 ) ) ).
% list.rel_distinct(1)
thf(fact_922_list_Orel__distinct_I2_J,axiom,
! [R: a > a > $o,Y21: a,Y22: list_a] :
~ ( list_all2_a_a @ R @ ( cons_a @ Y21 @ Y22 ) @ nil_a ) ).
% list.rel_distinct(2)
thf(fact_923_list__all2__append,axiom,
! [Xs: list_a,Ys: list_a,P2: a > a > $o,Us2: list_a,Vs2: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( list_all2_a_a @ P2 @ ( append_a @ Xs @ Us2 ) @ ( append_a @ Ys @ Vs2 ) )
= ( ( list_all2_a_a @ P2 @ Xs @ Ys )
& ( list_all2_a_a @ P2 @ Us2 @ Vs2 ) ) ) ) ).
% list_all2_append
thf(fact_924_list__all2__append1,axiom,
! [P2: a > a > $o,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( list_all2_a_a @ P2 @ ( append_a @ Xs @ Ys ) @ Zs )
= ( ? [Us3: list_a,Vs3: list_a] :
( ( Zs
= ( append_a @ Us3 @ Vs3 ) )
& ( ( size_size_list_a @ Us3 )
= ( size_size_list_a @ Xs ) )
& ( ( size_size_list_a @ Vs3 )
= ( size_size_list_a @ Ys ) )
& ( list_all2_a_a @ P2 @ Xs @ Us3 )
& ( list_all2_a_a @ P2 @ Ys @ Vs3 ) ) ) ) ).
% list_all2_append1
thf(fact_925_list__all2__append2,axiom,
! [P2: a > a > $o,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( list_all2_a_a @ P2 @ Xs @ ( append_a @ Ys @ Zs ) )
= ( ? [Us3: list_a,Vs3: list_a] :
( ( Xs
= ( append_a @ Us3 @ Vs3 ) )
& ( ( size_size_list_a @ Us3 )
= ( size_size_list_a @ Ys ) )
& ( ( size_size_list_a @ Vs3 )
= ( size_size_list_a @ Zs ) )
& ( list_all2_a_a @ P2 @ Us3 @ Ys )
& ( list_all2_a_a @ P2 @ Vs3 @ Zs ) ) ) ) ).
% list_all2_append2
thf(fact_926_essentially__equal__def,axiom,
( essent8953798148185448568xt_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,Fs14: list_a,Fs23: list_a] :
? [Fs15: list_a] :
( ( ( mset_a @ Fs14 )
= ( mset_a @ Fs15 ) )
& ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ G3 ) @ Fs15 @ Fs23 ) ) ) ) ).
% essentially_equal_def
thf(fact_927_essentially__equal__def,axiom,
( essent9005414202370111435t_unit
= ( ^ [G3: partia8223610829204095565t_unit,Fs14: list_a,Fs23: list_a] :
? [Fs15: list_a] :
( ( ( mset_a @ Fs14 )
= ( mset_a @ Fs15 ) )
& ( list_all2_a_a @ ( associ6879500422977059064t_unit @ G3 ) @ Fs15 @ Fs23 ) ) ) ) ).
% essentially_equal_def
thf(fact_928_domain_Oassoc__iff__assoc__mult,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( associ5860276527279195403xt_a_b @ R @ A3 @ B2 )
= ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ R ) @ A3 @ B2 ) ) ) ) ) ).
% domain.assoc_iff_assoc_mult
thf(fact_929_monoid__cancel_Oirrlist__listassoc__cong,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,Bs: list_a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ As ) )
=> ( irredu6211895646901577903xt_a_b @ G @ X ) )
=> ( ( list_all2_a_a @ ( associ5860276527279195403xt_a_b @ G ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Bs ) )
=> ( irredu6211895646901577903xt_a_b @ G @ X3 ) ) ) ) ) ) ) ).
% monoid_cancel.irrlist_listassoc_cong
thf(fact_930_monoid__cancel_Oirrlist__listassoc__cong,axiom,
! [G: partia8223610829204095565t_unit,As: list_a,Bs: list_a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ As ) )
=> ( irredu4023057619401689684t_unit @ G @ X ) )
=> ( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ G ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ G ) )
=> ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Bs ) )
=> ( irredu4023057619401689684t_unit @ G @ X3 ) ) ) ) ) ) ) ).
% monoid_cancel.irrlist_listassoc_cong
thf(fact_931_mult__of_Oee__factorsI,axiom,
! [A3: a,B2: a,As: list_a,Bs: list_a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ~ ( member_a @ A3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ~ ( member_a @ B2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs ) ) ) ) ) ) ) ) ).
% mult_of.ee_factorsI
thf(fact_932_mult__of_Ofmset__listassoc__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( list_all2_a_a @ ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ As )
= ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Bs ) ) ) ) ) ).
% mult_of.fmset_listassoc_cong
thf(fact_933_mult__of_Ofactors__unique,axiom,
! [Fs: list_a,A3: a,Fs3: list_a] :
( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A3 )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fs3 @ A3 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ~ ( member_a @ A3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs3 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ Fs3 ) ) ) ) ) ) ) ).
% mult_of.factors_unique
thf(fact_934_mult__of_OUnits__closed,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.Units_closed
thf(fact_935_mult__of_OUnits__assoc,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) ) ) ).
% mult_of.Units_assoc
thf(fact_936_mult__of_Ofmset__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ As )
= ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Bs ) ) ) ).
% mult_of.fmset_perm_cong
thf(fact_937_mult__of_Oprod__unit__l,axiom,
! [A3: a,B2: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A3 @ ( 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_938_mult__of_Oprod__unit__r,axiom,
! [A3: a,B2: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A3 @ 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 @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.prod_unit_r
thf(fact_939_mult__of_Ounit__factor,axiom,
! [A3: a,B2: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.unit_factor
thf(fact_940_mult__of_OUnits__inv__comm,axiom,
! [X4: a,Y: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X4 @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X4 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.Units_inv_comm
thf(fact_941_mult__of_Oassoc__unit__r,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( 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.assoc_unit_r
thf(fact_942_mult__of_Oassoc__unit__l,axiom,
! [A3: a,B2: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( member_a @ B2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.assoc_unit_l
thf(fact_943_mult__of_Ounit__divides,axiom,
! [U: a,A3: a] :
( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ U @ A3 ) ) ) ).
% mult_of.unit_divides
thf(fact_944_mult__of_Odivides__unit,axiom,
! [A3: a,U: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ U )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.divides_unit
thf(fact_945_mult__of_Ounit__wfactors,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ nil_a @ A3 ) ) ).
% mult_of.unit_wfactors
thf(fact_946_mult__of_OUnits__r__inv__ex,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X4 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.Units_r_inv_ex
thf(fact_947_mult__of_OUnits__l__inv__ex,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.Units_l_inv_ex
thf(fact_948_mult__of_OassociatedD2,axiom,
! [A3: a,B2: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( A3
= ( mult_a_ring_ext_a_b @ r @ B2 @ X ) ) ) ) ) ) ).
% mult_of.associatedD2
thf(fact_949_mult__of_OassociatedE2,axiom,
! [A3: a,B2: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ! [U2: a] :
( ( A3
= ( mult_a_ring_ext_a_b @ r @ B2 @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ~ ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.associatedE2
thf(fact_950_mult__of_OassociatedI2,axiom,
! [U: a,A3: a,B2: a] :
( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( A3
= ( mult_a_ring_ext_a_b @ r @ B2 @ U ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) ) ) ) ).
% mult_of.associatedI2
thf(fact_951_mult__of_OassociatedI2_H,axiom,
! [A3: a,B2: a,U: a] :
( ( A3
= ( mult_a_ring_ext_a_b @ r @ B2 @ U ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) ) ) ) ).
% mult_of.associatedI2'
thf(fact_952_mult__of_Oassociated__iff,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( A3
= ( mult_a_ring_ext_a_b @ r @ B2 @ X2 ) ) ) ) ) ) ) ).
% mult_of.associated_iff
thf(fact_953_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_954_mult__of_Oirreducible__prod__rI,axiom,
! [A3: a,B2: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A3 )
=> ( ( member_a @ B2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A3 @ ( 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 @ A3 @ B2 ) ) ) ) ) ) ).
% mult_of.irreducible_prod_rI
thf(fact_955_mult__of_Oirreducible__prod__lI,axiom,
! [B2: a,A3: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ B2 )
=> ( ( member_a @ A3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A3 @ ( 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 @ A3 @ B2 ) ) ) ) ) ) ).
% mult_of.irreducible_prod_lI
thf(fact_956_mult__of_Oirreducible__prodE,axiom,
! [A3: a,B2: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ ( 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 ) @ A3 )
=> ~ ( member_a @ B2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) )
=> ~ ( ( member_a @ A3 @ ( 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_957_mult__of_Ofactors__exist,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ~ ( member_a @ A3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [Fs4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs4 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fs4 @ A3 ) ) ) ) ).
% mult_of.factors_exist
thf(fact_958_mult__of_Oee__fmset,axiom,
! [As: list_a,Bs: list_a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ As )
= ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Bs ) ) ) ) ) ).
% mult_of.ee_fmset
thf(fact_959_mult__of_Oee__is__fmset,axiom,
! [As: list_a,Bs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
= ( ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ As )
= ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Bs ) ) ) ) ) ).
% mult_of.ee_is_fmset
thf(fact_960_mult__of_Ofmset__ee,axiom,
! [As: list_a,Bs: list_a] :
( ( ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ As )
= ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs ) ) ) ) ).
% mult_of.fmset_ee
thf(fact_961_mult__of_Ounit__wfactors__empty,axiom,
! [A3: a,Fs: list_a] :
( ( member_a @ A3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( Fs = nil_a ) ) ) ) ).
% mult_of.unit_wfactors_empty
thf(fact_962_mult__of_OUnits__m__closed,axiom,
! [X4: a,Y: a] :
( ( member_a @ X4 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X4 @ Y ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.Units_m_closed
thf(fact_963_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_964_mult__of_OUnits__l__cancel,axiom,
! [X4: a,Y: a,Z: a] :
( ( member_a @ X4 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X4 @ Y )
= ( mult_a_ring_ext_a_b @ r @ X4 @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% mult_of.Units_l_cancel
thf(fact_965_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ A3 @ B2 )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ B2 @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_966_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( member_a @ A3 @ ( units_a_Product_unit @ G ) )
=> ( ( associ6879500422977059064t_unit @ G @ A3 @ B2 )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ B2 @ ( units_a_Product_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_967_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A3 @ B2 )
=> ( ( member_a @ B2 @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ A3 @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_968_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A3 @ B2 )
=> ( ( member_a @ B2 @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ A3 @ ( units_a_Product_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_969_divides__irreducible__condition,axiom,
! [G: partia2175431115845679010xt_a_b,R2: a,A3: a] :
( ( irredu6211895646901577903xt_a_b @ G @ R2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ A3 @ R2 )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ G ) )
| ( associ5860276527279195403xt_a_b @ G @ A3 @ R2 ) ) ) ) ) ).
% divides_irreducible_condition
thf(fact_970_divides__irreducible__condition,axiom,
! [G: partia8223610829204095565t_unit,R2: a,A3: a] :
( ( irredu4023057619401689684t_unit @ G @ R2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ A3 @ R2 )
=> ( ( member_a @ A3 @ ( units_a_Product_unit @ G ) )
| ( associ6879500422977059064t_unit @ G @ A3 @ R2 ) ) ) ) ) ).
% divides_irreducible_condition
thf(fact_971_prime__def,axiom,
( prime_a_ring_ext_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,P5: a] :
( ~ ( member_a @ P5 @ ( units_a_ring_ext_a_b @ G3 ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( ( factor8216151070175719842xt_a_b @ G3 @ P5 @ ( mult_a_ring_ext_a_b @ G3 @ X2 @ Y3 ) )
=> ( ( factor8216151070175719842xt_a_b @ G3 @ P5 @ X2 )
| ( factor8216151070175719842xt_a_b @ G3 @ P5 @ Y3 ) ) ) ) ) ) ) ) ).
% prime_def
thf(fact_972_prime__def,axiom,
( prime_a_Product_unit
= ( ^ [G3: partia8223610829204095565t_unit,P5: a] :
( ~ ( member_a @ P5 @ ( units_a_Product_unit @ G3 ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ G3 ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G3 ) )
=> ( ( factor3040189038382604065t_unit @ G3 @ P5 @ ( mult_a_Product_unit @ G3 @ X2 @ Y3 ) )
=> ( ( factor3040189038382604065t_unit @ G3 @ P5 @ X2 )
| ( factor3040189038382604065t_unit @ G3 @ P5 @ Y3 ) ) ) ) ) ) ) ) ).
% prime_def
thf(fact_973_primeI,axiom,
! [P: a,G: partia2175431115845679010xt_a_b] :
( ~ ( member_a @ P @ ( units_a_ring_ext_a_b @ G ) )
=> ( ! [A2: a,B3: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ P @ ( mult_a_ring_ext_a_b @ G @ A2 @ B3 ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ P @ A2 )
| ( factor8216151070175719842xt_a_b @ G @ P @ B3 ) ) ) ) )
=> ( prime_a_ring_ext_a_b @ G @ P ) ) ) ).
% primeI
thf(fact_974_primeI,axiom,
! [P: a,G: partia8223610829204095565t_unit] :
( ~ ( member_a @ P @ ( units_a_Product_unit @ G ) )
=> ( ! [A2: a,B3: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ P @ ( mult_a_Product_unit @ G @ A2 @ B3 ) )
=> ( ( factor3040189038382604065t_unit @ G @ P @ A2 )
| ( factor3040189038382604065t_unit @ G @ P @ B3 ) ) ) ) )
=> ( prime_a_Product_unit @ G @ P ) ) ) ).
% primeI
thf(fact_975_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 ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ ( 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 @ X3 @ Xa ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ P @ X3 )
| ( factor8216151070175719842xt_a_b @ G @ P @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_976_primeE,axiom,
! [G: partia8223610829204095565t_unit,P: a] :
( ( prime_a_Product_unit @ G @ P )
=> ~ ( ~ ( member_a @ P @ ( units_a_Product_unit @ G ) )
=> ~ ! [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ G ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ P @ ( mult_a_Product_unit @ G @ X3 @ Xa ) )
=> ( ( factor3040189038382604065t_unit @ G @ P @ X3 )
| ( factor3040189038382604065t_unit @ G @ P @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_977_monoid__cancel_Oassociated__iff,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ A3 @ B2 )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ G ) )
& ( A3
= ( mult_a_ring_ext_a_b @ G @ B2 @ X2 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_978_monoid__cancel_Oassociated__iff,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( associ6879500422977059064t_unit @ G @ A3 @ B2 )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_Product_unit @ G ) )
& ( A3
= ( mult_a_Product_unit @ G @ B2 @ X2 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_979_monoid__cancel_OassociatedE2,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A3 @ B2 )
=> ( ! [U2: a] :
( ( A3
= ( mult_a_ring_ext_a_b @ G @ B2 @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_ring_ext_a_b @ G ) ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ~ ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_980_monoid__cancel_OassociatedE2,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A3 @ B2 )
=> ( ! [U2: a] :
( ( A3
= ( mult_a_Product_unit @ G @ B2 @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_Product_unit @ G ) ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ~ ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_981_monoid__cancel_OassociatedD2,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
& ( A3
= ( mult_a_ring_ext_a_b @ G @ B2 @ X ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_982_monoid__cancel_OassociatedD2,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ? [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ G ) )
& ( A3
= ( mult_a_Product_unit @ G @ B2 @ X ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_983_mult__of_Ofactorial__monoidI,axiom,
( ! [A2: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ~ ( member_a @ A2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [Fs5: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs5 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs5 @ A2 ) ) ) )
=> ( ! [A2: a,Fs4: list_a,Fs6: list_a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs4 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs6 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs4 @ A2 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs6 @ A2 )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Fs4 @ Fs6 ) ) ) ) ) )
=> ( factor2046533344642582127t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.factorial_monoidI
thf(fact_984_mult__of_Ofmset__wfactors__mult,axiom,
! [Cs: list_a,As: list_a,Bs: list_a,A3: a,B2: a,C: a] :
( ( ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Cs )
= ( plus_p2331992037799027419_set_a @ ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ As ) @ ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Bs ) ) )
=> ( ( member_a @ A3 @ ( 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 ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Cs @ C )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ C @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% mult_of.fmset_wfactors_mult
thf(fact_985_mult__of_Omult__factors__fmset,axiom,
! [As: list_a,A3: a,Bs: list_a,B2: a,Cs: list_a] :
( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Cs @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Cs )
= ( plus_p2331992037799027419_set_a @ ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ As ) @ ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Bs ) ) ) ) ) ) ) ) ) ).
% mult_of.mult_factors_fmset
thf(fact_986_Units__closed,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_987_Units__assoc,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A3 @ B2 ) ) ) ).
% Units_assoc
thf(fact_988_Units__pow__closed,axiom,
! [X4: a,D2: nat] :
( ( member_a @ X4 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X4 @ D2 ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_pow_closed
thf(fact_989_prod__unit__l,axiom,
! [A3: a,B2: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( 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_990_prod__unit__r,axiom,
! [A3: a,B2: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_991_unit__factor,axiom,
! [A3: a,B2: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_992_Units__inv__comm,axiom,
! [X4: a,Y: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X4 @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X4 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_inv_comm
thf(fact_993_Units__cong,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A3 @ B2 )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_cong
thf(fact_994_divides__unit,axiom,
! [A3: a,U: a] :
( ( factor8216151070175719842xt_a_b @ r @ A3 @ U )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% divides_unit
thf(fact_995_unit__divides,axiom,
! [U: a,A3: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ U @ A3 ) ) ) ).
% unit_divides
thf(fact_996_ideal__eq__carrier__iff,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( partia707051561876973205xt_a_b @ r )
= ( cgenid547466209912283029xt_a_b @ r @ A3 ) )
= ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ideal_eq_carrier_iff
thf(fact_997_ring__irreducibleE_I4_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_998_unit__wfactors,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( wfacto3557276942076956612xt_a_b @ r @ nil_a @ A3 ) ) ).
% unit_wfactors
thf(fact_999_Units__l__inv__ex,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_1000_Units__r__inv__ex,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X4 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_1001_ring__associated__iff,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A3 @ B2 )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
& ( A3
= ( mult_a_ring_ext_a_b @ r @ X2 @ B2 ) ) ) ) ) ) ) ).
% ring_associated_iff
thf(fact_1002_associatedI2_H,axiom,
! [A3: a,B2: a,U: a] :
( ( A3
= ( mult_a_ring_ext_a_b @ r @ B2 @ U ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A3 @ B2 ) ) ) ) ).
% associatedI2'
thf(fact_1003_associatedI2,axiom,
! [U: a,A3: a,B2: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( A3
= ( mult_a_ring_ext_a_b @ r @ B2 @ U ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A3 @ B2 ) ) ) ) ).
% associatedI2
thf(fact_1004_mult__of_Ofactorial__monoid__axioms,axiom,
factor2046533344642582127t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.factorial_monoid_axioms
thf(fact_1005_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_1006_divides__one,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A3 @ ( one_a_ring_ext_a_b @ r ) )
= ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% divides_one
thf(fact_1007_irreducible__prod__rI,axiom,
! [A3: a,B2: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A3 )
=> ( ( member_a @ B2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) ) ) ) ) ) ).
% irreducible_prod_rI
thf(fact_1008_irreducible__prod__lI,axiom,
! [B2: a,A3: a] :
( ( irredu6211895646901577903xt_a_b @ r @ B2 )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) ) ) ) ) ) ).
% irreducible_prod_lI
thf(fact_1009_ring__irreducibleE_I5_J,axiom,
! [R2: a,A3: a,B2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_1010_Units__m__closed,axiom,
! [X4: a,Y: a] :
( ( member_a @ X4 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X4 @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_1011_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_1012_mult__of_Omult__wfactors__fmset,axiom,
! [As: list_a,A3: a,Bs: list_a,B2: a,Cs: list_a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Cs @ ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Cs )
= ( plus_p2331992037799027419_set_a @ ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ As ) @ ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Bs ) ) ) ) ) ) ) ) ) ) ) ).
% mult_of.mult_wfactors_fmset
thf(fact_1013_Units__l__cancel,axiom,
! [X4: a,Y: a,Z: a] :
( ( member_a @ X4 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X4 @ Y )
= ( mult_a_ring_ext_a_b @ r @ X4 @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% Units_l_cancel
thf(fact_1014_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_1015_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_1016_ring_Ofinite__ring__finite__units,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ R ) )
=> ( finite_finite_a @ ( units_a_ring_ext_a_b @ R ) ) ) ) ).
% ring.finite_ring_finite_units
thf(fact_1017_factorial__monoid_Ogcdof__exists,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ G ) )
& ( isgcd_a_ring_ext_a_b @ G @ C2 @ A3 @ B2 ) ) ) ) ) ).
% factorial_monoid.gcdof_exists
thf(fact_1018_factorial__monoid_Ogcdof__exists,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ G ) )
& ( isgcd_a_Product_unit @ G @ C2 @ A3 @ B2 ) ) ) ) ) ).
% factorial_monoid.gcdof_exists
thf(fact_1019_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_1020_domain_OUnits__mult__eq__Units,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( units_a_Product_unit @ ( ring_mult_of_a_b @ R ) )
= ( units_a_ring_ext_a_b @ R ) ) ) ).
% domain.Units_mult_eq_Units
thf(fact_1021_factorial__domain_Oaxioms_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_f5272581269873410839in_a_b @ R )
=> ( factor2046533344642582127t_unit @ ( ring_mult_of_a_b @ R ) ) ) ).
% factorial_domain.axioms(2)
thf(fact_1022_factorial__monoid_Olcmof__exists,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ G ) )
& ( islcm_a_ring_ext_a_b @ G @ C2 @ A3 @ B2 ) ) ) ) ) ).
% factorial_monoid.lcmof_exists
thf(fact_1023_factorial__monoid_Olcmof__exists,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ G ) )
& ( islcm_a_Product_unit @ G @ C2 @ A3 @ B2 ) ) ) ) ) ).
% factorial_monoid.lcmof_exists
thf(fact_1024_factorial__monoid_Omult__wfactors__fmset,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,A3: a,Bs: list_a,B2: a,Cs: list_a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ As @ A3 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ Bs @ B2 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ Cs @ ( mult_a_ring_ext_a_b @ G @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( fmset_a_ring_ext_a_b @ G @ Cs )
= ( plus_p2331992037799027419_set_a @ ( fmset_a_ring_ext_a_b @ G @ As ) @ ( fmset_a_ring_ext_a_b @ G @ Bs ) ) ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.mult_wfactors_fmset
thf(fact_1025_factorial__monoid_Omult__wfactors__fmset,axiom,
! [G: partia8223610829204095565t_unit,As: list_a,A3: a,Bs: list_a,B2: a,Cs: list_a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( wfacto3536202916627062655t_unit @ G @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ G @ Bs @ B2 )
=> ( ( wfacto3536202916627062655t_unit @ G @ Cs @ ( mult_a_Product_unit @ G @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( fmset_a_Product_unit @ G @ Cs )
= ( plus_p2331992037799027419_set_a @ ( fmset_a_Product_unit @ G @ As ) @ ( fmset_a_Product_unit @ G @ Bs ) ) ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.mult_wfactors_fmset
thf(fact_1026_factorial__monoid_Omult__factors__fmset,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,A3: a,Bs: list_a,B2: a,Cs: list_a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( factor5638265376665762323xt_a_b @ G @ As @ A3 )
=> ( ( factor5638265376665762323xt_a_b @ G @ Bs @ B2 )
=> ( ( factor5638265376665762323xt_a_b @ G @ Cs @ ( mult_a_ring_ext_a_b @ G @ A3 @ B2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( fmset_a_ring_ext_a_b @ G @ Cs )
= ( plus_p2331992037799027419_set_a @ ( fmset_a_ring_ext_a_b @ G @ As ) @ ( fmset_a_ring_ext_a_b @ G @ Bs ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.mult_factors_fmset
thf(fact_1027_factorial__monoid_Omult__factors__fmset,axiom,
! [G: partia8223610829204095565t_unit,As: list_a,A3: a,Bs: list_a,B2: a,Cs: list_a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( factor4979495158039764464t_unit @ G @ As @ A3 )
=> ( ( factor4979495158039764464t_unit @ G @ Bs @ B2 )
=> ( ( factor4979495158039764464t_unit @ G @ Cs @ ( mult_a_Product_unit @ G @ A3 @ B2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( fmset_a_Product_unit @ G @ Cs )
= ( plus_p2331992037799027419_set_a @ ( fmset_a_Product_unit @ G @ As ) @ ( fmset_a_Product_unit @ G @ Bs ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.mult_factors_fmset
thf(fact_1028_domain_Oring__associated__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( associ5860276527279195403xt_a_b @ R @ A3 @ B2 )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ R ) )
& ( A3
= ( mult_a_ring_ext_a_b @ R @ X2 @ B2 ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_1029_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a,A3: a,B2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ R @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ R ) )
| ( member_a @ B2 @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_1030_factorial__monoid_Ofactors__irreducible__prime,axiom,
! [G: partia2175431115845679010xt_a_b,P: a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( irredu6211895646901577903xt_a_b @ G @ P )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ G ) )
=> ( prime_a_ring_ext_a_b @ G @ P ) ) ) ) ).
% factorial_monoid.factors_irreducible_prime
thf(fact_1031_factorial__monoid_Ofactors__irreducible__prime,axiom,
! [G: partia8223610829204095565t_unit,P: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( irredu4023057619401689684t_unit @ G @ P )
=> ( ( member_a @ P @ ( partia6735698275553448452t_unit @ G ) )
=> ( prime_a_Product_unit @ G @ P ) ) ) ) ).
% factorial_monoid.factors_irreducible_prime
thf(fact_1032_factorial__monoid_Oassociated__fcount,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ A3 @ B2 )
=> ( ( factor7111650050812153303xt_a_b @ G @ A3 )
= ( factor7111650050812153303xt_a_b @ G @ B2 ) ) ) ) ) ) ).
% factorial_monoid.associated_fcount
thf(fact_1033_factorial__monoid_Oassociated__fcount,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( associ6879500422977059064t_unit @ G @ A3 @ B2 )
=> ( ( factor4067924603488134956t_unit @ G @ A3 )
= ( factor4067924603488134956t_unit @ G @ B2 ) ) ) ) ) ) ).
% factorial_monoid.associated_fcount
thf(fact_1034_factorial__domain__def,axiom,
( ring_f5272581269873410839in_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R3 )
& ( factor2046533344642582127t_unit @ ( ring_mult_of_a_b @ R3 ) ) ) ) ) ).
% factorial_domain_def
thf(fact_1035_factorial__domain_Ointro,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( factor2046533344642582127t_unit @ ( ring_mult_of_a_b @ R ) )
=> ( ring_f5272581269873410839in_a_b @ R ) ) ) ).
% factorial_domain.intro
thf(fact_1036_factorial__monoid_Owfactors__exist,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [Fs4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs4 ) @ ( partia707051561876973205xt_a_b @ G ) )
& ( wfacto3557276942076956612xt_a_b @ G @ Fs4 @ A3 ) ) ) ) ).
% factorial_monoid.wfactors_exist
thf(fact_1037_factorial__monoid_Owfactors__exist,axiom,
! [G: partia8223610829204095565t_unit,A3: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ? [Fs4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs4 ) @ ( partia6735698275553448452t_unit @ G ) )
& ( wfacto3536202916627062655t_unit @ G @ Fs4 @ A3 ) ) ) ) ).
% factorial_monoid.wfactors_exist
thf(fact_1038_factorial__monoid_Odivides__fcount,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( factor8216151070175719842xt_a_b @ G @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ord_less_eq_nat @ ( factor7111650050812153303xt_a_b @ G @ A3 ) @ ( factor7111650050812153303xt_a_b @ G @ B2 ) ) ) ) ) ) ).
% factorial_monoid.divides_fcount
thf(fact_1039_factorial__monoid_Odivides__fcount,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ord_less_eq_nat @ ( factor4067924603488134956t_unit @ G @ A3 ) @ ( factor4067924603488134956t_unit @ G @ B2 ) ) ) ) ) ) ).
% factorial_monoid.divides_fcount
thf(fact_1040_factorial__monoid_Ofactorcount__exists,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [C2: nat] :
! [As2: list_a] :
( ( ( ord_less_eq_set_a @ ( set_a2 @ As2 ) @ ( partia707051561876973205xt_a_b @ G ) )
& ( wfacto3557276942076956612xt_a_b @ G @ As2 @ A3 ) )
=> ( C2
= ( size_size_list_a @ As2 ) ) ) ) ) ).
% factorial_monoid.factorcount_exists
thf(fact_1041_factorial__monoid_Ofactorcount__exists,axiom,
! [G: partia8223610829204095565t_unit,A3: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ? [C2: nat] :
! [As2: list_a] :
( ( ( ord_less_eq_set_a @ ( set_a2 @ As2 ) @ ( partia6735698275553448452t_unit @ G ) )
& ( wfacto3536202916627062655t_unit @ G @ As2 @ A3 ) )
=> ( C2
= ( size_size_list_a @ As2 ) ) ) ) ) ).
% factorial_monoid.factorcount_exists
thf(fact_1042_factorial__monoid_Ofactors__exist,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ~ ( member_a @ A3 @ ( units_a_ring_ext_a_b @ G ) )
=> ? [Fs4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs4 ) @ ( partia707051561876973205xt_a_b @ G ) )
& ( factor5638265376665762323xt_a_b @ G @ Fs4 @ A3 ) ) ) ) ) ).
% factorial_monoid.factors_exist
thf(fact_1043_factorial__monoid_Ofactors__exist,axiom,
! [G: partia8223610829204095565t_unit,A3: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ~ ( member_a @ A3 @ ( units_a_Product_unit @ G ) )
=> ? [Fs4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs4 ) @ ( partia6735698275553448452t_unit @ G ) )
& ( factor4979495158039764464t_unit @ G @ Fs4 @ A3 ) ) ) ) ) ).
% factorial_monoid.factors_exist
thf(fact_1044_factorial__monoid_Owfactors__unique,axiom,
! [G: partia2175431115845679010xt_a_b,Fs: list_a,A3: a,Fs3: list_a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ Fs @ A3 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ Fs3 @ A3 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs3 ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( essent8953798148185448568xt_a_b @ G @ Fs @ Fs3 ) ) ) ) ) ) ) ).
% factorial_monoid.wfactors_unique
thf(fact_1045_factorial__monoid_Owfactors__unique,axiom,
! [G: partia8223610829204095565t_unit,Fs: list_a,A3: a,Fs3: list_a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( wfacto3536202916627062655t_unit @ G @ Fs @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ G @ Fs3 @ A3 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs3 ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( essent9005414202370111435t_unit @ G @ Fs @ Fs3 ) ) ) ) ) ) ) ).
% factorial_monoid.wfactors_unique
thf(fact_1046_factorial__monoid_Oee__wfactorsI,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a,As: list_a,Bs: list_a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A3 @ B2 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ As @ A3 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( essent8953798148185448568xt_a_b @ G @ As @ Bs ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.ee_wfactorsI
thf(fact_1047_factorial__monoid_Oee__wfactorsI,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a,As: list_a,Bs: list_a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A3 @ B2 )
=> ( ( wfacto3536202916627062655t_unit @ G @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ G @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( essent9005414202370111435t_unit @ G @ As @ Bs ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.ee_wfactorsI
thf(fact_1048_factorial__monoid_Oee__wfactors,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,A3: a,Bs: list_a,B2: a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ As @ A3 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ A3 @ B2 )
= ( essent8953798148185448568xt_a_b @ G @ As @ Bs ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.ee_wfactors
thf(fact_1049_factorial__monoid_Oee__wfactors,axiom,
! [G: partia8223610829204095565t_unit,As: list_a,A3: a,Bs: list_a,B2: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( wfacto3536202916627062655t_unit @ G @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ G @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( associ6879500422977059064t_unit @ G @ A3 @ B2 )
= ( essent9005414202370111435t_unit @ G @ As @ Bs ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.ee_wfactors
thf(fact_1050_factorial__monoid_Ofactors__unique,axiom,
! [G: partia2175431115845679010xt_a_b,Fs: list_a,A3: a,Fs3: list_a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( factor5638265376665762323xt_a_b @ G @ Fs @ A3 )
=> ( ( factor5638265376665762323xt_a_b @ G @ Fs3 @ A3 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ~ ( member_a @ A3 @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs3 ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( essent8953798148185448568xt_a_b @ G @ Fs @ Fs3 ) ) ) ) ) ) ) ) ).
% factorial_monoid.factors_unique
thf(fact_1051_factorial__monoid_Ofactors__unique,axiom,
! [G: partia8223610829204095565t_unit,Fs: list_a,A3: a,Fs3: list_a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( factor4979495158039764464t_unit @ G @ Fs @ A3 )
=> ( ( factor4979495158039764464t_unit @ G @ Fs3 @ A3 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ~ ( member_a @ A3 @ ( units_a_Product_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs3 ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( essent9005414202370111435t_unit @ G @ Fs @ Fs3 ) ) ) ) ) ) ) ) ).
% factorial_monoid.factors_unique
thf(fact_1052_factorial__monoid_Ofactorcount__unique,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,A3: a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ As @ A3 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( factor7111650050812153303xt_a_b @ G @ A3 )
= ( size_size_list_a @ As ) ) ) ) ) ) ).
% factorial_monoid.factorcount_unique
thf(fact_1053_factorial__monoid_Ofactorcount__unique,axiom,
! [G: partia8223610829204095565t_unit,As: list_a,A3: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( wfacto3536202916627062655t_unit @ G @ As @ A3 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor4067924603488134956t_unit @ G @ A3 )
= ( size_size_list_a @ As ) ) ) ) ) ) ).
% factorial_monoid.factorcount_unique
thf(fact_1054_factorial__monoid_Oee__factorsI,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a,As: list_a,Bs: list_a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A3 @ B2 )
=> ( ( factor5638265376665762323xt_a_b @ G @ As @ A3 )
=> ( ~ ( member_a @ A3 @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( factor5638265376665762323xt_a_b @ G @ Bs @ B2 )
=> ( ~ ( member_a @ B2 @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( essent8953798148185448568xt_a_b @ G @ As @ Bs ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.ee_factorsI
thf(fact_1055_factorial__monoid_Oee__factorsI,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a,As: list_a,Bs: list_a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A3 @ B2 )
=> ( ( factor4979495158039764464t_unit @ G @ As @ A3 )
=> ( ~ ( member_a @ A3 @ ( units_a_Product_unit @ G ) )
=> ( ( factor4979495158039764464t_unit @ G @ Bs @ B2 )
=> ( ~ ( member_a @ B2 @ ( units_a_Product_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( essent9005414202370111435t_unit @ G @ As @ Bs ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.ee_factorsI
thf(fact_1056_le__add__diff__inverse2,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% le_add_diff_inverse2
thf(fact_1057_le__add__diff__inverse,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A3 @ B2 ) )
= A3 ) ) ).
% le_add_diff_inverse
thf(fact_1058_fmset__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( fmset_a_ring_ext_a_b @ r @ As )
= ( fmset_a_ring_ext_a_b @ r @ Bs ) ) ) ).
% fmset_perm_cong
thf(fact_1059_nat__pow__mult,axiom,
! [X4: a,N: nat,M2: nat] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X4 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X4 @ M2 ) )
= ( pow_a_1026414303147256608_b_nat @ r @ X4 @ ( plus_plus_nat @ N @ M2 ) ) ) ) ).
% nat_pow_mult
thf(fact_1060_length__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( size_size_list_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_append
thf(fact_1061_add__le__add__imp__diff__le,axiom,
! [I2: nat,K2: nat,N: nat,J3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J3 @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J3 @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J3 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1062_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_1063_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A3: nat,B2: nat] :
( ~ ( ord_less_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A3 @ B2 ) )
= A3 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1064_mult__of_Odivides__as__fmsubset,axiom,
! [As: list_a,A3: a,Bs: list_a,B2: a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
= ( subseteq_mset_set_a @ ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ As ) @ ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Bs ) ) ) ) ) ) ) ) ) ).
% mult_of.divides_as_fmsubset
thf(fact_1065_mult__of_Odivides__fmsubset,axiom,
! [A3: a,B2: a,As: list_a,Bs: list_a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( subseteq_mset_set_a @ ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ As ) @ ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Bs ) ) ) ) ) ) ) ) ) ).
% mult_of.divides_fmsubset
thf(fact_1066_mult__of_Ofmsubset__divides,axiom,
! [As: list_a,Bs: list_a,A3: a,B2: a] :
( ( subseteq_mset_set_a @ ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ As ) @ ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Bs ) )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% mult_of.fmsubset_divides
thf(fact_1067_length__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys7: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys7 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P2 @ Ys7 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_1068_finite__maxlen,axiom,
! [M: set_list_a] :
( ( finite_finite_list_a @ M )
=> ? [N2: nat] :
! [X3: list_a] :
( ( member_list_a @ X3 @ M )
=> ( ord_less_nat @ ( size_size_list_a @ X3 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_1069_factorial__monoid_Odivides__as__fmsubset,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,A3: a,Bs: list_a,B2: a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ As @ A3 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ A3 @ B2 )
= ( subseteq_mset_set_a @ ( fmset_a_ring_ext_a_b @ G @ As ) @ ( fmset_a_ring_ext_a_b @ G @ Bs ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.divides_as_fmsubset
thf(fact_1070_factorial__monoid_Odivides__as__fmsubset,axiom,
! [G: partia8223610829204095565t_unit,As: list_a,A3: a,Bs: list_a,B2: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( wfacto3536202916627062655t_unit @ G @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ G @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ A3 @ B2 )
= ( subseteq_mset_set_a @ ( fmset_a_Product_unit @ G @ As ) @ ( fmset_a_Product_unit @ G @ Bs ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.divides_as_fmsubset
thf(fact_1071_factorial__monoid_Odivides__fmsubset,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a,As: list_a,Bs: list_a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( factor8216151070175719842xt_a_b @ G @ A3 @ B2 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ As @ A3 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( subseteq_mset_set_a @ ( fmset_a_ring_ext_a_b @ G @ As ) @ ( fmset_a_ring_ext_a_b @ G @ Bs ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.divides_fmsubset
thf(fact_1072_factorial__monoid_Odivides__fmsubset,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a,As: list_a,Bs: list_a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( factor3040189038382604065t_unit @ G @ A3 @ B2 )
=> ( ( wfacto3536202916627062655t_unit @ G @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ G @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( subseteq_mset_set_a @ ( fmset_a_Product_unit @ G @ As ) @ ( fmset_a_Product_unit @ G @ Bs ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.divides_fmsubset
thf(fact_1073_boundD__carrier,axiom,
! [N: nat,F: nat > a,M2: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F )
=> ( ( ord_less_nat @ N @ M2 )
=> ( member_a @ ( F @ M2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_1074_mult__of_Omultlist__prime__pos,axiom,
! [A3: a,As: list_a] :
( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A3 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ As @ ( one_a_ring_ext_a_b @ r ) ) )
=> ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ As ) )
& ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ ( nth_a @ As @ I4 ) ) ) ) ) ) ) ).
% mult_of.multlist_prime_pos
thf(fact_1075_nth__append__length,axiom,
! [Xs: list_a,X4: a,Ys: list_a] :
( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X4 @ Ys ) ) @ ( size_size_list_a @ Xs ) )
= X4 ) ).
% nth_append_length
thf(fact_1076_nth__append__length__plus,axiom,
! [Xs: list_a,Ys: list_a,N: nat] :
( ( nth_a @ ( append_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ N ) )
= ( nth_a @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_1077_nth__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
= ( nth_a @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
= ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1078_list__all2__conv__all__nth,axiom,
( list_all2_a_a
= ( ^ [P6: a > a > $o,Xs5: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs5 )
= ( size_size_list_a @ Ys2 ) )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_a @ Xs5 ) )
=> ( P6 @ ( nth_a @ Xs5 @ I5 ) @ ( nth_a @ Ys2 @ I5 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_1079_list__all2__all__nthI,axiom,
! [A3: list_a,B2: list_a,P2: a > a > $o] :
( ( ( size_size_list_a @ A3 )
= ( size_size_list_a @ B2 ) )
=> ( ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ A3 ) )
=> ( P2 @ ( nth_a @ A3 @ N2 ) @ ( nth_a @ B2 @ N2 ) ) )
=> ( list_all2_a_a @ P2 @ A3 @ B2 ) ) ) ).
% list_all2_all_nthI
thf(fact_1080_list__all2__nthD2,axiom,
! [P2: a > a > $o,Xs: list_a,Ys: list_a,P: nat] :
( ( list_all2_a_a @ P2 @ Xs @ Ys )
=> ( ( ord_less_nat @ P @ ( size_size_list_a @ Ys ) )
=> ( P2 @ ( nth_a @ Xs @ P ) @ ( nth_a @ Ys @ P ) ) ) ) ).
% list_all2_nthD2
thf(fact_1081_list__all2__nthD,axiom,
! [P2: a > a > $o,Xs: list_a,Ys: list_a,P: nat] :
( ( list_all2_a_a @ P2 @ Xs @ Ys )
=> ( ( ord_less_nat @ P @ ( size_size_list_a @ Xs ) )
=> ( P2 @ ( nth_a @ Xs @ P ) @ ( nth_a @ Ys @ P ) ) ) ) ).
% list_all2_nthD
thf(fact_1082_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_a,Z3: list_a] : ( Y5 = Z3 ) )
= ( ^ [Xs5: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs5 )
= ( size_size_list_a @ Ys2 ) )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_a @ Xs5 ) )
=> ( ( nth_a @ Xs5 @ I5 )
= ( nth_a @ Ys2 @ I5 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1083_Skolem__list__nth,axiom,
! [K2: nat,P2: nat > a > $o] :
( ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ? [X6: a] : ( P2 @ I5 @ X6 ) ) )
= ( ? [Xs5: list_a] :
( ( ( size_size_list_a @ Xs5 )
= K2 )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ( P2 @ I5 @ ( nth_a @ Xs5 @ I5 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1084_nth__equalityI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I4 )
= ( nth_a @ Ys @ I4 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_1085_nth__mem,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1086_list__ball__nth,axiom,
! [N: nat,Xs: list_a,P2: a > $o] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( P2 @ X ) )
=> ( P2 @ ( nth_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1087_in__set__conv__nth,axiom,
! [X4: a,Xs: list_a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
= ( ? [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I5 )
= X4 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1088_all__nth__imp__all__set,axiom,
! [Xs: list_a,P2: a > $o,X4: a] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( P2 @ ( nth_a @ Xs @ I4 ) ) )
=> ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ( P2 @ X4 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1089_all__set__conv__all__nth,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( P2 @ X2 ) ) )
= ( ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_a @ Xs ) )
=> ( P2 @ ( nth_a @ Xs @ I5 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1090_mult__of_Onunit__factors,axiom,
! [A3: a,As: list_a] :
( ~ ( member_a @ A3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ As ) ) ) ) ).
% mult_of.nunit_factors
thf(fact_1091_mult__of_Oproperfactor__fmset,axiom,
! [A3: a,B2: a,As: list_a,Bs: list_a] :
( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( subseteq_mset_set_a @ ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ As ) @ ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Bs ) ) ) ) ) ) ) ) ) ).
% mult_of.properfactor_fmset
thf(fact_1092_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_1093_mult__of_Oproperfactor__divides,axiom,
! [A3: a,B2: a] :
( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) ) ).
% mult_of.properfactor_divides
thf(fact_1094_properfactor__of__zero_I1_J,axiom,
! [B2: a] :
( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ~ ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ B2 @ ( zero_a_b @ r ) ) ) ).
% properfactor_of_zero(1)
thf(fact_1095_mult__of_Oproperfactor__prod__r,axiom,
! [A3: a,B2: a,C: a] :
( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( 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 ) ) )
=> ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ ( mult_a_ring_ext_a_b @ r @ B2 @ C ) ) ) ) ) ) ).
% mult_of.properfactor_prod_r
thf(fact_1096_mult__of_Oproperfactor__prod__l,axiom,
! [A3: a,B2: a,C: a] :
( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( 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 ) ) )
=> ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ ( mult_a_ring_ext_a_b @ r @ C @ B2 ) ) ) ) ) ) ).
% mult_of.properfactor_prod_l
thf(fact_1097_mult__of_Oproperfactor__cong__l,axiom,
! [X5: a,X4: a,Y: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ X5 @ X4 )
=> ( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ X4 @ Y )
=> ( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ X5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ X5 @ Y ) ) ) ) ) ) ).
% mult_of.properfactor_cong_l
thf(fact_1098_mult__of_Oproperfactor__cong__r,axiom,
! [X4: a,Y: a,Y2: a] :
( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ X4 @ Y )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ Y @ Y2 )
=> ( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ X4 @ Y2 ) ) ) ) ) ) ).
% mult_of.properfactor_cong_r
thf(fact_1099_mult__of_Oproperfactor__trans1,axiom,
! [A3: a,B2: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ B2 @ C )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ C ) ) ) ) ) ).
% mult_of.properfactor_trans1
thf(fact_1100_mult__of_Oproperfactor__trans2,axiom,
! [A3: a,B2: a,C: a] :
( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ B2 @ C )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ C ) ) ) ) ) ).
% mult_of.properfactor_trans2
thf(fact_1101_mult__of_Oproperfactor__unitE,axiom,
! [U: a,A3: a] :
( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ U )
=> ~ ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.properfactor_unitE
thf(fact_1102_mult__of_OproperfactorI3,axiom,
! [P: a,A3: a,B2: a] :
( ( P
= ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) )
=> ( ~ ( member_a @ B2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ P ) ) ) ) ) ).
% mult_of.properfactorI3
thf(fact_1103_nunit__factors,axiom,
! [A3: a,As: list_a] :
( ~ ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( factor5638265376665762323xt_a_b @ r @ As @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ As ) ) ) ) ).
% nunit_factors
thf(fact_1104_mult__of_Oproperfactor__fcount,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ord_less_nat @ ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ A3 ) @ ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ B2 ) ) ) ) ) ).
% mult_of.properfactor_fcount
thf(fact_1105_mult__of_Oproperfactor__fmset__ne,axiom,
! [A3: a,B2: a,As: list_a,Bs: list_a] :
( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ As )
!= ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Bs ) ) ) ) ) ) ) ) ) ).
% mult_of.properfactor_fmset_ne
thf(fact_1106_mult__of_Ofmset__properfactor,axiom,
! [As: list_a,Bs: list_a,A3: a,B2: a] :
( ( subseteq_mset_set_a @ ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ As ) @ ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Bs ) )
=> ( ( ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ As )
!= ( fmset_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ Bs ) )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) ) ) ) ) ) ) ) ) ).
% mult_of.fmset_properfactor
thf(fact_1107_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_1108_nth__Cons__0,axiom,
! [X4: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X4 @ Xs ) @ zero_zero_nat )
= X4 ) ).
% nth_Cons_0
thf(fact_1109_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_1110_local_Onat__pow__0,axiom,
! [X4: a] :
( ( pow_a_1026414303147256608_b_nat @ r @ X4 @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ r ) ) ).
% local.nat_pow_0
thf(fact_1111_mult__of_Oproperfactor__mult__l,axiom,
! [A3: a,B2: a,C: a] :
( ( member_a @ A3 @ ( 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 ) ) )
=> ( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ A3 ) @ ( mult_a_ring_ext_a_b @ r @ C @ B2 ) )
= ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) ) ) ) ) ).
% mult_of.properfactor_mult_l
thf(fact_1112_mult__of_Oproperfactor__mult__lI,axiom,
! [A3: a,B2: a,C: a] :
( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ A3 ) @ ( mult_a_ring_ext_a_b @ r @ C @ B2 ) ) ) ) ) ).
% mult_of.properfactor_mult_lI
thf(fact_1113_mult__of_Oproperfactor__mult__r,axiom,
! [A3: a,B2: a,C: a] :
( ( member_a @ A3 @ ( 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 ) ) )
=> ( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ C ) @ ( mult_a_ring_ext_a_b @ r @ B2 @ C ) )
= ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) ) ) ) ) ).
% mult_of.properfactor_mult_r
thf(fact_1114_mult__of_Oproperfactor__mult__rI,axiom,
! [A3: a,B2: a,C: a] :
( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ C ) @ ( mult_a_ring_ext_a_b @ r @ B2 @ C ) ) ) ) ) ).
% mult_of.properfactor_mult_rI
thf(fact_1115_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1116_length__pos__if__in__set,axiom,
! [X4: a,Xs: list_a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1117_irreducible__def,axiom,
( irredu6211895646901577903xt_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,A4: a] :
( ~ ( member_a @ A4 @ ( units_a_ring_ext_a_b @ G3 ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( ( proper19828929941537682xt_a_b @ G3 @ X2 @ A4 )
=> ( member_a @ X2 @ ( units_a_ring_ext_a_b @ G3 ) ) ) ) ) ) ) ).
% irreducible_def
thf(fact_1118_irreducible__def,axiom,
( irredu4023057619401689684t_unit
= ( ^ [G3: partia8223610829204095565t_unit,A4: a] :
( ~ ( member_a @ A4 @ ( units_a_Product_unit @ G3 ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ G3 ) )
=> ( ( proper6663671550266415409t_unit @ G3 @ X2 @ A4 )
=> ( member_a @ X2 @ ( units_a_Product_unit @ G3 ) ) ) ) ) ) ) ).
% irreducible_def
thf(fact_1119_irreducibleI,axiom,
! [A3: a,G: partia2175431115845679010xt_a_b] :
( ~ ( member_a @ A3 @ ( units_a_ring_ext_a_b @ G ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( proper19828929941537682xt_a_b @ G @ B3 @ A3 )
=> ( member_a @ B3 @ ( units_a_ring_ext_a_b @ G ) ) ) )
=> ( irredu6211895646901577903xt_a_b @ G @ A3 ) ) ) ).
% irreducibleI
thf(fact_1120_irreducibleI,axiom,
! [A3: a,G: partia8223610829204095565t_unit] :
( ~ ( member_a @ A3 @ ( units_a_Product_unit @ G ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( proper6663671550266415409t_unit @ G @ B3 @ A3 )
=> ( member_a @ B3 @ ( units_a_Product_unit @ G ) ) ) )
=> ( irredu4023057619401689684t_unit @ G @ A3 ) ) ) ).
% irreducibleI
thf(fact_1121_irreducibleE,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a] :
( ( irredu6211895646901577903xt_a_b @ G @ A3 )
=> ~ ( ~ ( member_a @ A3 @ ( units_a_ring_ext_a_b @ G ) )
=> ~ ! [B8: a] :
( ( ( member_a @ B8 @ ( partia707051561876973205xt_a_b @ G ) )
& ( proper19828929941537682xt_a_b @ G @ B8 @ A3 ) )
=> ( member_a @ B8 @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ).
% irreducibleE
thf(fact_1122_irreducibleE,axiom,
! [G: partia8223610829204095565t_unit,A3: a] :
( ( irredu4023057619401689684t_unit @ G @ A3 )
=> ~ ( ~ ( member_a @ A3 @ ( units_a_Product_unit @ G ) )
=> ~ ! [B8: a] :
( ( ( member_a @ B8 @ ( partia6735698275553448452t_unit @ G ) )
& ( proper6663671550266415409t_unit @ G @ B8 @ A3 ) )
=> ( member_a @ B8 @ ( units_a_Product_unit @ G ) ) ) ) ) ).
% irreducibleE
thf(fact_1123_irreducibleD,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( irredu6211895646901577903xt_a_b @ G @ A3 )
=> ( ( proper19828929941537682xt_a_b @ G @ B2 @ A3 )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ B2 @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ).
% irreducibleD
thf(fact_1124_irreducibleD,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a] :
( ( irredu4023057619401689684t_unit @ G @ A3 )
=> ( ( proper6663671550266415409t_unit @ G @ B2 @ A3 )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ B2 @ ( units_a_Product_unit @ G ) ) ) ) ) ).
% irreducibleD
thf(fact_1125_monoid__cancel_Oproperfactor__mult__l,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( proper19828929941537682xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ C @ A3 ) @ ( mult_a_ring_ext_a_b @ G @ C @ B2 ) )
= ( proper19828929941537682xt_a_b @ G @ A3 @ B2 ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_l
thf(fact_1126_monoid__cancel_Oproperfactor__mult__l,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a,C: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( proper6663671550266415409t_unit @ G @ ( mult_a_Product_unit @ G @ C @ A3 ) @ ( mult_a_Product_unit @ G @ C @ B2 ) )
= ( proper6663671550266415409t_unit @ G @ A3 @ B2 ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_l
thf(fact_1127_monoid__cancel_Oproperfactor__mult__lI,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( proper19828929941537682xt_a_b @ G @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( proper19828929941537682xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ C @ A3 ) @ ( mult_a_ring_ext_a_b @ G @ C @ B2 ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_lI
thf(fact_1128_monoid__cancel_Oproperfactor__mult__lI,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a,C: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( proper6663671550266415409t_unit @ G @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( proper6663671550266415409t_unit @ G @ ( mult_a_Product_unit @ G @ C @ A3 ) @ ( mult_a_Product_unit @ G @ C @ B2 ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_lI
thf(fact_1129_semiring_Onat__pow__zero,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat] :
( ( semiring_a_b @ R )
=> ( ( N != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ R @ ( zero_a_b @ R ) @ N )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.nat_pow_zero
thf(fact_1130_domain_Oproperfactor__of__zero_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,B2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ~ ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ R ) @ B2 @ ( zero_a_b @ R ) ) ) ) ).
% domain.properfactor_of_zero(1)
thf(fact_1131_factorial__monoid_Oproperfactor__fcount,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( proper19828929941537682xt_a_b @ G @ A3 @ B2 )
=> ( ord_less_nat @ ( factor7111650050812153303xt_a_b @ G @ A3 ) @ ( factor7111650050812153303xt_a_b @ G @ B2 ) ) ) ) ) ) ).
% factorial_monoid.properfactor_fcount
thf(fact_1132_factorial__monoid_Oproperfactor__fcount,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( proper6663671550266415409t_unit @ G @ A3 @ B2 )
=> ( ord_less_nat @ ( factor4067924603488134956t_unit @ G @ A3 ) @ ( factor4067924603488134956t_unit @ G @ B2 ) ) ) ) ) ) ).
% factorial_monoid.properfactor_fcount
thf(fact_1133_nth__equal__first__eq,axiom,
! [X4: a,Xs: list_a,N: nat] :
( ~ ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( ( nth_a @ ( cons_a @ X4 @ Xs ) @ N )
= X4 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1134_factorial__monoid_Oproperfactor__fmset__ne,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a,As: list_a,Bs: list_a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( proper19828929941537682xt_a_b @ G @ A3 @ B2 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ As @ A3 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( fmset_a_ring_ext_a_b @ G @ As )
!= ( fmset_a_ring_ext_a_b @ G @ Bs ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.properfactor_fmset_ne
thf(fact_1135_factorial__monoid_Oproperfactor__fmset__ne,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a,As: list_a,Bs: list_a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( proper6663671550266415409t_unit @ G @ A3 @ B2 )
=> ( ( wfacto3536202916627062655t_unit @ G @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ G @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( fmset_a_Product_unit @ G @ As )
!= ( fmset_a_Product_unit @ G @ Bs ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.properfactor_fmset_ne
thf(fact_1136_factorial__monoid_Ofmset__properfactor,axiom,
! [G: partia2175431115845679010xt_a_b,As: list_a,Bs: list_a,A3: a,B2: a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( subseteq_mset_set_a @ ( fmset_a_ring_ext_a_b @ G @ As ) @ ( fmset_a_ring_ext_a_b @ G @ Bs ) )
=> ( ( ( fmset_a_ring_ext_a_b @ G @ As )
!= ( fmset_a_ring_ext_a_b @ G @ Bs ) )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ As @ A3 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( proper19828929941537682xt_a_b @ G @ A3 @ B2 ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.fmset_properfactor
thf(fact_1137_factorial__monoid_Ofmset__properfactor,axiom,
! [G: partia8223610829204095565t_unit,As: list_a,Bs: list_a,A3: a,B2: a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( subseteq_mset_set_a @ ( fmset_a_Product_unit @ G @ As ) @ ( fmset_a_Product_unit @ G @ Bs ) )
=> ( ( ( fmset_a_Product_unit @ G @ As )
!= ( fmset_a_Product_unit @ G @ Bs ) )
=> ( ( wfacto3536202916627062655t_unit @ G @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ G @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( proper6663671550266415409t_unit @ G @ A3 @ B2 ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.fmset_properfactor
thf(fact_1138_factorial__monoid_Oproperfactor__fmset,axiom,
! [G: partia2175431115845679010xt_a_b,A3: a,B2: a,As: list_a,Bs: list_a] :
( ( factor1412514010551229652xt_a_b @ G )
=> ( ( proper19828929941537682xt_a_b @ G @ A3 @ B2 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ As @ A3 )
=> ( ( wfacto3557276942076956612xt_a_b @ G @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( subseteq_mset_set_a @ ( fmset_a_ring_ext_a_b @ G @ As ) @ ( fmset_a_ring_ext_a_b @ G @ Bs ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.properfactor_fmset
thf(fact_1139_factorial__monoid_Oproperfactor__fmset,axiom,
! [G: partia8223610829204095565t_unit,A3: a,B2: a,As: list_a,Bs: list_a] :
( ( factor2046533344642582127t_unit @ G )
=> ( ( proper6663671550266415409t_unit @ G @ A3 @ B2 )
=> ( ( wfacto3536202916627062655t_unit @ G @ As @ A3 )
=> ( ( wfacto3536202916627062655t_unit @ G @ Bs @ B2 )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( subseteq_mset_set_a @ ( fmset_a_Product_unit @ G @ As ) @ ( fmset_a_Product_unit @ G @ Bs ) ) ) ) ) ) ) ) ) ) ).
% factorial_monoid.properfactor_fmset
thf(fact_1140_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_1141_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_1142_properfactor__divides,axiom,
! [A3: a,B2: a] :
( ( proper19828929941537682xt_a_b @ r @ A3 @ B2 )
=> ( factor8216151070175719842xt_a_b @ r @ A3 @ B2 ) ) ).
% properfactor_divides
thf(fact_1143_properfactor__of__zero_I2_J,axiom,
! [B2: a] :
( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( proper19828929941537682xt_a_b @ r @ B2 @ ( zero_a_b @ r ) )
= ( B2
!= ( zero_a_b @ r ) ) ) ) ).
% properfactor_of_zero(2)
thf(fact_1144_properfactor__prod__r,axiom,
! [A3: a,B2: a,C: a] :
( ( proper19828929941537682xt_a_b @ r @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ A3 @ ( mult_a_ring_ext_a_b @ r @ B2 @ C ) ) ) ) ) ) ).
% properfactor_prod_r
thf(fact_1145_properfactor__prod__l,axiom,
! [A3: a,B2: a,C: a] :
( ( proper19828929941537682xt_a_b @ r @ A3 @ B2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ A3 @ ( mult_a_ring_ext_a_b @ r @ C @ B2 ) ) ) ) ) ) ).
% properfactor_prod_l
thf(fact_1146_properfactor__unitE,axiom,
! [U: a,A3: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( proper19828929941537682xt_a_b @ r @ A3 @ U )
=> ~ ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% properfactor_unitE
thf(fact_1147_properfactor__cong__r,axiom,
! [X4: a,Y: a,Y2: a] :
( ( proper19828929941537682xt_a_b @ r @ X4 @ Y )
=> ( ( associ5860276527279195403xt_a_b @ r @ Y @ Y2 )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ X4 @ Y2 ) ) ) ) ) ) ).
% properfactor_cong_r
thf(fact_1148_properfactor__cong__l,axiom,
! [X5: a,X4: a,Y: a] :
( ( associ5860276527279195403xt_a_b @ r @ X5 @ X4 )
=> ( ( proper19828929941537682xt_a_b @ r @ X4 @ Y )
=> ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ X5 @ Y ) ) ) ) ) ) ).
% properfactor_cong_l
thf(fact_1149_properfactor__trans2,axiom,
! [A3: a,B2: a,C: a] :
( ( proper19828929941537682xt_a_b @ r @ A3 @ B2 )
=> ( ( factor8216151070175719842xt_a_b @ r @ B2 @ C )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ A3 @ C ) ) ) ) ) ).
% properfactor_trans2
thf(fact_1150_properfactor__trans1,axiom,
! [A3: a,B2: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A3 @ B2 )
=> ( ( proper19828929941537682xt_a_b @ r @ B2 @ C )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ A3 @ C ) ) ) ) ) ).
% properfactor_trans1
thf(fact_1151_properfactor__mult__imp__properfactor,axiom,
! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ r ) @ B2 @ A3 )
=> ( proper19828929941537682xt_a_b @ r @ B2 @ A3 ) ) ) ) ).
% properfactor_mult_imp_properfactor
thf(fact_1152_domain_Oproperfactor__of__zero_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,B2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( proper19828929941537682xt_a_b @ R @ B2 @ ( zero_a_b @ R ) )
= ( B2
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.properfactor_of_zero(2)
thf(fact_1153_domain_Oproperfactor__mult__imp__properfactor,axiom,
! [R: partia2175431115845679010xt_a_b,A3: a,B2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( proper6663671550266415409t_unit @ ( ring_mult_of_a_b @ R ) @ B2 @ A3 )
=> ( proper19828929941537682xt_a_b @ R @ B2 @ A3 ) ) ) ) ) ).
% domain.properfactor_mult_imp_properfactor
thf(fact_1154_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_1155_mult__of_Ofactors__cong__unit,axiom,
! [U: a,A3: a,As: list_a] :
( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ~ ( member_a @ A3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A3 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ ( list_update_a @ As @ zero_zero_nat @ ( mult_a_ring_ext_a_b @ r @ ( nth_a @ As @ zero_zero_nat ) @ U ) ) @ ( mult_a_ring_ext_a_b @ r @ A3 @ U ) ) ) ) ) ) ).
% mult_of.factors_cong_unit
thf(fact_1156_list__update__overwrite,axiom,
! [Xs: list_a,I2: nat,X4: a,Y: a] :
( ( list_update_a @ ( list_update_a @ Xs @ I2 @ X4 ) @ I2 @ Y )
= ( list_update_a @ Xs @ I2 @ Y ) ) ).
% list_update_overwrite
thf(fact_1157_list__update__nonempty,axiom,
! [Xs: list_a,K2: nat,X4: a] :
( ( ( list_update_a @ Xs @ K2 @ X4 )
= nil_a )
= ( Xs = nil_a ) ) ).
% list_update_nonempty
thf(fact_1158_length__list__update,axiom,
! [Xs: list_a,I2: nat,X4: a] :
( ( size_size_list_a @ ( list_update_a @ Xs @ I2 @ X4 ) )
= ( size_size_list_a @ Xs ) ) ).
% length_list_update
thf(fact_1159_list__update__id,axiom,
! [Xs: list_a,I2: nat] :
( ( list_update_a @ Xs @ I2 @ ( nth_a @ Xs @ I2 ) )
= Xs ) ).
% list_update_id
thf(fact_1160_nth__list__update__neq,axiom,
! [I2: nat,J3: nat,Xs: list_a,X4: a] :
( ( I2 != J3 )
=> ( ( nth_a @ ( list_update_a @ Xs @ I2 @ X4 ) @ J3 )
= ( nth_a @ Xs @ J3 ) ) ) ).
% nth_list_update_neq
thf(fact_1161_list__update__beyond,axiom,
! [Xs: list_a,I2: nat,X4: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I2 )
=> ( ( list_update_a @ Xs @ I2 @ X4 )
= Xs ) ) ).
% list_update_beyond
thf(fact_1162_mult__of_Ounitfactor__ee,axiom,
! [U: a,As: list_a] :
( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ ( list_update_a @ As @ zero_zero_nat @ ( mult_a_ring_ext_a_b @ r @ ( nth_a @ As @ zero_zero_nat ) @ U ) ) @ As ) ) ) ).
% mult_of.unitfactor_ee
thf(fact_1163_list__update__length,axiom,
! [Xs: list_a,X4: a,Ys: list_a,Y: a] :
( ( list_update_a @ ( append_a @ Xs @ ( cons_a @ X4 @ Ys ) ) @ ( size_size_list_a @ Xs ) @ Y )
= ( append_a @ Xs @ ( cons_a @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_1164_nth__list__update__eq,axiom,
! [I2: nat,Xs: list_a,X4: a] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( list_update_a @ Xs @ I2 @ X4 ) @ I2 )
= X4 ) ) ).
% nth_list_update_eq
thf(fact_1165_nat__pow__eone,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X4 @ one_one_nat )
= X4 ) ) ).
% nat_pow_eone
thf(fact_1166_set__swap,axiom,
! [I2: nat,Xs: list_a,J3: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_a @ Xs ) )
=> ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs @ I2 @ ( nth_a @ Xs @ J3 ) ) @ J3 @ ( nth_a @ Xs @ I2 ) ) )
= ( set_a2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1167_nth__Cons__pos,axiom,
! [N: nat,X4: a,Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_a @ ( cons_a @ X4 @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1168_list__all2__update__cong,axiom,
! [P2: a > a > $o,Xs: list_a,Ys: list_a,X4: a,Y: a,I2: nat] :
( ( list_all2_a_a @ P2 @ Xs @ Ys )
=> ( ( P2 @ X4 @ Y )
=> ( list_all2_a_a @ P2 @ ( list_update_a @ Xs @ I2 @ X4 ) @ ( list_update_a @ Ys @ I2 @ Y ) ) ) ) ).
% list_all2_update_cong
thf(fact_1169_set__update__subsetI,axiom,
! [Xs: list_a,A: set_a,X4: a,I2: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A )
=> ( ( member_a @ X4 @ A )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs @ I2 @ X4 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1170_list__update__swap,axiom,
! [I2: nat,I6: nat,Xs: list_a,X4: a,X5: a] :
( ( I2 != I6 )
=> ( ( list_update_a @ ( list_update_a @ Xs @ I2 @ X4 ) @ I6 @ X5 )
= ( list_update_a @ ( list_update_a @ Xs @ I6 @ X5 ) @ I2 @ X4 ) ) ) ).
% list_update_swap
thf(fact_1171_list__update_Osimps_I1_J,axiom,
! [I2: nat,V2: a] :
( ( list_update_a @ nil_a @ I2 @ V2 )
= nil_a ) ).
% list_update.simps(1)
thf(fact_1172_list__update__code_I1_J,axiom,
! [I2: nat,Y: a] :
( ( list_update_a @ nil_a @ I2 @ Y )
= nil_a ) ).
% list_update_code(1)
thf(fact_1173_list__update__code_I2_J,axiom,
! [X4: a,Xs: list_a,Y: a] :
( ( list_update_a @ ( cons_a @ X4 @ Xs ) @ zero_zero_nat @ Y )
= ( cons_a @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1174_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_1175_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_1176_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1177_set__update__memI,axiom,
! [N: nat,Xs: list_a,X4: a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ X4 @ ( set_a2 @ ( list_update_a @ Xs @ N @ X4 ) ) ) ) ).
% set_update_memI
thf(fact_1178_list__update__append1,axiom,
! [I2: nat,Xs: list_a,Ys: list_a,X4: a] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ I2 @ X4 )
= ( append_a @ ( list_update_a @ Xs @ I2 @ X4 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1179_nth__list__update,axiom,
! [I2: nat,Xs: list_a,J3: nat,X4: a] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( ( I2 = J3 )
=> ( ( nth_a @ ( list_update_a @ Xs @ I2 @ X4 ) @ J3 )
= X4 ) )
& ( ( I2 != J3 )
=> ( ( nth_a @ ( list_update_a @ Xs @ I2 @ X4 ) @ J3 )
= ( nth_a @ Xs @ J3 ) ) ) ) ) ).
% nth_list_update
thf(fact_1180_list__update__same__conv,axiom,
! [I2: nat,Xs: list_a,X4: a] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( ( list_update_a @ Xs @ I2 @ X4 )
= Xs )
= ( ( nth_a @ Xs @ I2 )
= X4 ) ) ) ).
% list_update_same_conv
thf(fact_1181_list__update__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a,X4: a] :
( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ N @ X4 )
= ( append_a @ ( list_update_a @ Xs @ N @ X4 ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ N @ X4 )
= ( append_a @ Xs @ ( list_update_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ X4 ) ) ) ) ) ).
% list_update_append
thf(fact_1182_nth__Cons_H,axiom,
! [N: nat,X4: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X4 @ Xs ) @ N )
= X4 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X4 @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1183_nth__non__equal__first__eq,axiom,
! [X4: a,Y: a,Xs: list_a,N: nat] :
( ( X4 != Y )
=> ( ( ( nth_a @ ( cons_a @ X4 @ Xs ) @ N )
= Y )
= ( ( ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1184_field_Oorder__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( order_a_Product_unit @ ( multip3210463924028840165of_a_b @ R ) )
= ( minus_minus_nat @ ( order_a_ring_ext_a_b @ R ) @ one_one_nat ) ) ) ) ).
% field.order_mult_of
thf(fact_1185_Group_Onat__pow__0,axiom,
! [G: partia8223610829204095565t_unit,X4: a] :
( ( pow_a_1875594501834816709it_nat @ G @ X4 @ zero_zero_nat )
= ( one_a_Product_unit @ G ) ) ).
% Group.nat_pow_0
thf(fact_1186_Group_Onat__pow__0,axiom,
! [G: partia2175431115845679010xt_a_b,X4: a] :
( ( pow_a_1026414303147256608_b_nat @ G @ X4 @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ G ) ) ).
% Group.nat_pow_0
thf(fact_1187_diff__add__zero,axiom,
! [A3: multiset_set_a,B2: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A3 @ ( plus_p2331992037799027419_set_a @ A3 @ B2 ) )
= zero_z5079479921072680283_set_a ) ).
% diff_add_zero
thf(fact_1188_diff__add__zero,axiom,
! [A3: nat,B2: nat] :
( ( minus_minus_nat @ A3 @ ( plus_plus_nat @ A3 @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1189_mult__of_OUnits__pow__closed,axiom,
! [X4: a,D2: nat] :
( ( member_a @ X4 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X4 @ D2 ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.Units_pow_closed
thf(fact_1190_mult__of_Ogroup__commutes__pow,axiom,
! [X4: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X4 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X4 ) )
=> ( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X4 @ N ) @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X4 @ N ) ) ) ) ) ) ).
% mult_of.group_commutes_pow
thf(fact_1191_mult__of_Onat__pow__comm,axiom,
! [X4: a,N: nat,M2: nat] :
( ( member_a @ X4 @ ( 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 ) @ X4 @ N ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X4 @ M2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X4 @ M2 ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X4 @ N ) ) ) ) ).
% mult_of.nat_pow_comm
thf(fact_1192_mult__of_Onat__pow__distrib,axiom,
! [X4: a,Y: a,N: nat] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ X4 @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X4 @ N ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ Y @ N ) ) ) ) ) ).
% mult_of.nat_pow_distrib
thf(fact_1193_mult__of_Opow__mult__distrib,axiom,
! [X4: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X4 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X4 ) )
=> ( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ X4 @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X4 @ N ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ Y @ N ) ) ) ) ) ) ).
% mult_of.pow_mult_distrib
thf(fact_1194_units__of__pow,axiom,
! [X4: a,N: nat] :
( ( member_a @ X4 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( units_8174867845824275201xt_a_b @ r ) @ X4 @ N )
= ( pow_a_1026414303147256608_b_nat @ r @ X4 @ N ) ) ) ).
% units_of_pow
thf(fact_1195_mult__of_Onat__pow__mult,axiom,
! [X4: a,N: nat,M2: nat] :
( ( member_a @ X4 @ ( 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 ) @ X4 @ N ) @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X4 @ M2 ) )
= ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X4 @ ( plus_plus_nat @ N @ M2 ) ) ) ) ).
% mult_of.nat_pow_mult
thf(fact_1196_mult__of_Oprime__pow__divides__iff,axiom,
! [P: a,A3: a,B2: a,N: nat] :
( ( member_a @ P @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A3 @ ( 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 @ A3 )
=> ( ( 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 @ A3 @ 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_1197_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1198_add__le__cancel__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_1199_add__le__cancel__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_1200_zero__diff,axiom,
! [A3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1201_diff__zero,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% diff_zero
thf(fact_1202_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ A3 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1203_add__diff__cancel__right_H,axiom,
! [A3: multiset_set_a,B2: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ A3 @ B2 ) @ B2 )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_1204_add__diff__cancel__right_H,axiom,
! [A3: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ B2 )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_1205_add__diff__cancel__right,axiom,
! [A3: multiset_set_a,C: multiset_set_a,B2: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ A3 @ C ) @ ( plus_p2331992037799027419_set_a @ B2 @ C ) )
= ( minus_706656509937749387_set_a @ A3 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_1206_add__diff__cancel__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A3 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_1207_add__diff__cancel__left_H,axiom,
! [A3: multiset_set_a,B2: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ A3 @ B2 ) @ A3 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_1208_add__diff__cancel__left_H,axiom,
! [A3: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B2 ) @ A3 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_1209_add__diff__cancel__left,axiom,
! [C: multiset_set_a,A3: multiset_set_a,B2: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ C @ A3 ) @ ( plus_p2331992037799027419_set_a @ C @ B2 ) )
= ( minus_706656509937749387_set_a @ A3 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_1210_add__diff__cancel__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A3 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_1211_add__le__same__cancel1,axiom,
! [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A3 ) @ B2 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1212_add__le__same__cancel2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1213_le__add__same__cancel1,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ A3 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_1214_le__add__same__cancel2,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ B2 @ A3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_1215_mult__of_Onat__pow__closed,axiom,
! [X4: a,N: nat] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X4 @ N ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.nat_pow_closed
thf(fact_1216_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_1217_mult__of_Onat__pow__eone,axiom,
! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X4 @ one_one_nat )
= X4 ) ) ).
% mult_of.nat_pow_eone
thf(fact_1218_mult__of_Onat__pow__0,axiom,
! [X4: a] :
( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X4 @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ r ) ) ).
% mult_of.nat_pow_0
thf(fact_1219_Ring__Divisibility_Onat__pow__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ R ) )
= ( pow_a_1026414303147256608_b_nat @ R ) ) ).
% Ring_Divisibility.nat_pow_mult_of
thf(fact_1220_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1221_zero__le,axiom,
! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).
% zero_le
thf(fact_1222_add__le__imp__le__right,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_1223_add__le__imp__le__left,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A3 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_1224_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B6: nat] :
? [C4: nat] :
( B6
= ( plus_plus_nat @ A4 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_1225_add__right__mono,axiom,
! [A3: multiset_set_a,B2: multiset_set_a,C: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A3 @ B2 )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A3 @ C ) @ ( plus_p2331992037799027419_set_a @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_1226_add__right__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_1227_less__eqE,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ~ ! [C2: nat] :
( B2
!= ( plus_plus_nat @ A3 @ C2 ) ) ) ).
% less_eqE
thf(fact_1228_add__left__mono,axiom,
! [A3: multiset_set_a,B2: multiset_set_a,C: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A3 @ B2 )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ C @ A3 ) @ ( plus_p2331992037799027419_set_a @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_1229_add__left__mono,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_1230_add__mono,axiom,
! [A3: multiset_set_a,B2: multiset_set_a,C: multiset_set_a,D2: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A3 @ B2 )
=> ( ( ord_le7905258569527593284_set_a @ C @ D2 )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A3 @ C ) @ ( plus_p2331992037799027419_set_a @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_1231_add__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_1232_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: multiset_set_a,J3: multiset_set_a,K2: multiset_set_a,L: multiset_set_a] :
( ( ( ord_le7905258569527593284_set_a @ I2 @ J3 )
& ( ord_le7905258569527593284_set_a @ K2 @ L ) )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ I2 @ K2 ) @ ( plus_p2331992037799027419_set_a @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1233_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J3: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J3 )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1234_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: multiset_set_a,J3: multiset_set_a,K2: multiset_set_a,L: multiset_set_a] :
( ( ( I2 = J3 )
& ( ord_le7905258569527593284_set_a @ K2 @ L ) )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ I2 @ K2 ) @ ( plus_p2331992037799027419_set_a @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1235_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J3: nat,K2: nat,L: nat] :
( ( ( I2 = J3 )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1236_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: multiset_set_a,J3: multiset_set_a,K2: multiset_set_a,L: multiset_set_a] :
( ( ( ord_le7905258569527593284_set_a @ I2 @ J3 )
& ( K2 = L ) )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ I2 @ K2 ) @ ( plus_p2331992037799027419_set_a @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1237_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J3: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J3 )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1238_diff__diff__eq,axiom,
! [A3: multiset_set_a,B2: multiset_set_a,C: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ A3 @ B2 ) @ C )
= ( minus_706656509937749387_set_a @ A3 @ ( plus_p2331992037799027419_set_a @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_1239_diff__diff__eq,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B2 ) @ C )
= ( minus_minus_nat @ A3 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_1240_add__implies__diff,axiom,
! [C: multiset_set_a,B2: multiset_set_a,A3: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ C @ B2 )
= A3 )
=> ( C
= ( minus_706656509937749387_set_a @ A3 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_1241_add__implies__diff,axiom,
! [C: nat,B2: nat,A3: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A3 )
=> ( C
= ( minus_minus_nat @ A3 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_1242_add__decreasing,axiom,
! [A3: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_1243_add__increasing,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing
thf(fact_1244_add__decreasing2,axiom,
! [C: nat,A3: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_1245_add__increasing2,axiom,
! [C: nat,B2: nat,A3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A3 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_1246_add__nonneg__nonneg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1247_add__nonpos__nonpos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1248_add__nonneg__eq__0__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1249_add__nonpos__eq__0__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1250_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J3: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J3 )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1251_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J3: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J3 )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1252_add__le__less__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_1253_add__less__le__mono,axiom,
! [A3: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_1254_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A3 )
= C )
= ( B2
= ( plus_plus_nat @ C @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1255_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ A3 @ ( minus_minus_nat @ B2 @ A3 ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1256_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A3 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1257_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A3 )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A3 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1258_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A3 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1259_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A3 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1260_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1261_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A3 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1262_le__add__diff,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A3 ) ) ) ).
% le_add_diff
thf(fact_1263_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A3 ) @ A3 )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1264_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_1265_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_1266_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_1267_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_1268_add__strict__increasing2,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1269_add__strict__increasing,axiom,
! [A3: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1270_add__pos__nonneg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1271_add__nonpos__neg,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1272_add__nonneg__pos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1273_add__neg__nonpos,axiom,
! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A3 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1274_mult__of_Ounits__of__pow,axiom,
! [X4: a,N: nat] :
( ( member_a @ X4 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( units_7501539392726747778t_unit @ ( ring_mult_of_a_b @ r ) ) @ X4 @ N )
= ( pow_a_1875594501834816709it_nat @ ( ring_mult_of_a_b @ r ) @ X4 @ N ) ) ) ).
% mult_of.units_of_pow
% Conjectures (1)
thf(conj_0,conjecture,
( ( eval_a_b @ r @ ( lagran2649660974587678107al_a_b @ r @ s @ x ) @ x )
= ( one_a_ring_ext_a_b @ r ) ) ).
%------------------------------------------------------------------------------