TPTP Problem File: SLH0905^1.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_00035_001189__17113458_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1451 ( 259 unt; 174 typ; 0 def)
% Number of atoms : 4745 (1337 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 17911 ( 177 ~; 32 |; 164 &;14637 @)
% ( 0 <=>;2901 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 358 ( 358 >; 0 *; 0 +; 0 <<)
% Number of symbols : 156 ( 155 usr; 11 con; 0-4 aty)
% Number of variables : 3678 ( 56 ^;3544 !; 78 ?;3678 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:36:23.443
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
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% Explicit typings (155)
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ord_le8861187494160871172list_a: set_list_a > set_list_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_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno6951661231331188332t_unit: partia2670972154091845814t_unit > list_list_a > list_a > $o ).
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_Opolynomial_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno1315193887021588240t_unit: partia2670972154091845814t_unit > set_list_a > list_list_a > $o ).
thf(sy_c_Polynomials_Opolynomial_001tf__a_001tf__b,type,
polynomial_a_b: partia2175431115845679010xt_a_b > set_a > list_a > $o ).
thf(sy_c_Polynomials_Oring_Oconst__term_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
const_6738166269504826821t_unit: partia2670972154091845814t_unit > list_list_a > list_a ).
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_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
eval_l34571156754992824t_unit: partia2670972154091845814t_unit > list_list_a > list_a > list_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_Opoly__of__const_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
poly_o8716471131768098070t_unit: partia2670972154091845814t_unit > list_a > list_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_Ouniv__poly_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
univ_p7953238456130426574t_unit: partia2670972154091845814t_unit > set_list_a > partia2956882679547061052t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001tf__a_001tf__b,type,
univ_poly_a_b: partia2175431115845679010xt_a_b > set_a > partia2670972154091845814t_unit ).
thf(sy_c_Polynomials_Ovar_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
var_li8453953174693405341t_unit: partia2670972154091845814t_unit > list_list_a ).
thf(sy_c_Polynomials_Ovar_001tf__a_001tf__b,type,
var_a_b: partia2175431115845679010xt_a_b > list_a ).
thf(sy_c_QuotRing_Oring__iso_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_i7414513579304222626t_unit: partia2670972154091845814t_unit > partia2670972154091845814t_unit > set_list_a_list_a ).
thf(sy_c_QuotRing_Oring__iso_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_i7048835797181109658it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > set_list_a_a ).
thf(sy_c_QuotRing_Oring__iso_001tf__a_001tf__b_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_i4557880751517319194t_unit: partia2175431115845679010xt_a_b > partia2670972154091845814t_unit > set_a_list_a ).
thf(sy_c_QuotRing_Oring__iso_001tf__a_001tf__b_001tf__a_001tf__b,type,
ring_iso_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > set_a_a ).
thf(sy_c_Ring_Oa__inv_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
a_inv_8944721093294617173t_unit: partia2670972154091845814t_unit > list_a > list_a ).
thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
a_inv_a_b: partia2175431115845679010xt_a_b > a > a ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
a_minu3984020753470702548t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Ring_Oa__minus_001tf__a_001tf__b,type,
a_minus_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oabelian__group_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
abelia3891852623213500406t_unit: partia2670972154091845814t_unit > $o ).
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_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
abelia226231641709521465t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Oabelian__monoid_001tf__a_001tf__b,type,
abelian_monoid_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oadd__pow_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__Int__Oint,type,
add_po2638046716968164713it_int: partia2670972154091845814t_unit > int > list_a > list_a ).
thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Int__Oint,type,
add_pow_a_b_int: partia2175431115845679010xt_a_b > int > a > a ).
thf(sy_c_Ring_Oring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_l6212528067271185461t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Oring_001tf__a_001tf__b,type,
ring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
add_li7652885771158616974t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
add_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oring_Omore_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
more_l6502722259342772890t_unit: partia2670972154091845814t_unit > product_unit ).
thf(sy_c_Ring_Oring_Omore_001tf__a_001tf__b,type,
more_a_b: partia2175431115845679010xt_a_b > b ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
zero_l4142658623432671053t_unit: partia2670972154091845814t_unit > list_a ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h7399960747407462284t_unit: partia2670972154091845814t_unit > partia2670972154091845814t_unit > set_list_a_list_a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h2895973938487309444it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > set_list_a_a ).
thf(sy_c_Ring_Oring__hom_001tf__a_001tf__b_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h405018892823518980t_unit: partia2175431115845679010xt_a_b > partia2670972154091845814t_unit > set_a_list_a ).
thf(sy_c_Ring_Oring__hom_001tf__a_001tf__b_001tf__a_001tf__b,type,
ring_hom_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > set_a_a ).
thf(sy_c_Ring_Osemiring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
semiri2871908745932252451t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
semiring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Osemiring__axioms_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
semiri6743050769115779782t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Osemiring__axioms_001tf__a_001tf__b,type,
semiring_axioms_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r932985474545269838t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Subrings_Osubcring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subcri7763218559781929323t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001tf__a_001tf__b,type,
subcring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubdomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subdom7821232466298058046t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001tf__a_001tf__b,type,
subdomain_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subfie1779122896746047282t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subrin6918843898125473962t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubring_001tf__a_001tf__b,type,
subring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member_list_a_list_a: ( list_a > list_a ) > set_list_a_list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
member_list_a_a: ( list_a > a ) > set_list_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
member_a_list_a: ( a > list_a ) > set_a_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_y,type,
y: a ).
% Relevant facts (1276)
thf(fact_0_assms,axiom,
member_a @ x @ ( partia707051561876973205xt_a_b @ r ) ).
% assms
thf(fact_1_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_2_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_3_eval__var,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X )
= X ) ) ).
% eval_var
thf(fact_4_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_5_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_6_is__abelian__group,axiom,
abelian_group_a_b @ r ).
% is_abelian_group
thf(fact_7_ring_Opoly__of__const_Ocong,axiom,
poly_of_const_a_b = poly_of_const_a_b ).
% ring.poly_of_const.cong
thf(fact_8_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_9_ring_Oeval_Ocong,axiom,
eval_a_b = eval_a_b ).
% ring.eval.cong
thf(fact_10_associated__sym,axiom,
! [A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( associ5860276527279195403xt_a_b @ r @ B @ A ) ) ).
% associated_sym
thf(fact_11_monoid__axioms,axiom,
monoid8385113658579753027xt_a_b @ r ).
% monoid_axioms
thf(fact_12_associated__trans,axiom,
! [A: a,B: a,C: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ( associ5860276527279195403xt_a_b @ r @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ C ) ) ) ) ) ).
% associated_trans
thf(fact_13_assoc__subst,axiom,
! [A: a,B: a,F: a > a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ! [A2: a,B2: a] :
( ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ A2 @ B2 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ ( F @ B2 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% assoc_subst
thf(fact_14_associated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ A ) ) ).
% associated_refl
thf(fact_15_ring_Oeval__var,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( var_a_b @ R ) @ X )
= X ) ) ) ).
% ring.eval_var
thf(fact_16_ring_Oeval__var,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( var_li8453953174693405341t_unit @ R ) @ X )
= X ) ) ) ).
% ring.eval_var
thf(fact_17_ring_Oonepideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( principalideal_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% ring.onepideal
thf(fact_18_ring_Oonepideal,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).
% ring.onepideal
thf(fact_19_ring_Ocgenideal__self,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ R @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_20_ring_Ocgenideal__self,axiom,
! [R: partia2670972154091845814t_unit,I: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ I @ ( cgenid9131348535277946915t_unit @ R @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_21_monoid_Oassoc__subst,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,F: a > a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ! [A2: a,B2: a] :
( ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
& ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
& ( associ5860276527279195403xt_a_b @ G @ A2 @ B2 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ G ) )
& ( member_a @ ( F @ B2 ) @ ( partia707051561876973205xt_a_b @ G ) )
& ( associ5860276527279195403xt_a_b @ G @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ) ).
% monoid.assoc_subst
thf(fact_22_monoid_Oassoc__subst,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,F: list_a > list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ! [A2: list_a,B2: list_a] :
( ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
& ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ G ) )
& ( associ8407585678920448409t_unit @ G @ A2 @ B2 ) )
=> ( ( member_list_a @ ( F @ A2 ) @ ( partia5361259788508890537t_unit @ G ) )
& ( member_list_a @ ( F @ B2 ) @ ( partia5361259788508890537t_unit @ G ) )
& ( associ8407585678920448409t_unit @ G @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( associ8407585678920448409t_unit @ G @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ) ).
% monoid.assoc_subst
thf(fact_23_monoid_Oassociated__refl,axiom,
! [G: partia2175431115845679010xt_a_b,A: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A @ A ) ) ) ).
% monoid.associated_refl
thf(fact_24_monoid_Oassociated__refl,axiom,
! [G: partia2670972154091845814t_unit,A: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( associ8407585678920448409t_unit @ G @ A @ A ) ) ) ).
% monoid.associated_refl
thf(fact_25_monoid_Oassociated__trans,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ( associ5860276527279195403xt_a_b @ G @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A @ C ) ) ) ) ) ) ).
% monoid.associated_trans
thf(fact_26_monoid_Oassociated__trans,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ( associ8407585678920448409t_unit @ G @ B @ C )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( associ8407585678920448409t_unit @ G @ A @ C ) ) ) ) ) ) ).
% monoid.associated_trans
thf(fact_27_properfactor__cong__r,axiom,
! [X: a,Y: a,Y2: a] :
( ( proper19828929941537682xt_a_b @ r @ X @ Y )
=> ( ( associ5860276527279195403xt_a_b @ r @ Y @ Y2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ X @ Y2 ) ) ) ) ) ) ).
% properfactor_cong_r
thf(fact_28_properfactor__cong__l,axiom,
! [X2: a,X: a,Y: a] :
( ( associ5860276527279195403xt_a_b @ r @ X2 @ X )
=> ( ( proper19828929941537682xt_a_b @ r @ X @ Y )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ X2 @ Y ) ) ) ) ) ) ).
% properfactor_cong_l
thf(fact_29_mult__cong__r,axiom,
! [B: a,B3: a,A: a] :
( ( associ5860276527279195403xt_a_b @ r @ B @ B3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B3 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_30_semiring_Oaxioms_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( monoid8385113658579753027xt_a_b @ R ) ) ).
% semiring.axioms(2)
thf(fact_31_semiring_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( abelian_monoid_a_b @ R ) ) ).
% semiring.axioms(1)
thf(fact_32_m__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_33_properfactor__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( proper19828929941537682xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% properfactor_prod_r
thf(fact_34_m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_35_monoid_Oproperfactor__prod__r,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( proper19828929941537682xt_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( proper19828929941537682xt_a_b @ G @ A @ ( mult_a_ring_ext_a_b @ G @ B @ C ) ) ) ) ) ) ) ).
% monoid.properfactor_prod_r
thf(fact_36_monoid_Oproperfactor__prod__r,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( proper8313688649498433056t_unit @ G @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( proper8313688649498433056t_unit @ G @ A @ ( mult_l7073676228092353617t_unit @ G @ B @ C ) ) ) ) ) ) ) ).
% monoid.properfactor_prod_r
thf(fact_37_ring_Oring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_38_ring_Oring__simprules_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_39_ring_Oring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_40_ring_Oring__simprules_I11_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ R @ X @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_41_ring__hom__mult,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_42_ring__hom__mult,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_43_ring__hom__mult,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_44_ring__hom__mult,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_45_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_46_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_47_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_48_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ R @ X @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_49_ring__iso__memE_I2_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_50_ring__iso__memE_I2_J,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_51_ring__iso__memE_I2_J,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_52_ring__iso__memE_I2_J,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_53_monoid_Omult__cong__r,axiom,
! [G: partia2175431115845679010xt_a_b,B: a,B3: a,A: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ B @ B3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( mult_a_ring_ext_a_b @ G @ A @ B3 ) ) ) ) ) ) ) ).
% monoid.mult_cong_r
thf(fact_54_monoid_Omult__cong__r,axiom,
! [G: partia2670972154091845814t_unit,B: list_a,B3: list_a,A: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ B @ B3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( associ8407585678920448409t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) @ ( mult_l7073676228092353617t_unit @ G @ A @ B3 ) ) ) ) ) ) ) ).
% monoid.mult_cong_r
thf(fact_55_monoid_Oproperfactor__cong__r,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Y2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( proper19828929941537682xt_a_b @ G @ X @ Y )
=> ( ( associ5860276527279195403xt_a_b @ G @ Y @ Y2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( proper19828929941537682xt_a_b @ G @ X @ Y2 ) ) ) ) ) ) ) ).
% monoid.properfactor_cong_r
thf(fact_56_monoid_Oproperfactor__cong__r,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Y2: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( proper8313688649498433056t_unit @ G @ X @ Y )
=> ( ( associ8407585678920448409t_unit @ G @ Y @ Y2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( proper8313688649498433056t_unit @ G @ X @ Y2 ) ) ) ) ) ) ) ).
% monoid.properfactor_cong_r
thf(fact_57_monoid_Oproperfactor__cong__l,axiom,
! [G: partia2175431115845679010xt_a_b,X2: a,X: a,Y: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ X2 @ X )
=> ( ( proper19828929941537682xt_a_b @ G @ X @ Y )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( proper19828929941537682xt_a_b @ G @ X2 @ Y ) ) ) ) ) ) ) ).
% monoid.properfactor_cong_l
thf(fact_58_monoid_Oproperfactor__cong__l,axiom,
! [G: partia2670972154091845814t_unit,X2: list_a,X: list_a,Y: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ X2 @ X )
=> ( ( proper8313688649498433056t_unit @ G @ X @ Y )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( proper8313688649498433056t_unit @ G @ X2 @ Y ) ) ) ) ) ) ) ).
% monoid.properfactor_cong_l
thf(fact_59_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A3: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_63_principalideal_Ois__principalideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ( principalideal_a_b @ I2 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_64_ring__hom__closed,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_65_ring__hom__closed,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a] :
( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_66_ring__hom__closed,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a] :
( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_67_ring__hom__closed,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a] :
( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_68_monoid_Oassociated__sym,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( associ5860276527279195403xt_a_b @ G @ B @ A ) ) ) ).
% monoid.associated_sym
thf(fact_69_ring_Ois__abelian__group,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( abelian_group_a_b @ R ) ) ).
% ring.is_abelian_group
thf(fact_70_ring__iso__memE_I1_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_71_ring__iso__memE_I1_J,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_72_ring__iso__memE_I1_J,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_73_ring__iso__memE_I1_J,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a] :
( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_74_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_75_ring_Ois__monoid,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( monoid8385113658579753027xt_a_b @ R ) ) ).
% ring.is_monoid
thf(fact_76_monoid__cancelI,axiom,
( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A2 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A2 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B2 @ C2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_77_add__pow__ldistr__int,axiom,
! [A: a,B: a,K: int] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ K @ A ) @ B )
= ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_ldistr_int
thf(fact_78_add__pow__rdistr__int,axiom,
! [A: a,B: a,K: int] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ K @ B ) )
= ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_rdistr_int
thf(fact_79_associatedI2,axiom,
! [U: a,A: a,B: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2
thf(fact_80_associatedI2_H,axiom,
! [A: a,B: a,U: a] :
( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2'
thf(fact_81_monoid_Om__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% monoid.m_closed
thf(fact_82_monoid_Om__closed,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ X @ Y ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% monoid.m_closed
thf(fact_83_monoid_Om__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ G @ X @ ( mult_a_ring_ext_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).
% monoid.m_assoc
thf(fact_84_monoid_Om__assoc,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ G @ X @ ( mult_l7073676228092353617t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).
% monoid.m_assoc
thf(fact_85_poly__of__const__in__carrier,axiom,
! [S2: a] :
( ( member_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( poly_of_const_a_b @ r @ S2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% poly_of_const_in_carrier
thf(fact_86_l__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_87_r__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_88_semiring__def,axiom,
( semiring_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ R2 )
& ( monoid8385113658579753027xt_a_b @ R2 )
& ( semiring_axioms_a_b @ R2 ) ) ) ) ).
% semiring_def
thf(fact_89_a__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_90_a__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) ) ) ) ).
% a_comm
thf(fact_91_a__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
= ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_92_add_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( add_a_b @ r @ A @ C )
= ( add_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.r_cancel
thf(fact_93_add_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( add_a_b @ r @ C @ A )
= ( add_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.l_cancel
thf(fact_94_Units__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_95_Units__assoc,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ).
% Units_assoc
thf(fact_96_add_Oint__pow__mult__distrib,axiom,
! [X: a,Y: a,I: int] :
( ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ) ).
% add.int_pow_mult_distrib
thf(fact_97_add_Oint__pow__distrib,axiom,
! [X: a,Y: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ).
% add.int_pow_distrib
thf(fact_98_prod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_99_prod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_100_properfactor__unitE,axiom,
! [U: a,A: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( proper19828929941537682xt_a_b @ r @ A @ U )
=> ~ ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% properfactor_unitE
thf(fact_101_a__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_102_local_Oadd_Oright__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X )
= ( add_a_b @ r @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_103_add_Oint__pow__closed,axiom,
! [X: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_pow_a_b_int @ r @ I @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.int_pow_closed
thf(fact_104_Units__m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_105_Units__l__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% Units_l_cancel
thf(fact_106_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_107_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_108_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_109_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_110_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_111_monoid_OUnits__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% monoid.Units_closed
thf(fact_112_monoid_OUnits__closed,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
=> ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) ) ) ) ).
% monoid.Units_closed
thf(fact_113_monoid_OUnits__m__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.Units_m_closed
thf(fact_114_monoid_OUnits__m__closed,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ G ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ X @ Y ) @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ).
% monoid.Units_m_closed
thf(fact_115_monoid__cancel_OassociatedD2,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( units_a_ring_ext_a_b @ G ) )
& ( A
= ( mult_a_ring_ext_a_b @ G @ B @ X4 ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_116_monoid__cancel_OassociatedD2,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( units_2932844235741507942t_unit @ G ) )
& ( A
= ( mult_l7073676228092353617t_unit @ G @ B @ X4 ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_117_monoid__cancel_OassociatedE2,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ! [U2: a] :
( ( A
= ( mult_a_ring_ext_a_b @ G @ B @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_ring_ext_a_b @ G ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ~ ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_118_monoid__cancel_OassociatedE2,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ! [U2: list_a] :
( ( A
= ( mult_l7073676228092353617t_unit @ G @ B @ U2 ) )
=> ~ ( member_list_a @ U2 @ ( units_2932844235741507942t_unit @ G ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ~ ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_119_monoid__cancel_Oassociated__iff,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ G ) )
& ( A
= ( mult_a_ring_ext_a_b @ G @ B @ X3 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_120_monoid__cancel_Oassociated__iff,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ G ) )
& ( A
= ( mult_l7073676228092353617t_unit @ G @ B @ X3 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_121_monoid__cancel_Oaxioms_I1_J,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( monoid8385113658579753027xt_a_b @ G ) ) ).
% monoid_cancel.axioms(1)
thf(fact_122_monoid_OUnits__l__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
= ( mult_a_ring_ext_a_b @ G @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ).
% monoid.Units_l_cancel
thf(fact_123_monoid_OUnits__l__cancel,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ X @ Y )
= ( mult_l7073676228092353617t_unit @ G @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ).
% monoid.Units_l_cancel
thf(fact_124_ring_Oring__simprules_I22_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_125_ring_Oring__simprules_I22_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ R @ Y @ ( add_li7652885771158616974t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_126_ring_Oring__simprules_I10_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y )
= ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_127_ring_Oring__simprules_I10_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_128_ring_Oring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_129_ring_Oring__simprules_I7_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_130_ring_Oring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_131_ring_Oring__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_132_monoid_OUnits__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A @ B ) ) ) ) ).
% monoid.Units_assoc
thf(fact_133_abelian__groupE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y )
= ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_134_abelian__groupE_I4_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia3891852623213500406t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_135_abelian__groupE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_136_abelian__groupE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( abelia3891852623213500406t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_137_abelian__groupE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_138_abelian__groupE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia3891852623213500406t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_139_ring__hom__add,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_140_ring__hom__add,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_141_ring__hom__add,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_142_ring__hom__add,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_143_abelian__monoidE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y )
= ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_144_abelian__monoidE_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_145_abelian__monoidE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_146_abelian__monoidE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_147_abelian__monoidE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_148_abelian__monoidE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_149_abelian__monoid_Oa__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X @ Y )
= ( add_a_b @ G @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_150_abelian__monoid_Oa__comm,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X @ Y )
= ( add_li7652885771158616974t_unit @ G @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_151_abelian__monoid_Oa__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( add_a_b @ G @ X @ Y ) @ Z )
= ( add_a_b @ G @ X @ ( add_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_152_abelian__monoid_Oa__assoc,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( add_li7652885771158616974t_unit @ G @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ G @ X @ ( add_li7652885771158616974t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_153_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X @ ( add_a_b @ G @ Y @ Z ) )
= ( add_a_b @ G @ Y @ ( add_a_b @ G @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_154_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X @ ( add_li7652885771158616974t_unit @ G @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ G @ Y @ ( add_li7652885771158616974t_unit @ G @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_155_abelian__monoid_Oa__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( add_a_b @ G @ X @ Y ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_156_abelian__monoid_Oa__closed,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ G @ X @ Y ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_157_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_158_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ R @ Y @ ( add_li7652885771158616974t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_159_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y )
= ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_160_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_161_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_162_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_163_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_164_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_165_ring__iso__memE_I3_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_166_ring__iso__memE_I3_J,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_167_ring__iso__memE_I3_J,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_168_ring__iso__memE_I3_J,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_169_monoid__cancel_Or__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,C: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ A @ C )
= ( mult_a_ring_ext_a_b @ G @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_170_monoid__cancel_Or__cancel,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,C: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ A @ C )
= ( mult_l7073676228092353617t_unit @ G @ B @ C ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_171_monoid__cancel_Ol__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,C: a,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ C @ A )
= ( mult_a_ring_ext_a_b @ G @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_172_monoid__cancel_Ol__cancel,axiom,
! [G: partia2670972154091845814t_unit,C: list_a,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ C @ A )
= ( mult_l7073676228092353617t_unit @ G @ C @ B ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_173_monoid_Oprod__unit__l,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ) ).
% monoid.prod_unit_l
thf(fact_174_monoid_Oprod__unit__l,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ) ) ).
% monoid.prod_unit_l
thf(fact_175_monoid_Oprod__unit__r,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ) ).
% monoid.prod_unit_r
thf(fact_176_monoid_Oprod__unit__r,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ) ) ).
% monoid.prod_unit_r
thf(fact_177_monoid_Oproperfactor__unitE,axiom,
! [G: partia2175431115845679010xt_a_b,U: a,A: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( proper19828929941537682xt_a_b @ G @ A @ U )
=> ~ ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% monoid.properfactor_unitE
thf(fact_178_monoid_Oproperfactor__unitE,axiom,
! [G: partia2670972154091845814t_unit,U: list_a,A: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( proper8313688649498433056t_unit @ G @ A @ U )
=> ~ ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% monoid.properfactor_unitE
thf(fact_179_ring_Oring__simprules_I23_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_180_ring_Oring__simprules_I23_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ Z @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_181_ring_Oring__simprules_I13_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_182_ring_Oring__simprules_I13_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_183_semiring_Ol__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_184_semiring_Ol__distr,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_185_semiring_Or__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_186_semiring_Or__distr,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ Z @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_187_ring_Opoly__of__const__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,S2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ S2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( poly_of_const_a_b @ R @ S2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% ring.poly_of_const_in_carrier
thf(fact_188_ring_Opoly__of__const__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,S2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ S2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( poly_o8716471131768098070t_unit @ R @ S2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ).
% ring.poly_of_const_in_carrier
thf(fact_189_semiring_Oaxioms_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( semiring_axioms_a_b @ R ) ) ).
% semiring.axioms(3)
thf(fact_190_monoid_OassociatedI2,axiom,
! [G: partia2175431115845679010xt_a_b,U: a,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ G @ B @ U ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A @ B ) ) ) ) ) ).
% monoid.associatedI2
thf(fact_191_monoid_OassociatedI2,axiom,
! [G: partia2670972154091845814t_unit,U: list_a,A: list_a,B: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( A
= ( mult_l7073676228092353617t_unit @ G @ B @ U ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( associ8407585678920448409t_unit @ G @ A @ B ) ) ) ) ) ).
% monoid.associatedI2
thf(fact_192_monoid_OassociatedI2_H,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,U: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( A
= ( mult_a_ring_ext_a_b @ G @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( associ5860276527279195403xt_a_b @ G @ A @ B ) ) ) ) ) ).
% monoid.associatedI2'
thf(fact_193_monoid_OassociatedI2_H,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,U: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( A
= ( mult_l7073676228092353617t_unit @ G @ B @ U ) )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( associ8407585678920448409t_unit @ G @ A @ B ) ) ) ) ) ).
% monoid.associatedI2'
thf(fact_194_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,B3: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( mult_a_ring_ext_a_b @ G @ A @ B3 ) )
=> ( associ5860276527279195403xt_a_b @ G @ B @ B3 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_195_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,B3: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( associ8407585678920448409t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) @ ( mult_l7073676228092353617t_unit @ G @ A @ B3 ) )
=> ( associ8407585678920448409t_unit @ G @ B @ B3 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_196_monoid_Omonoid__cancelI,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ G @ C2 @ A2 )
= ( mult_a_ring_ext_a_b @ G @ C2 @ B2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ G @ A2 @ C2 )
= ( mult_a_ring_ext_a_b @ G @ B2 @ C2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ G ) ) ) ) ).
% monoid.monoid_cancelI
thf(fact_197_monoid_Omonoid__cancelI,axiom,
! [G: partia2670972154091845814t_unit] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ! [A2: list_a,B2: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ G @ C2 @ A2 )
= ( mult_l7073676228092353617t_unit @ G @ C2 @ B2 ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: list_a,B2: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ G @ A2 @ C2 )
= ( mult_l7073676228092353617t_unit @ G @ B2 @ C2 ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ G ) ) ) ) ).
% monoid.monoid_cancelI
thf(fact_198_monoid__cancel_Oproperfactor__mult__lI,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( proper19828929941537682xt_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( proper19828929941537682xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ C @ A ) @ ( mult_a_ring_ext_a_b @ G @ C @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_lI
thf(fact_199_monoid__cancel_Oproperfactor__mult__lI,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( proper8313688649498433056t_unit @ G @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( proper8313688649498433056t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ C @ A ) @ ( mult_l7073676228092353617t_unit @ G @ C @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_lI
thf(fact_200_monoid__cancel_Oproperfactor__mult__l,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( proper19828929941537682xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ C @ A ) @ ( mult_a_ring_ext_a_b @ G @ C @ B ) )
= ( proper19828929941537682xt_a_b @ G @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_l
thf(fact_201_monoid__cancel_Oproperfactor__mult__l,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( proper8313688649498433056t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ C @ A ) @ ( mult_l7073676228092353617t_unit @ G @ C @ B ) )
= ( proper8313688649498433056t_unit @ G @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_l
thf(fact_202_ringI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ R )
=> ( ( monoid8385113658579753027xt_a_b @ R )
=> ( ! [X4: a,Y3: a,Z2: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X4 @ Y3 ) @ Z2 )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X4 @ Z2 ) @ ( mult_a_ring_ext_a_b @ R @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: a,Y3: a,Z2: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z2 @ ( add_a_b @ R @ X4 @ Y3 ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z2 @ X4 ) @ ( mult_a_ring_ext_a_b @ R @ Z2 @ Y3 ) ) ) ) ) )
=> ( ring_a_b @ R ) ) ) ) ) ).
% ringI
thf(fact_203_ringI,axiom,
! [R: partia2670972154091845814t_unit] :
( ( abelia3891852623213500406t_unit @ R )
=> ( ( monoid5589397312508706001t_unit @ R )
=> ( ! [X4: list_a,Y3: list_a,Z2: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y3 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X4 @ Z2 ) @ ( mult_l7073676228092353617t_unit @ R @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: list_a,Y3: list_a,Z2: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ Z2 @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y3 ) )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z2 @ X4 ) @ ( mult_l7073676228092353617t_unit @ R @ Z2 @ Y3 ) ) ) ) ) )
=> ( ring_l6212528067271185461t_unit @ R ) ) ) ) ) ).
% ringI
thf(fact_204_ring_Oadd__pow__ldistr__int,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,K: int] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_pow_a_b_int @ R @ K @ A ) @ B )
= ( add_pow_a_b_int @ R @ K @ ( mult_a_ring_ext_a_b @ R @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_ldistr_int
thf(fact_205_ring_Oadd__pow__ldistr__int,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,K: int] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( add_po2638046716968164713it_int @ R @ K @ A ) @ B )
= ( add_po2638046716968164713it_int @ R @ K @ ( mult_l7073676228092353617t_unit @ R @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_ldistr_int
thf(fact_206_ring_Oadd__pow__rdistr__int,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,K: int] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ A @ ( add_pow_a_b_int @ R @ K @ B ) )
= ( add_pow_a_b_int @ R @ K @ ( mult_a_ring_ext_a_b @ R @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_rdistr_int
thf(fact_207_ring_Oadd__pow__rdistr__int,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,K: int] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ A @ ( add_po2638046716968164713it_int @ R @ K @ B ) )
= ( add_po2638046716968164713it_int @ R @ K @ ( mult_l7073676228092353617t_unit @ R @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_rdistr_int
thf(fact_208_semiring_Ointro,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ R )
=> ( ( monoid8385113658579753027xt_a_b @ R )
=> ( ( semiring_axioms_a_b @ R )
=> ( semiring_a_b @ R ) ) ) ) ).
% semiring.intro
thf(fact_209_irreducible__prod__rI,axiom,
! [A: a,B: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_rI
thf(fact_210_line__extension__mem__iff,axiom,
! [U: a,K2: set_a,A: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ K2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ A ) @ Y4 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_211_add_Oint__pow__mult,axiom,
! [X: a,I: int,J: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ ( plus_plus_int @ I @ J ) @ X )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ J @ X ) ) ) ) ).
% add.int_pow_mult
thf(fact_212_Units__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_213_Units__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X4 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_214_add_Oint__pow__pow,axiom,
! [X: a,M: int,N: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ M @ ( add_pow_a_b_int @ r @ N @ X ) )
= ( add_pow_a_b_int @ r @ ( times_times_int @ N @ M ) @ X ) ) ) ).
% add.int_pow_pow
thf(fact_215_local_Ominus__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_216_add_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X4 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_217_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X4 )
= X4 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_218_add_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X4 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_219_add_Oinv__comm,axiom,
! [X: a,Y: a] :
( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_220_inv__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% inv_unique
thf(fact_221_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X4 )
= X4 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_222_Units__inv__comm,axiom,
! [X: a,Y: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_inv_comm
thf(fact_223_monoid_Oone__closed,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ).
% monoid.one_closed
thf(fact_224_monoid_Oone__closed,axiom,
! [G: partia2670972154091845814t_unit] :
( ( monoid5589397312508706001t_unit @ G )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) ) ) ).
% monoid.one_closed
thf(fact_225_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_226_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_227_add_Oint__pow__one,axiom,
! [Z: int] :
( ( add_pow_a_b_int @ r @ Z @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% add.int_pow_one
thf(fact_228_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_229_monoid_Or__one,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X @ ( one_a_ring_ext_a_b @ G ) )
= X ) ) ) ).
% monoid.r_one
thf(fact_230_monoid_Or__one,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ X @ ( one_li8328186300101108157t_unit @ G ) )
= X ) ) ) ).
% monoid.r_one
thf(fact_231_monoid_Ol__one,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X )
= X ) ) ) ).
% monoid.l_one
thf(fact_232_monoid_Ol__one,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( one_li8328186300101108157t_unit @ G ) @ X )
= X ) ) ) ).
% monoid.l_one
thf(fact_233_add_Ol__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ A )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_234_add_Ol__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ X @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_235_add_Or__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_236_add_Or__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ A @ X ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_237_l__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
= X ) ) ).
% l_zero
thf(fact_238_r__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
= X ) ) ).
% r_zero
thf(fact_239_l__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_240_r__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_241_l__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
= X ) ) ).
% l_one
thf(fact_242_r__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
= X ) ) ).
% r_one
thf(fact_243_ring__hom__one,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_hom_one
thf(fact_244_ring__iso__memE_I4_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_245_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_246_ring_Oring__simprules_I6_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% ring.ring_simprules(6)
thf(fact_247_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_248_ring_Oring__simprules_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% ring.ring_simprules(2)
thf(fact_249_monoid_OUnits__one__closed,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( units_a_ring_ext_a_b @ G ) ) ) ).
% monoid.Units_one_closed
thf(fact_250_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_251_abelian__groupE_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( abelia3891852623213500406t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% abelian_groupE(2)
thf(fact_252_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_253_abelian__monoid_Ozero__closed,axiom,
! [G: partia2670972154091845814t_unit] :
( ( abelia226231641709521465t_unit @ G )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_254_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_255_abelian__monoidE_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( abelia226231641709521465t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_256_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_257_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_258_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_259_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_260_ring__hom__zero,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( ring_a_b @ R )
=> ( ( ring_a_b @ S )
=> ( ( H @ ( zero_a_b @ R ) )
= ( zero_a_b @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_261_ring__hom__zero,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( ring_a_b @ R )
=> ( ( ring_l6212528067271185461t_unit @ S )
=> ( ( H @ ( zero_a_b @ R ) )
= ( zero_l4142658623432671053t_unit @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_262_ring__hom__zero,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ring_a_b @ S )
=> ( ( H @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_a_b @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_263_ring__hom__zero,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit] :
( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ring_l6212528067271185461t_unit @ S )
=> ( ( H @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_264_Group_Omonoid__def,axiom,
( monoid8385113658579753027xt_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b] :
( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G2 @ X3 @ Y4 ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) )
& ! [X3: a,Y4: a,Z3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( mult_a_ring_ext_a_b @ G2 @ ( mult_a_ring_ext_a_b @ G2 @ X3 @ Y4 ) @ Z3 )
= ( mult_a_ring_ext_a_b @ G2 @ X3 @ ( mult_a_ring_ext_a_b @ G2 @ Y4 @ Z3 ) ) ) ) ) )
& ( member_a @ ( one_a_ring_ext_a_b @ G2 ) @ ( partia707051561876973205xt_a_b @ G2 ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( mult_a_ring_ext_a_b @ G2 @ ( one_a_ring_ext_a_b @ G2 ) @ X3 )
= X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( mult_a_ring_ext_a_b @ G2 @ X3 @ ( one_a_ring_ext_a_b @ G2 ) )
= X3 ) ) ) ) ) ).
% Group.monoid_def
thf(fact_265_Group_Omonoid__def,axiom,
( monoid5589397312508706001t_unit
= ( ^ [G2: partia2670972154091845814t_unit] :
( ! [X3: list_a,Y4: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ G2 @ X3 @ Y4 ) @ ( partia5361259788508890537t_unit @ G2 ) ) ) )
& ! [X3: list_a,Y4: list_a,Z3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( mult_l7073676228092353617t_unit @ G2 @ ( mult_l7073676228092353617t_unit @ G2 @ X3 @ Y4 ) @ Z3 )
= ( mult_l7073676228092353617t_unit @ G2 @ X3 @ ( mult_l7073676228092353617t_unit @ G2 @ Y4 @ Z3 ) ) ) ) ) )
& ( member_list_a @ ( one_li8328186300101108157t_unit @ G2 ) @ ( partia5361259788508890537t_unit @ G2 ) )
& ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( mult_l7073676228092353617t_unit @ G2 @ ( one_li8328186300101108157t_unit @ G2 ) @ X3 )
= X3 ) )
& ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( mult_l7073676228092353617t_unit @ G2 @ X3 @ ( one_li8328186300101108157t_unit @ G2 ) )
= X3 ) ) ) ) ) ).
% Group.monoid_def
thf(fact_266_monoidI,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X4 @ Y3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X4: a,Y3: a,Z2: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X4 @ Y3 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ G @ X4 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X4 )
= X4 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X4 @ ( one_a_ring_ext_a_b @ G ) )
= X4 ) )
=> ( monoid8385113658579753027xt_a_b @ G ) ) ) ) ) ) ).
% monoidI
thf(fact_267_monoidI,axiom,
! [G: partia2670972154091845814t_unit] :
( ! [X4: list_a,Y3: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ X4 @ Y3 ) @ ( partia5361259788508890537t_unit @ G ) ) ) )
=> ( ( member_list_a @ ( one_li8328186300101108157t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) )
=> ( ! [X4: list_a,Y3: list_a,Z2: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ X4 @ Y3 ) @ Z2 )
= ( mult_l7073676228092353617t_unit @ G @ X4 @ ( mult_l7073676228092353617t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( one_li8328186300101108157t_unit @ G ) @ X4 )
= X4 ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ X4 @ ( one_li8328186300101108157t_unit @ G ) )
= X4 ) )
=> ( monoid5589397312508706001t_unit @ G ) ) ) ) ) ) ).
% monoidI
thf(fact_268_monoid_Oone__unique,axiom,
! [G: partia2175431115845679010xt_a_b,U: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ U @ X4 )
= X4 ) )
=> ( U
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.one_unique
thf(fact_269_monoid_Oone__unique,axiom,
! [G: partia2670972154091845814t_unit,U: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ G ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ U @ X4 )
= X4 ) )
=> ( U
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ).
% monoid.one_unique
thf(fact_270_monoid_Oinv__unique,axiom,
! [G: partia2175431115845679010xt_a_b,Y: a,X: a,Y2: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y2 )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% monoid.inv_unique
thf(fact_271_monoid_Oinv__unique,axiom,
! [G: partia2670972154091845814t_unit,Y: list_a,X: list_a,Y2: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ Y @ X )
= ( one_li8328186300101108157t_unit @ G ) )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ X @ Y2 )
= ( one_li8328186300101108157t_unit @ G ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% monoid.inv_unique
thf(fact_272_Group_Omonoid_Ointro,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X4 @ Y3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) )
=> ( ! [X4: a,Y3: a,Z2: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X4 @ Y3 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ G @ X4 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X4 )
= X4 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X4 @ ( one_a_ring_ext_a_b @ G ) )
= X4 ) )
=> ( monoid8385113658579753027xt_a_b @ G ) ) ) ) ) ) ).
% Group.monoid.intro
thf(fact_273_Group_Omonoid_Ointro,axiom,
! [G: partia2670972154091845814t_unit] :
( ! [X4: list_a,Y3: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ X4 @ Y3 ) @ ( partia5361259788508890537t_unit @ G ) ) ) )
=> ( ! [X4: list_a,Y3: list_a,Z2: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ X4 @ Y3 ) @ Z2 )
= ( mult_l7073676228092353617t_unit @ G @ X4 @ ( mult_l7073676228092353617t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ( member_list_a @ ( one_li8328186300101108157t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( one_li8328186300101108157t_unit @ G ) @ X4 )
= X4 ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ X4 @ ( one_li8328186300101108157t_unit @ G ) )
= X4 ) )
=> ( monoid5589397312508706001t_unit @ G ) ) ) ) ) ) ).
% Group.monoid.intro
thf(fact_274_ring_Oring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X )
= X ) ) ) ).
% ring.ring_simprules(12)
thf(fact_275_ring_Oring__simprules_I12_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ X )
= X ) ) ) ).
% ring.ring_simprules(12)
thf(fact_276_ring_Oring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% ring.ring_simprules(8)
thf(fact_277_ring_Oring__simprules_I8_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% ring.ring_simprules(8)
thf(fact_278_ring_Oring__simprules_I15_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
= X ) ) ) ).
% ring.ring_simprules(15)
thf(fact_279_ring_Oring__simprules_I15_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= X ) ) ) ).
% ring.ring_simprules(15)
thf(fact_280_monoid_OUnits__inv__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% monoid.Units_inv_comm
thf(fact_281_monoid_OUnits__inv__comm,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ X @ Y )
= ( one_li8328186300101108157t_unit @ G ) )
=> ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ Y @ X )
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ) ).
% monoid.Units_inv_comm
thf(fact_282_ring_Oring__simprules_I24_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_283_ring_Oring__simprules_I24_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_284_ring_Oring__simprules_I25_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_285_ring_Oring__simprules_I25_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_286_abelian__groupI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X4 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Z2: a] :
( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X4 @ Y3 ) @ Z2 )
= ( add_a_b @ R @ X4 @ ( add_a_b @ R @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X4 @ Y3 )
= ( add_a_b @ R @ Y3 @ X4 ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X4 )
= X4 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( add_a_b @ R @ Xa @ X4 )
= ( zero_a_b @ R ) ) ) )
=> ( abelian_group_a_b @ R ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_287_abelian__groupI,axiom,
! [R: partia2670972154091845814t_unit] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y3 ) @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ! [Z2: list_a] :
( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y3 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ X4 @ ( add_li7652885771158616974t_unit @ R @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X4 @ Y3 )
= ( add_li7652885771158616974t_unit @ R @ Y3 @ X4 ) ) ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X4 )
= X4 ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ ( partia5361259788508890537t_unit @ R ) )
& ( ( add_li7652885771158616974t_unit @ R @ Xa @ X4 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) )
=> ( abelia3891852623213500406t_unit @ R ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_288_abelian__groupE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% abelian_groupE(5)
thf(fact_289_abelian__groupE_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( abelia3891852623213500406t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% abelian_groupE(5)
thf(fact_290_abelian__groupE_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( add_a_b @ R @ X4 @ X )
= ( zero_a_b @ R ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_291_abelian__groupE_I6_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( abelia3891852623213500406t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
& ( ( add_li7652885771158616974t_unit @ R @ X4 @ X )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_292_abelian__monoidI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X4 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ! [X4: a,Y3: a,Z2: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X4 @ Y3 ) @ Z2 )
= ( add_a_b @ R @ X4 @ ( add_a_b @ R @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X4 )
= X4 ) )
=> ( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X4 @ Y3 )
= ( add_a_b @ R @ Y3 @ X4 ) ) ) )
=> ( abelian_monoid_a_b @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_293_abelian__monoidI,axiom,
! [R: partia2670972154091845814t_unit] :
( ! [X4: list_a,Y3: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y3 ) @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ! [X4: list_a,Y3: list_a,Z2: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y3 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ X4 @ ( add_li7652885771158616974t_unit @ R @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X4 )
= X4 ) )
=> ( ! [X4: list_a,Y3: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X4 @ Y3 )
= ( add_li7652885771158616974t_unit @ R @ Y3 @ X4 ) ) ) )
=> ( abelia226231641709521465t_unit @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_294_abelian__monoid_Ominus__unique,axiom,
! [G: partia2175431115845679010xt_a_b,Y: a,X: a,Y2: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( ( add_a_b @ G @ Y @ X )
= ( zero_a_b @ G ) )
=> ( ( ( add_a_b @ G @ X @ Y2 )
= ( zero_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_295_abelian__monoid_Ominus__unique,axiom,
! [G: partia2670972154091845814t_unit,Y: list_a,X: list_a,Y2: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( ( add_li7652885771158616974t_unit @ G @ Y @ X )
= ( zero_l4142658623432671053t_unit @ G ) )
=> ( ( ( add_li7652885771158616974t_unit @ G @ X @ Y2 )
= ( zero_l4142658623432671053t_unit @ G ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_296_abelian__monoid_Or__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X @ ( zero_a_b @ G ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_297_abelian__monoid_Or__zero,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X @ ( zero_l4142658623432671053t_unit @ G ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_298_abelian__monoid_Ol__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( zero_a_b @ G ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_299_abelian__monoid_Ol__zero,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( zero_l4142658623432671053t_unit @ G ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_300_abelian__monoidE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_301_abelian__monoidE_I4_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_302_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_303_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_304_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_305_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_306_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_307_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_308_semiring_Or__null,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.r_null
thf(fact_309_semiring_Or__null,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.r_null
thf(fact_310_semiring_Ol__null,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.l_null
thf(fact_311_semiring_Ol__null,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.l_null
thf(fact_312_irreducibleD,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( irredu6211895646901577903xt_a_b @ G @ A )
=> ( ( proper19828929941537682xt_a_b @ G @ B @ A )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ).
% irreducibleD
thf(fact_313_irreducibleD,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( irredu4230924414530676029t_unit @ G @ A )
=> ( ( proper8313688649498433056t_unit @ G @ B @ A )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ).
% irreducibleD
thf(fact_314_irreducibleE,axiom,
! [G: partia2175431115845679010xt_a_b,A: a] :
( ( irredu6211895646901577903xt_a_b @ G @ A )
=> ~ ( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
=> ~ ! [B4: a] :
( ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ G ) )
& ( proper19828929941537682xt_a_b @ G @ B4 @ A ) )
=> ( member_a @ B4 @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ).
% irreducibleE
thf(fact_315_irreducibleE,axiom,
! [G: partia2670972154091845814t_unit,A: list_a] :
( ( irredu4230924414530676029t_unit @ G @ A )
=> ~ ( ~ ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) )
=> ~ ! [B4: list_a] :
( ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ G ) )
& ( proper8313688649498433056t_unit @ G @ B4 @ A ) )
=> ( member_list_a @ B4 @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ).
% irreducibleE
thf(fact_316_irreducibleI,axiom,
! [A: a,G: partia2175431115845679010xt_a_b] :
( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
=> ( ! [B2: a] :
( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( proper19828929941537682xt_a_b @ G @ B2 @ A )
=> ( member_a @ B2 @ ( units_a_ring_ext_a_b @ G ) ) ) )
=> ( irredu6211895646901577903xt_a_b @ G @ A ) ) ) ).
% irreducibleI
thf(fact_317_irreducibleI,axiom,
! [A: list_a,G: partia2670972154091845814t_unit] :
( ~ ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) )
=> ( ! [B2: list_a] :
( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( proper8313688649498433056t_unit @ G @ B2 @ A )
=> ( member_list_a @ B2 @ ( units_2932844235741507942t_unit @ G ) ) ) )
=> ( irredu4230924414530676029t_unit @ G @ A ) ) ) ).
% irreducibleI
thf(fact_318_irreducible__def,axiom,
( irredu6211895646901577903xt_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b,A4: a] :
( ~ ( member_a @ A4 @ ( units_a_ring_ext_a_b @ G2 ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( proper19828929941537682xt_a_b @ G2 @ X3 @ A4 )
=> ( member_a @ X3 @ ( units_a_ring_ext_a_b @ G2 ) ) ) ) ) ) ) ).
% irreducible_def
thf(fact_319_irreducible__def,axiom,
( irredu4230924414530676029t_unit
= ( ^ [G2: partia2670972154091845814t_unit,A4: list_a] :
( ~ ( member_list_a @ A4 @ ( units_2932844235741507942t_unit @ G2 ) )
& ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( proper8313688649498433056t_unit @ G2 @ X3 @ A4 )
=> ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ G2 ) ) ) ) ) ) ) ).
% irreducible_def
thf(fact_320_monoid_OUnits__l__inv__ex,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ X4 @ X )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.Units_l_inv_ex
thf(fact_321_monoid_OUnits__l__inv__ex,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
& ( ( mult_l7073676228092353617t_unit @ G @ X4 @ X )
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ).
% monoid.Units_l_inv_ex
thf(fact_322_monoid_OUnits__r__inv__ex,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ X @ X4 )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.Units_r_inv_ex
thf(fact_323_monoid_OUnits__r__inv__ex,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
& ( ( mult_l7073676228092353617t_unit @ G @ X @ X4 )
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ).
% monoid.Units_r_inv_ex
thf(fact_324_monoid__cancel_Oirreducible__cong,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,A5: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( irredu6211895646901577903xt_a_b @ G @ A )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ A5 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( irredu6211895646901577903xt_a_b @ G @ A5 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_325_monoid__cancel_Oirreducible__cong,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,A5: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( irredu4230924414530676029t_unit @ G @ A )
=> ( ( associ8407585678920448409t_unit @ G @ A @ A5 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ G ) )
=> ( irredu4230924414530676029t_unit @ G @ A5 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_326_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H: a > a,S: partia2175431115845679010xt_a_b] :
( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X4 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X4 @ Y3 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X4 @ Y3 ) )
= ( add_a_b @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_327_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H: a > list_a,S: partia2670972154091845814t_unit] :
( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H @ X4 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R @ X4 @ Y3 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ! [X4: a,Y3: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X4 @ Y3 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_328_ring__hom__memI,axiom,
! [R: partia2670972154091845814t_unit,H: list_a > a,S: partia2175431115845679010xt_a_b] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H @ X4 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X4: list_a,Y3: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X4 @ Y3 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ! [X4: list_a,Y3: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y3 ) )
= ( add_a_b @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ( ( H @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_329_ring__hom__memI,axiom,
! [R: partia2670972154091845814t_unit,H: list_a > list_a,S: partia2670972154091845814t_unit] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H @ X4 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X4: list_a,Y3: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X4 @ Y3 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ! [X4: list_a,Y3: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y3 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ( ( H @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_330_semiring__axioms_Ointro,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X4: a,Y3: a,Z2: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X4 @ Y3 ) @ Z2 )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X4 @ Z2 ) @ ( mult_a_ring_ext_a_b @ R @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: a,Y3: a,Z2: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z2 @ ( add_a_b @ R @ X4 @ Y3 ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z2 @ X4 ) @ ( mult_a_ring_ext_a_b @ R @ Z2 @ Y3 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X4 )
= ( zero_a_b @ R ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X4 @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) )
=> ( semiring_axioms_a_b @ R ) ) ) ) ) ).
% semiring_axioms.intro
thf(fact_331_semiring__axioms_Ointro,axiom,
! [R: partia2670972154091845814t_unit] :
( ! [X4: list_a,Y3: list_a,Z2: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y3 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X4 @ Z2 ) @ ( mult_l7073676228092353617t_unit @ R @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: list_a,Y3: list_a,Z2: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ Z2 @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y3 ) )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z2 @ X4 ) @ ( mult_l7073676228092353617t_unit @ R @ Z2 @ Y3 ) ) ) ) ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X4 )
= ( zero_l4142658623432671053t_unit @ R ) ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X4 @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) )
=> ( semiri6743050769115779782t_unit @ R ) ) ) ) ) ).
% semiring_axioms.intro
thf(fact_332_semiring__axioms__def,axiom,
( semiring_axioms_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b] :
( ! [X3: a,Y4: a,Z3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( add_a_b @ R2 @ X3 @ Y4 ) @ Z3 )
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ X3 @ Z3 ) @ ( mult_a_ring_ext_a_b @ R2 @ Y4 @ Z3 ) ) ) ) ) )
& ! [X3: a,Y4: a,Z3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ Z3 @ ( add_a_b @ R2 @ X3 @ Y4 ) )
= ( add_a_b @ R2 @ ( mult_a_ring_ext_a_b @ R2 @ Z3 @ X3 ) @ ( mult_a_ring_ext_a_b @ R2 @ Z3 @ Y4 ) ) ) ) ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ ( zero_a_b @ R2 ) @ X3 )
= ( zero_a_b @ R2 ) ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( mult_a_ring_ext_a_b @ R2 @ X3 @ ( zero_a_b @ R2 ) )
= ( zero_a_b @ R2 ) ) ) ) ) ) ).
% semiring_axioms_def
thf(fact_333_semiring__axioms__def,axiom,
( semiri6743050769115779782t_unit
= ( ^ [R2: partia2670972154091845814t_unit] :
( ! [X3: list_a,Y4: list_a,Z3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y4 ) @ Z3 )
= ( add_li7652885771158616974t_unit @ R2 @ ( mult_l7073676228092353617t_unit @ R2 @ X3 @ Z3 ) @ ( mult_l7073676228092353617t_unit @ R2 @ Y4 @ Z3 ) ) ) ) ) )
& ! [X3: list_a,Y4: list_a,Z3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ Z3 @ ( add_li7652885771158616974t_unit @ R2 @ X3 @ Y4 ) )
= ( add_li7652885771158616974t_unit @ R2 @ ( mult_l7073676228092353617t_unit @ R2 @ Z3 @ X3 ) @ ( mult_l7073676228092353617t_unit @ R2 @ Z3 @ Y4 ) ) ) ) ) )
& ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ ( zero_l4142658623432671053t_unit @ R2 ) @ X3 )
= ( zero_l4142658623432671053t_unit @ R2 ) ) )
& ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( mult_l7073676228092353617t_unit @ R2 @ X3 @ ( zero_l4142658623432671053t_unit @ R2 ) )
= ( zero_l4142658623432671053t_unit @ R2 ) ) ) ) ) ) ).
% semiring_axioms_def
thf(fact_334_monoid_Oirreducible__prod__rI,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( irredu6211895646901577903xt_a_b @ G @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( irredu6211895646901577903xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A @ B ) ) ) ) ) ) ) ).
% monoid.irreducible_prod_rI
thf(fact_335_monoid_Oirreducible__prod__rI,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( irredu4230924414530676029t_unit @ G @ A )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( irredu4230924414530676029t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) ) ) ) ) ) ) ).
% monoid.irreducible_prod_rI
thf(fact_336_const__term__def,axiom,
! [P2: list_a] :
( ( const_term_a_b @ r @ P2 )
= ( eval_a_b @ r @ P2 @ ( zero_a_b @ r ) ) ) ).
% const_term_def
thf(fact_337_ring_Oline__extension__mem__iff,axiom,
! [R: partia2670972154091845814t_unit,U: list_a,K2: set_list_a,A: list_a,E: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ R @ K2 @ A @ E ) )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ K2 )
& ? [Y4: list_a] :
( ( member_list_a @ Y4 @ E )
& ( U
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X3 @ A ) @ Y4 ) ) ) ) ) ) ) ).
% ring.line_extension_mem_iff
thf(fact_338_ring_Oline__extension__mem__iff,axiom,
! [R: partia2175431115845679010xt_a_b,U: a,K2: set_a,A: a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ K2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ E )
& ( U
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ A ) @ Y4 ) ) ) ) ) ) ) ).
% ring.line_extension_mem_iff
thf(fact_339_group__l__invI,axiom,
( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ Xa @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) )
=> ( group_a_ring_ext_a_b @ r ) ) ).
% group_l_invI
thf(fact_340_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_341_line__extension__in__carrier,axiom,
! [K2: set_a,A: a,E: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_342_ring_Oequality,axiom,
! [R3: partia2175431115845679010xt_a_b,R4: partia2175431115845679010xt_a_b] :
( ( ( partia707051561876973205xt_a_b @ R3 )
= ( partia707051561876973205xt_a_b @ R4 ) )
=> ( ( ( mult_a_ring_ext_a_b @ R3 )
= ( mult_a_ring_ext_a_b @ R4 ) )
=> ( ( ( one_a_ring_ext_a_b @ R3 )
= ( one_a_ring_ext_a_b @ R4 ) )
=> ( ( ( zero_a_b @ R3 )
= ( zero_a_b @ R4 ) )
=> ( ( ( add_a_b @ R3 )
= ( add_a_b @ R4 ) )
=> ( ( ( more_a_b @ R3 )
= ( more_a_b @ R4 ) )
=> ( R3 = R4 ) ) ) ) ) ) ) ).
% ring.equality
thf(fact_343_ring_Oequality,axiom,
! [R3: partia2670972154091845814t_unit,R4: partia2670972154091845814t_unit] :
( ( ( partia5361259788508890537t_unit @ R3 )
= ( partia5361259788508890537t_unit @ R4 ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R3 )
= ( mult_l7073676228092353617t_unit @ R4 ) )
=> ( ( ( one_li8328186300101108157t_unit @ R3 )
= ( one_li8328186300101108157t_unit @ R4 ) )
=> ( ( ( zero_l4142658623432671053t_unit @ R3 )
= ( zero_l4142658623432671053t_unit @ R4 ) )
=> ( ( ( add_li7652885771158616974t_unit @ R3 )
= ( add_li7652885771158616974t_unit @ R4 ) )
=> ( ( ( more_l6502722259342772890t_unit @ R3 )
= ( more_l6502722259342772890t_unit @ R4 ) )
=> ( R3 = R4 ) ) ) ) ) ) ) ).
% ring.equality
thf(fact_344_inv__char,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ X )
= Y ) ) ) ) ) ).
% inv_char
thf(fact_345_inv__unique_H,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( Y
= ( m_inv_a_ring_ext_a_b @ r @ X ) ) ) ) ) ) ).
% inv_unique'
thf(fact_346_l__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_347_inv__eq__imp__eq,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( m_inv_a_ring_ext_a_b @ r @ Y ) )
=> ( X = Y ) ) ) ) ).
% inv_eq_imp_eq
thf(fact_348_r__neg2,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_349_r__neg1,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_350_local_Ominus__add,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_351_a__transpose__inv,axiom,
! [X: a,Y: a,Z: a] :
( ( ( add_a_b @ r @ X @ Y )
= Z )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_352_add_Oinv__solve__right_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
= A )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right'
thf(fact_353_add_Oinv__solve__right,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right
thf(fact_354_add_Oinv__solve__left_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
= A )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left'
thf(fact_355_add_Oinv__solve__left,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left
thf(fact_356_add_Oinv__mult__group,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_357_r__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% r_minus
thf(fact_358_l__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% l_minus
thf(fact_359_add_Oint__pow__inv,axiom,
! [X: a,I: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ I @ ( a_inv_a_b @ r @ X ) )
= ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% add.int_pow_inv
thf(fact_360_inv__eq__one__eq,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( one_a_ring_ext_a_b @ r ) )
= ( X
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% inv_eq_one_eq
thf(fact_361_const__term__not__zero,axiom,
! [P2: list_a] :
( ( ( const_term_a_b @ r @ P2 )
!= ( zero_a_b @ r ) )
=> ( P2 != nil_a ) ) ).
% const_term_not_zero
thf(fact_362_r__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_363_minus__equality,axiom,
! [Y: a,X: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X )
= Y ) ) ) ) ).
% minus_equality
thf(fact_364_inv__eq__neg__one__eq,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) )
= ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% inv_eq_neg_one_eq
thf(fact_365_group_Oinv__inv,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( m_inv_a_Product_unit @ G @ ( m_inv_a_Product_unit @ G @ X ) )
= X ) ) ) ).
% group.inv_inv
thf(fact_366_group_Oinv__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( m_inv_a_ring_ext_a_b @ G @ ( m_inv_a_ring_ext_a_b @ G @ X ) )
= X ) ) ) ).
% group.inv_inv
thf(fact_367_group_Oinv__inv,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( m_inv_2802811658206063947t_unit @ G @ ( m_inv_2802811658206063947t_unit @ G @ X ) )
= X ) ) ) ).
% group.inv_inv
thf(fact_368_univ__poly__zero__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) ) ).
% univ_poly_zero_closed
thf(fact_369_local_Ominus__minus,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
= X ) ) ).
% local.minus_minus
thf(fact_370_a__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_371_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_372_inv__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) )
= ( one_a_ring_ext_a_b @ r ) ) ).
% inv_one
thf(fact_373_Units__inv__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ X ) )
= X ) ) ).
% Units_inv_inv
thf(fact_374_Units__inv__Units,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_inv_Units
thf(fact_375_add_Oinv__eq__1__iff,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X )
= ( zero_a_b @ r ) )
= ( X
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_376_Units__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_inv_closed
thf(fact_377_Units__minus__one__closed,axiom,
member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_minus_one_closed
thf(fact_378_inv__neg__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) )
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ).
% inv_neg_one
thf(fact_379_Units__l__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% Units_l_inv
thf(fact_380_Units__r__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( m_inv_a_ring_ext_a_b @ r @ X ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% Units_r_inv
thf(fact_381_group_Oinv__closed,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ ( m_inv_a_Product_unit @ G @ X ) @ ( partia6735698275553448452t_unit @ G ) ) ) ) ).
% group.inv_closed
thf(fact_382_group_Oinv__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ G @ X ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% group.inv_closed
thf(fact_383_group_Oinv__closed,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ G @ X ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ).
% group.inv_closed
thf(fact_384_univ__poly__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K2 ) )
= nil_a ) ).
% univ_poly_zero
thf(fact_385_ring_Oconst__term_Ocong,axiom,
const_term_a_b = const_term_a_b ).
% ring.const_term.cong
thf(fact_386_group_Ois__group,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( group_a_ring_ext_a_b @ G )
=> ( group_a_ring_ext_a_b @ G ) ) ).
% group.is_group
thf(fact_387_group_Ois__group,axiom,
! [G: partia8223610829204095565t_unit] :
( ( group_a_Product_unit @ G )
=> ( group_a_Product_unit @ G ) ) ).
% group.is_group
thf(fact_388_group_Oinv__solve__right_H,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ B @ ( m_inv_a_Product_unit @ G @ C ) )
= A )
= ( B
= ( mult_a_Product_unit @ G @ A @ C ) ) ) ) ) ) ) ).
% group.inv_solve_right'
thf(fact_389_group_Oinv__solve__right_H,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ B @ ( m_inv_a_ring_ext_a_b @ G @ C ) )
= A )
= ( B
= ( mult_a_ring_ext_a_b @ G @ A @ C ) ) ) ) ) ) ) ).
% group.inv_solve_right'
thf(fact_390_group_Oinv__solve__right_H,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ B @ ( m_inv_2802811658206063947t_unit @ G @ C ) )
= A )
= ( B
= ( mult_l7073676228092353617t_unit @ G @ A @ C ) ) ) ) ) ) ) ).
% group.inv_solve_right'
thf(fact_391_group_Oinv__solve__right,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( A
= ( mult_a_Product_unit @ G @ B @ ( m_inv_a_Product_unit @ G @ C ) ) )
= ( B
= ( mult_a_Product_unit @ G @ A @ C ) ) ) ) ) ) ) ).
% group.inv_solve_right
thf(fact_392_group_Oinv__solve__right,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ G @ B @ ( m_inv_a_ring_ext_a_b @ G @ C ) ) )
= ( B
= ( mult_a_ring_ext_a_b @ G @ A @ C ) ) ) ) ) ) ) ).
% group.inv_solve_right
thf(fact_393_group_Oinv__solve__right,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( A
= ( mult_l7073676228092353617t_unit @ G @ B @ ( m_inv_2802811658206063947t_unit @ G @ C ) ) )
= ( B
= ( mult_l7073676228092353617t_unit @ G @ A @ C ) ) ) ) ) ) ) ).
% group.inv_solve_right
thf(fact_394_group_Oinv__solve__left_H,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ ( m_inv_a_Product_unit @ G @ B ) @ C )
= A )
= ( C
= ( mult_a_Product_unit @ G @ B @ A ) ) ) ) ) ) ) ).
% group.inv_solve_left'
thf(fact_395_group_Oinv__solve__left_H,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ ( m_inv_a_ring_ext_a_b @ G @ B ) @ C )
= A )
= ( C
= ( mult_a_ring_ext_a_b @ G @ B @ A ) ) ) ) ) ) ) ).
% group.inv_solve_left'
thf(fact_396_group_Oinv__solve__left_H,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ ( m_inv_2802811658206063947t_unit @ G @ B ) @ C )
= A )
= ( C
= ( mult_l7073676228092353617t_unit @ G @ B @ A ) ) ) ) ) ) ) ).
% group.inv_solve_left'
thf(fact_397_group_Oinv__solve__left,axiom,
! [G: partia8223610829204095565t_unit,A: a,B: a,C: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( A
= ( mult_a_Product_unit @ G @ ( m_inv_a_Product_unit @ G @ B ) @ C ) )
= ( C
= ( mult_a_Product_unit @ G @ B @ A ) ) ) ) ) ) ) ).
% group.inv_solve_left
thf(fact_398_group_Oinv__solve__left,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ G @ ( m_inv_a_ring_ext_a_b @ G @ B ) @ C ) )
= ( C
= ( mult_a_ring_ext_a_b @ G @ B @ A ) ) ) ) ) ) ) ).
% group.inv_solve_left
thf(fact_399_group_Oinv__solve__left,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( A
= ( mult_l7073676228092353617t_unit @ G @ ( m_inv_2802811658206063947t_unit @ G @ B ) @ C ) )
= ( C
= ( mult_l7073676228092353617t_unit @ G @ B @ A ) ) ) ) ) ) ) ).
% group.inv_solve_left
thf(fact_400_group_Oinv__mult__group,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( m_inv_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X @ Y ) )
= ( mult_a_Product_unit @ G @ ( m_inv_a_Product_unit @ G @ Y ) @ ( m_inv_a_Product_unit @ G @ X ) ) ) ) ) ) ).
% group.inv_mult_group
thf(fact_401_group_Oinv__mult__group,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( m_inv_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ G @ ( m_inv_a_ring_ext_a_b @ G @ Y ) @ ( m_inv_a_ring_ext_a_b @ G @ X ) ) ) ) ) ) ).
% group.inv_mult_group
thf(fact_402_group_Oinv__mult__group,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( m_inv_2802811658206063947t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ G @ ( m_inv_2802811658206063947t_unit @ G @ Y ) @ ( m_inv_2802811658206063947t_unit @ G @ X ) ) ) ) ) ) ).
% group.inv_mult_group
thf(fact_403_group_Oinv__eq__1__iff,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( m_inv_a_Product_unit @ G @ X )
= ( one_a_Product_unit @ G ) )
= ( X
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% group.inv_eq_1_iff
thf(fact_404_group_Oinv__eq__1__iff,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ G @ X )
= ( one_a_ring_ext_a_b @ G ) )
= ( X
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% group.inv_eq_1_iff
thf(fact_405_group_Oinv__eq__1__iff,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( ( m_inv_2802811658206063947t_unit @ G @ X )
= ( one_li8328186300101108157t_unit @ G ) )
= ( X
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ).
% group.inv_eq_1_iff
thf(fact_406_ring_Oinv__neg__one,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( m_inv_a_ring_ext_a_b @ R @ ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) )
= ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) ) ) ).
% ring.inv_neg_one
thf(fact_407_group_OUnits,axiom,
! [G: partia8223610829204095565t_unit] :
( ( group_a_Product_unit @ G )
=> ( ord_less_eq_set_a @ ( partia6735698275553448452t_unit @ G ) @ ( units_a_Product_unit @ G ) ) ) ).
% group.Units
thf(fact_408_group_OUnits,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( group_a_ring_ext_a_b @ G )
=> ( ord_less_eq_set_a @ ( partia707051561876973205xt_a_b @ G ) @ ( units_a_ring_ext_a_b @ G ) ) ) ).
% group.Units
thf(fact_409_group_OUnits,axiom,
! [G: partia2670972154091845814t_unit] :
( ( group_5191797799156037398t_unit @ G )
=> ( ord_le8861187494160871172list_a @ ( partia5361259788508890537t_unit @ G ) @ ( units_2932844235741507942t_unit @ G ) ) ) ).
% group.Units
thf(fact_410_ring_Oconst__term__not__zero,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ( const_6738166269504826821t_unit @ R @ P2 )
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( P2 != nil_list_a ) ) ) ).
% ring.const_term_not_zero
thf(fact_411_ring_Oconst__term__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( ( const_term_a_b @ R @ P2 )
!= ( zero_a_b @ R ) )
=> ( P2 != nil_a ) ) ) ).
% ring.const_term_not_zero
thf(fact_412_group_Ois__monoid,axiom,
! [G: partia8223610829204095565t_unit] :
( ( group_a_Product_unit @ G )
=> ( monoid2746444814143937472t_unit @ G ) ) ).
% group.is_monoid
thf(fact_413_group_Ois__monoid,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( group_a_ring_ext_a_b @ G )
=> ( monoid8385113658579753027xt_a_b @ G ) ) ).
% group.is_monoid
thf(fact_414_group_Ol__inv,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( m_inv_a_Product_unit @ G @ X ) @ X )
= ( one_a_Product_unit @ G ) ) ) ) ).
% group.l_inv
thf(fact_415_group_Ol__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( m_inv_a_ring_ext_a_b @ G @ X ) @ X )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ).
% group.l_inv
thf(fact_416_group_Ol__inv,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( m_inv_2802811658206063947t_unit @ G @ X ) @ X )
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ).
% group.l_inv
thf(fact_417_group_Or__inv,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ X @ ( m_inv_a_Product_unit @ G @ X ) )
= ( one_a_Product_unit @ G ) ) ) ) ).
% group.r_inv
thf(fact_418_group_Or__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X @ ( m_inv_a_ring_ext_a_b @ G @ X ) )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ).
% group.r_inv
thf(fact_419_group_Or__inv,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ X @ ( m_inv_2802811658206063947t_unit @ G @ X ) )
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ).
% group.r_inv
thf(fact_420_group_Oinv__equality,axiom,
! [G: partia8223610829204095565t_unit,Y: a,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ Y @ X )
= ( one_a_Product_unit @ G ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( m_inv_a_Product_unit @ G @ X )
= Y ) ) ) ) ) ).
% group.inv_equality
thf(fact_421_group_Oinv__equality,axiom,
! [G: partia2175431115845679010xt_a_b,Y: a,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( m_inv_a_ring_ext_a_b @ G @ X )
= Y ) ) ) ) ) ).
% group.inv_equality
thf(fact_422_group_Oinv__equality,axiom,
! [G: partia2670972154091845814t_unit,Y: list_a,X: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ Y @ X )
= ( one_li8328186300101108157t_unit @ G ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( m_inv_2802811658206063947t_unit @ G @ X )
= Y ) ) ) ) ) ).
% group.inv_equality
thf(fact_423_ring_Oinv__eq__neg__one__eq,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ R ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ R @ X )
= ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) )
= ( X
= ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) ) ) ) ) ).
% ring.inv_eq_neg_one_eq
thf(fact_424_ring_Oring__simprules_I20_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( a_inv_a_b @ R @ X ) )
= X ) ) ) ).
% ring.ring_simprules(20)
thf(fact_425_ring_Oring__simprules_I20_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) )
= X ) ) ) ).
% ring.ring_simprules(20)
thf(fact_426_ring_Oring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_inv_a_b @ R @ X ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_427_ring_Oring__simprules_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_428_ring_Ominus__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( a_inv_a_b @ R @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ).
% ring.minus_zero
thf(fact_429_ring_Ominus__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% ring.minus_zero
thf(fact_430_abelian__group_Ominus__minus,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_inv_a_b @ G @ ( a_inv_a_b @ G @ X ) )
= X ) ) ) ).
% abelian_group.minus_minus
thf(fact_431_abelian__group_Ominus__minus,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( a_inv_8944721093294617173t_unit @ G @ ( a_inv_8944721093294617173t_unit @ G @ X ) )
= X ) ) ) ).
% abelian_group.minus_minus
thf(fact_432_abelian__group_Oa__inv__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( a_inv_a_b @ G @ X ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% abelian_group.a_inv_closed
thf(fact_433_abelian__group_Oa__inv__closed,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ G @ X ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ).
% abelian_group.a_inv_closed
thf(fact_434_monoid_Oinv__one,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( m_inv_a_Product_unit @ G @ ( one_a_Product_unit @ G ) )
= ( one_a_Product_unit @ G ) ) ) ).
% monoid.inv_one
thf(fact_435_monoid_Oinv__one,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( m_inv_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) )
= ( one_a_ring_ext_a_b @ G ) ) ) ).
% monoid.inv_one
thf(fact_436_Group_Ogroup_Oright__cancel,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a,Z: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ Y @ X )
= ( mult_a_Product_unit @ G @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% Group.group.right_cancel
thf(fact_437_Group_Ogroup_Oright__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
= ( mult_a_ring_ext_a_b @ G @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% Group.group.right_cancel
thf(fact_438_Group_Ogroup_Oright__cancel,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ Y @ X )
= ( mult_l7073676228092353617t_unit @ G @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% Group.group.right_cancel
thf(fact_439_monoid_OUnits__inv__inv,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( ( m_inv_a_Product_unit @ G @ ( m_inv_a_Product_unit @ G @ X ) )
= X ) ) ) ).
% monoid.Units_inv_inv
thf(fact_440_monoid_OUnits__inv__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( m_inv_a_ring_ext_a_b @ G @ ( m_inv_a_ring_ext_a_b @ G @ X ) )
= X ) ) ) ).
% monoid.Units_inv_inv
thf(fact_441_monoid_OUnits__inv__Units,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( member_a @ ( m_inv_a_Product_unit @ G @ X ) @ ( units_a_Product_unit @ G ) ) ) ) ).
% monoid.Units_inv_Units
thf(fact_442_monoid_OUnits__inv__Units,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ G @ X ) @ ( units_a_ring_ext_a_b @ G ) ) ) ) ).
% monoid.Units_inv_Units
thf(fact_443_monoid_Oinv__eq__imp__eq,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ Y @ ( units_a_Product_unit @ G ) )
=> ( ( ( m_inv_a_Product_unit @ G @ X )
= ( m_inv_a_Product_unit @ G @ Y ) )
=> ( X = Y ) ) ) ) ) ).
% monoid.inv_eq_imp_eq
thf(fact_444_monoid_Oinv__eq__imp__eq,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ G @ X )
= ( m_inv_a_ring_ext_a_b @ G @ Y ) )
=> ( X = Y ) ) ) ) ) ).
% monoid.inv_eq_imp_eq
thf(fact_445_group_OUnits__eq,axiom,
! [G: partia8223610829204095565t_unit] :
( ( group_a_Product_unit @ G )
=> ( ( units_a_Product_unit @ G )
= ( partia6735698275553448452t_unit @ G ) ) ) ).
% group.Units_eq
thf(fact_446_group_OUnits__eq,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( units_a_ring_ext_a_b @ G )
= ( partia707051561876973205xt_a_b @ G ) ) ) ).
% group.Units_eq
thf(fact_447_group_OUnits__eq,axiom,
! [G: partia2670972154091845814t_unit] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( units_2932844235741507942t_unit @ G )
= ( partia5361259788508890537t_unit @ G ) ) ) ).
% group.Units_eq
thf(fact_448_ring_Oline__extension__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,A: list_a,E: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ R @ K2 @ A @ E ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_449_ring_Oline__extension__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,A: a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_450_ring_Oring__simprules_I19_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ ( a_inv_a_b @ R @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_451_ring_Oring__simprules_I19_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_452_ring_Oring__simprules_I18_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ ( add_a_b @ R @ X @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_453_ring_Oring__simprules_I18_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_454_ring_Oring__simprules_I17_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_455_ring_Oring__simprules_I17_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_456_ring_Ol__minus,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( a_inv_a_b @ R @ X ) @ Y )
= ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_457_ring_Ol__minus,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ Y )
= ( a_inv_8944721093294617173t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_458_ring_Or__minus,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( a_inv_a_b @ R @ Y ) )
= ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_459_ring_Or__minus,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ Y ) )
= ( a_inv_8944721093294617173t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_460_abelian__group_Or__neg1,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ ( add_a_b @ G @ X @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg1
thf(fact_461_abelian__group_Or__neg1,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( a_inv_8944721093294617173t_unit @ G @ X ) @ ( add_li7652885771158616974t_unit @ G @ X @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg1
thf(fact_462_abelian__group_Or__neg2,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X @ ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg2
thf(fact_463_abelian__group_Or__neg2,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X @ ( add_li7652885771158616974t_unit @ G @ ( a_inv_8944721093294617173t_unit @ G @ X ) @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg2
thf(fact_464_abelian__group_Ominus__add,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_inv_a_b @ G @ ( add_a_b @ G @ X @ Y ) )
= ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ ( a_inv_a_b @ G @ Y ) ) ) ) ) ) ).
% abelian_group.minus_add
thf(fact_465_abelian__group_Ominus__add,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( a_inv_8944721093294617173t_unit @ G @ ( add_li7652885771158616974t_unit @ G @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ G @ ( a_inv_8944721093294617173t_unit @ G @ X ) @ ( a_inv_8944721093294617173t_unit @ G @ Y ) ) ) ) ) ) ).
% abelian_group.minus_add
thf(fact_466_monoid_OUnits__inv__closed,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( member_a @ ( m_inv_a_Product_unit @ G @ X ) @ ( partia6735698275553448452t_unit @ G ) ) ) ) ).
% monoid.Units_inv_closed
thf(fact_467_monoid_OUnits__inv__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ G @ X ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% monoid.Units_inv_closed
thf(fact_468_monoid_OUnits__inv__closed,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ G @ X ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ).
% monoid.Units_inv_closed
thf(fact_469_monoid_Oinv__eq__one__eq,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( ( ( m_inv_a_Product_unit @ G @ X )
= ( one_a_Product_unit @ G ) )
= ( X
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% monoid.inv_eq_one_eq
thf(fact_470_monoid_Oinv__eq__one__eq,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ G @ X )
= ( one_a_ring_ext_a_b @ G ) )
= ( X
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% monoid.inv_eq_one_eq
thf(fact_471_ring_OUnits__minus__one__closed,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) @ ( units_a_ring_ext_a_b @ R ) ) ) ).
% ring.Units_minus_one_closed
thf(fact_472_groupI,axiom,
! [G: partia8223610829204095565t_unit] :
( ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ ( mult_a_Product_unit @ G @ X4 @ Y3 ) @ ( partia6735698275553448452t_unit @ G ) ) ) )
=> ( ( member_a @ ( one_a_Product_unit @ G ) @ ( partia6735698275553448452t_unit @ G ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia6735698275553448452t_unit @ G ) )
=> ! [Z2: a] :
( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X4 @ Y3 ) @ Z2 )
= ( mult_a_Product_unit @ G @ X4 @ ( mult_a_Product_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( one_a_Product_unit @ G ) @ X4 )
= X4 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia6735698275553448452t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ Xa @ X4 )
= ( one_a_Product_unit @ G ) ) ) )
=> ( group_a_Product_unit @ G ) ) ) ) ) ) ).
% groupI
thf(fact_473_groupI,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X4 @ Y3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ! [Z2: a] :
( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X4 @ Y3 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ G @ X4 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X4 )
= X4 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ Xa @ X4 )
= ( one_a_ring_ext_a_b @ G ) ) ) )
=> ( group_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% groupI
thf(fact_474_groupI,axiom,
! [G: partia2670972154091845814t_unit] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
=> ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ X4 @ Y3 ) @ ( partia5361259788508890537t_unit @ G ) ) ) )
=> ( ( member_list_a @ ( one_li8328186300101108157t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
=> ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ! [Z2: list_a] :
( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ X4 @ Y3 ) @ Z2 )
= ( mult_l7073676228092353617t_unit @ G @ X4 @ ( mult_l7073676228092353617t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( one_li8328186300101108157t_unit @ G ) @ X4 )
= X4 ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ ( partia5361259788508890537t_unit @ G ) )
& ( ( mult_l7073676228092353617t_unit @ G @ Xa @ X4 )
= ( one_li8328186300101108157t_unit @ G ) ) ) )
=> ( group_5191797799156037398t_unit @ G ) ) ) ) ) ) ).
% groupI
thf(fact_475_group_Or__cancel__one_H,axiom,
! [G: partia8223610829204095565t_unit,X: a,A: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( X
= ( mult_a_Product_unit @ G @ A @ X ) )
= ( A
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.r_cancel_one'
thf(fact_476_group_Or__cancel__one_H,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,A: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( X
= ( mult_a_ring_ext_a_b @ G @ A @ X ) )
= ( A
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% group.r_cancel_one'
thf(fact_477_group_Or__cancel__one_H,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,A: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( X
= ( mult_l7073676228092353617t_unit @ G @ A @ X ) )
= ( A
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ) ).
% group.r_cancel_one'
thf(fact_478_group_Ol__cancel__one_H,axiom,
! [G: partia8223610829204095565t_unit,X: a,A: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( X
= ( mult_a_Product_unit @ G @ X @ A ) )
= ( A
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.l_cancel_one'
thf(fact_479_group_Ol__cancel__one_H,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,A: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( X
= ( mult_a_ring_ext_a_b @ G @ X @ A ) )
= ( A
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% group.l_cancel_one'
thf(fact_480_group_Ol__cancel__one_H,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,A: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( X
= ( mult_l7073676228092353617t_unit @ G @ X @ A ) )
= ( A
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ) ).
% group.l_cancel_one'
thf(fact_481_group_Or__cancel__one,axiom,
! [G: partia8223610829204095565t_unit,X: a,A: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ A @ X )
= X )
= ( A
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.r_cancel_one
thf(fact_482_group_Or__cancel__one,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,A: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ A @ X )
= X )
= ( A
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% group.r_cancel_one
thf(fact_483_group_Or__cancel__one,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,A: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ A @ X )
= X )
= ( A
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ) ).
% group.r_cancel_one
thf(fact_484_group_Ol__cancel__one,axiom,
! [G: partia8223610829204095565t_unit,X: a,A: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ X @ A )
= X )
= ( A
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.l_cancel_one
thf(fact_485_group_Ol__cancel__one,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,A: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X @ A )
= X )
= ( A
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% group.l_cancel_one
thf(fact_486_group_Ol__cancel__one,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,A: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ X @ A )
= X )
= ( A
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ) ).
% group.l_cancel_one
thf(fact_487_group_Or__inv__ex,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ X @ X4 )
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% group.r_inv_ex
thf(fact_488_group_Or__inv__ex,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ X @ X4 )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% group.r_inv_ex
thf(fact_489_group_Or__inv__ex,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
& ( ( mult_l7073676228092353617t_unit @ G @ X @ X4 )
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ).
% group.r_inv_ex
thf(fact_490_group_Ol__inv__ex,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( group_a_Product_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ X4 @ X )
= ( one_a_Product_unit @ G ) ) ) ) ) ).
% group.l_inv_ex
thf(fact_491_group_Ol__inv__ex,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ X4 @ X )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% group.l_inv_ex
thf(fact_492_group_Ol__inv__ex,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
& ( ( mult_l7073676228092353617t_unit @ G @ X4 @ X )
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ).
% group.l_inv_ex
thf(fact_493_group_Oinv__comm,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a] :
( ( group_a_Product_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ X @ Y )
= ( one_a_Product_unit @ G ) )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ Y @ X )
= ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% group.inv_comm
thf(fact_494_group_Oinv__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% group.inv_comm
thf(fact_495_group_Oinv__comm,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ X @ Y )
= ( one_li8328186300101108157t_unit @ G ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ Y @ X )
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ) ).
% group.inv_comm
thf(fact_496_ring_Oeval_Osimps_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( eval_l34571156754992824t_unit @ R @ nil_list_a )
= ( ^ [Uu: list_a] : ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.eval.simps(1)
thf(fact_497_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_498_ring_Oring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_499_ring_Oring__simprules_I9_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ X )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_500_ring_Oring__simprules_I16_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( a_inv_a_b @ R @ X ) )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_501_ring_Oring__simprules_I16_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ X ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_502_abelian__group_Ol__neg,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ X )
= ( zero_a_b @ G ) ) ) ) ).
% abelian_group.l_neg
thf(fact_503_abelian__group_Ol__neg,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( a_inv_8944721093294617173t_unit @ G @ X ) @ X )
= ( zero_l4142658623432671053t_unit @ G ) ) ) ) ).
% abelian_group.l_neg
thf(fact_504_abelian__group_Or__neg,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X @ ( a_inv_a_b @ G @ X ) )
= ( zero_a_b @ G ) ) ) ) ).
% abelian_group.r_neg
thf(fact_505_abelian__group_Or__neg,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X @ ( a_inv_8944721093294617173t_unit @ G @ X ) )
= ( zero_l4142658623432671053t_unit @ G ) ) ) ) ).
% abelian_group.r_neg
thf(fact_506_abelian__group_Ominus__equality,axiom,
! [G: partia2175431115845679010xt_a_b,Y: a,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( ( add_a_b @ G @ Y @ X )
= ( zero_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_inv_a_b @ G @ X )
= Y ) ) ) ) ) ).
% abelian_group.minus_equality
thf(fact_507_abelian__group_Ominus__equality,axiom,
! [G: partia2670972154091845814t_unit,Y: list_a,X: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( ( add_li7652885771158616974t_unit @ G @ Y @ X )
= ( zero_l4142658623432671053t_unit @ G ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( a_inv_8944721093294617173t_unit @ G @ X )
= Y ) ) ) ) ) ).
% abelian_group.minus_equality
thf(fact_508_monoid_Oinv__unique_H,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ X @ Y )
= ( one_a_Product_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ Y @ X )
= ( one_a_Product_unit @ G ) )
=> ( Y
= ( m_inv_a_Product_unit @ G @ X ) ) ) ) ) ) ) ).
% monoid.inv_unique'
thf(fact_509_monoid_Oinv__unique_H,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
= ( one_a_ring_ext_a_b @ G ) )
=> ( Y
= ( m_inv_a_ring_ext_a_b @ G @ X ) ) ) ) ) ) ) ).
% monoid.inv_unique'
thf(fact_510_monoid_Oinv__unique_H,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ X @ Y )
= ( one_li8328186300101108157t_unit @ G ) )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ Y @ X )
= ( one_li8328186300101108157t_unit @ G ) )
=> ( Y
= ( m_inv_2802811658206063947t_unit @ G @ X ) ) ) ) ) ) ) ).
% monoid.inv_unique'
thf(fact_511_monoid_Oinv__char,axiom,
! [G: partia8223610829204095565t_unit,X: a,Y: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ X @ Y )
= ( one_a_Product_unit @ G ) )
=> ( ( ( mult_a_Product_unit @ G @ Y @ X )
= ( one_a_Product_unit @ G ) )
=> ( ( m_inv_a_Product_unit @ G @ X )
= Y ) ) ) ) ) ) ).
% monoid.inv_char
thf(fact_512_monoid_Oinv__char,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
= ( one_a_ring_ext_a_b @ G ) )
=> ( ( m_inv_a_ring_ext_a_b @ G @ X )
= Y ) ) ) ) ) ) ).
% monoid.inv_char
thf(fact_513_monoid_Oinv__char,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ X @ Y )
= ( one_li8328186300101108157t_unit @ G ) )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ Y @ X )
= ( one_li8328186300101108157t_unit @ G ) )
=> ( ( m_inv_2802811658206063947t_unit @ G @ X )
= Y ) ) ) ) ) ) ).
% monoid.inv_char
thf(fact_514_ring_Oline__extension_Ocong,axiom,
embedd971793762689825387on_a_b = embedd971793762689825387on_a_b ).
% ring.line_extension.cong
thf(fact_515_monoid_OUnits__l__inv,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ ( m_inv_a_Product_unit @ G @ X ) @ X )
= ( one_a_Product_unit @ G ) ) ) ) ).
% monoid.Units_l_inv
thf(fact_516_monoid_OUnits__l__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ ( m_inv_a_ring_ext_a_b @ G @ X ) @ X )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ).
% monoid.Units_l_inv
thf(fact_517_monoid_OUnits__l__inv,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ ( m_inv_2802811658206063947t_unit @ G @ X ) @ X )
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ).
% monoid.Units_l_inv
thf(fact_518_monoid_OUnits__r__inv,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( ( mult_a_Product_unit @ G @ X @ ( m_inv_a_Product_unit @ G @ X ) )
= ( one_a_Product_unit @ G ) ) ) ) ).
% monoid.Units_r_inv
thf(fact_519_monoid_OUnits__r__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( mult_a_ring_ext_a_b @ G @ X @ ( m_inv_a_ring_ext_a_b @ G @ X ) )
= ( one_a_ring_ext_a_b @ G ) ) ) ) ).
% monoid.Units_r_inv
thf(fact_520_monoid_OUnits__r__inv,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( mult_l7073676228092353617t_unit @ G @ X @ ( m_inv_2802811658206063947t_unit @ G @ X ) )
= ( one_li8328186300101108157t_unit @ G ) ) ) ) ).
% monoid.Units_r_inv
thf(fact_521_monoid_Ogroup__l__invI,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia6735698275553448452t_unit @ G ) )
& ( ( mult_a_Product_unit @ G @ Xa @ X4 )
= ( one_a_Product_unit @ G ) ) ) )
=> ( group_a_Product_unit @ G ) ) ) ).
% monoid.group_l_invI
thf(fact_522_monoid_Ogroup__l__invI,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ G ) )
& ( ( mult_a_ring_ext_a_b @ G @ Xa @ X4 )
= ( one_a_ring_ext_a_b @ G ) ) ) )
=> ( group_a_ring_ext_a_b @ G ) ) ) ).
% monoid.group_l_invI
thf(fact_523_monoid_Ogroup__l__invI,axiom,
! [G: partia2670972154091845814t_unit] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ ( partia5361259788508890537t_unit @ G ) )
& ( ( mult_l7073676228092353617t_unit @ G @ Xa @ X4 )
= ( one_li8328186300101108157t_unit @ G ) ) ) )
=> ( group_5191797799156037398t_unit @ G ) ) ) ).
% monoid.group_l_invI
thf(fact_524_ring_Oconst__term__def,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( const_6738166269504826821t_unit @ R @ P2 )
= ( eval_l34571156754992824t_unit @ R @ P2 @ ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.const_term_def
thf(fact_525_ring_Oconst__term__def,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( const_term_a_b @ R @ P2 )
= ( eval_a_b @ R @ P2 @ ( zero_a_b @ R ) ) ) ) ).
% ring.const_term_def
thf(fact_526_is__root__def,axiom,
! [P2: list_a,X: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
= ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P2 @ X )
= ( zero_a_b @ r ) )
& ( P2 != nil_a ) ) ) ).
% is_root_def
thf(fact_527_minus__eq,axiom,
! [X: a,Y: a] :
( ( a_minus_a_b @ r @ X @ Y )
= ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).
% minus_eq
thf(fact_528_add_Oone__in__subset,axiom,
! [H2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H2 != bot_bot_set_a )
=> ( ! [X4: a] :
( ( member_a @ X4 @ H2 )
=> ( member_a @ ( a_inv_a_b @ r @ X4 ) @ H2 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ H2 )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ H2 )
=> ( member_a @ ( add_a_b @ r @ X4 @ Xa2 ) @ H2 ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H2 ) ) ) ) ) ).
% add.one_in_subset
thf(fact_529_a__lcos__mult__one,axiom,
! [M2: set_a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M2 )
= M2 ) ) ).
% a_lcos_mult_one
thf(fact_530_a__lcos__m__assoc,axiom,
! [M2: set_a,G3: a,H: a] :
( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G3 @ ( a_l_coset_a_b @ r @ H @ M2 ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G3 @ H ) @ M2 ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_531_subringI,axiom,
! [H2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ H2 )
=> ( ! [H3: a] :
( ( member_a @ H3 @ H2 )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ H2 ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ H2 ) ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ H2 ) ) )
=> ( subring_a_b @ H2 @ r ) ) ) ) ) ) ).
% subringI
thf(fact_532_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_533_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_534_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_535_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_536_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_537_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_538_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_539_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_540_a__l__coset__subset__G,axiom,
! [H2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_541_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_542_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_543_minus__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_544_r__right__minus__eq,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( A = B ) ) ) ) ).
% r_right_minus_eq
thf(fact_545_carrier__polynomial__shell,axiom,
! [K2: set_a,P2: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% carrier_polynomial_shell
thf(fact_546_monoid_Ocarrier__not__empty,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( partia707051561876973205xt_a_b @ G )
!= bot_bot_set_a ) ) ).
% monoid.carrier_not_empty
thf(fact_547_monoid_Ocarrier__not__empty,axiom,
! [G: partia2670972154091845814t_unit] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( partia5361259788508890537t_unit @ G )
!= bot_bot_set_list_a ) ) ).
% monoid.carrier_not_empty
thf(fact_548_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_549_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_550_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B5: int] : ( times_times_int @ B5 @ A4 ) ) ) ).
% mult.commute
thf(fact_551_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B5: nat] : ( times_times_nat @ B5 @ A4 ) ) ) ).
% mult.commute
thf(fact_552_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_553_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_554_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_555_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_556_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_557_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_558_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_559_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_560_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_561_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_562_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B5: int] : ( plus_plus_int @ B5 @ A4 ) ) ) ).
% add.commute
thf(fact_563_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B5: nat] : ( plus_plus_nat @ B5 @ A4 ) ) ) ).
% add.commute
thf(fact_564_group__add__class_Oadd_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% group_add_class.add.right_cancel
thf(fact_565_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_566_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_567_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_568_group__cancel_Oadd2,axiom,
! [B6: int,K: int,B: int,A: int] :
( ( B6
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B6 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_569_group__cancel_Oadd2,axiom,
! [B6: nat,K: nat,B: nat,A: nat] :
( ( B6
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B6 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_570_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_571_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_572_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_573_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_574_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_575_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_576_ring_Oring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_minus_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_577_ring_Oring__simprules_I4_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_578_a__minus__def,axiom,
( a_minus_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,X3: a,Y4: a] : ( add_a_b @ R2 @ X3 @ ( a_inv_a_b @ R2 @ Y4 ) ) ) ) ).
% a_minus_def
thf(fact_579_a__minus__def,axiom,
( a_minu3984020753470702548t_unit
= ( ^ [R2: partia2670972154091845814t_unit,X3: list_a,Y4: list_a] : ( add_li7652885771158616974t_unit @ R2 @ X3 @ ( a_inv_8944721093294617173t_unit @ R2 @ Y4 ) ) ) ) ).
% a_minus_def
thf(fact_580_abelian__group_Ominus__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( a_minus_a_b @ G @ X @ Y ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_group.minus_closed
thf(fact_581_abelian__group_Ominus__closed,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ G @ X @ Y ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_group.minus_closed
thf(fact_582_ring_Oring__simprules_I14_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( a_minus_a_b @ R @ X @ Y )
= ( add_a_b @ R @ X @ ( a_inv_a_b @ R @ Y ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_583_ring_Oring__simprules_I14_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( a_minu3984020753470702548t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_584_abelian__group_Ominus__eq,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( a_minu3984020753470702548t_unit @ G @ X @ Y )
= ( add_li7652885771158616974t_unit @ G @ X @ ( a_inv_8944721093294617173t_unit @ G @ Y ) ) ) ) ).
% abelian_group.minus_eq
thf(fact_585_abelian__group_Ominus__eq,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_group_a_b @ G )
=> ( ( a_minus_a_b @ G @ X @ Y )
= ( add_a_b @ G @ X @ ( a_inv_a_b @ G @ Y ) ) ) ) ).
% abelian_group.minus_eq
thf(fact_586_group_Oone__in__subset,axiom,
! [G: partia8223610829204095565t_unit,H2: set_a] :
( ( group_a_Product_unit @ G )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( H2 != bot_bot_set_a )
=> ( ! [X4: a] :
( ( member_a @ X4 @ H2 )
=> ( member_a @ ( m_inv_a_Product_unit @ G @ X4 ) @ H2 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ H2 )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ H2 )
=> ( member_a @ ( mult_a_Product_unit @ G @ X4 @ Xa2 ) @ H2 ) ) )
=> ( member_a @ ( one_a_Product_unit @ G ) @ H2 ) ) ) ) ) ) ).
% group.one_in_subset
thf(fact_587_group_Oone__in__subset,axiom,
! [G: partia2175431115845679010xt_a_b,H2: set_a] :
( ( group_a_ring_ext_a_b @ G )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( H2 != bot_bot_set_a )
=> ( ! [X4: a] :
( ( member_a @ X4 @ H2 )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ G @ X4 ) @ H2 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ H2 )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ G @ X4 @ Xa2 ) @ H2 ) ) )
=> ( member_a @ ( one_a_ring_ext_a_b @ G ) @ H2 ) ) ) ) ) ) ).
% group.one_in_subset
thf(fact_588_group_Oone__in__subset,axiom,
! [G: partia2670972154091845814t_unit,H2: set_list_a] :
( ( group_5191797799156037398t_unit @ G )
=> ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( H2 != bot_bot_set_list_a )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ H2 )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ G @ X4 ) @ H2 ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ H2 )
=> ! [Xa2: list_a] :
( ( member_list_a @ Xa2 @ H2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ X4 @ Xa2 ) @ H2 ) ) )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ G ) @ H2 ) ) ) ) ) ) ).
% group.one_in_subset
thf(fact_589_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_590_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_591_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_592_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_593_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_594_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_595_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_596_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_597_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_598_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_599_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_600_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_601_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_602_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
? [C3: nat] :
( B5
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_603_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_604_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_605_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_606_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_607_units__of__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_Product_unit @ ( units_8174867845824275201xt_a_b @ r ) @ X )
= ( m_inv_a_ring_ext_a_b @ r @ X ) ) ) ).
% units_of_inv
thf(fact_608_units__group,axiom,
group_a_Product_unit @ ( units_8174867845824275201xt_a_b @ r ) ).
% units_group
thf(fact_609_ring_OsubringI,axiom,
! [R: partia2175431115845679010xt_a_b,H2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H2 )
=> ( ! [H3: a] :
( ( member_a @ H3 @ H2 )
=> ( member_a @ ( a_inv_a_b @ R @ H3 ) @ H2 ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H22 ) @ H2 ) ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( member_a @ ( add_a_b @ R @ H1 @ H22 ) @ H2 ) ) )
=> ( subring_a_b @ H2 @ R ) ) ) ) ) ) ) ).
% ring.subringI
thf(fact_610_ring_OsubringI,axiom,
! [R: partia2670972154091845814t_unit,H2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ H2 )
=> ( ! [H3: list_a] :
( ( member_list_a @ H3 @ H2 )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H3 ) @ H2 ) )
=> ( ! [H1: list_a,H22: list_a] :
( ( member_list_a @ H1 @ H2 )
=> ( ( member_list_a @ H22 @ H2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 ) @ H2 ) ) )
=> ( ! [H1: list_a,H22: list_a] :
( ( member_list_a @ H1 @ H2 )
=> ( ( member_list_a @ H22 @ H2 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H22 ) @ H2 ) ) )
=> ( subrin6918843898125473962t_unit @ H2 @ R ) ) ) ) ) ) ) ).
% ring.subringI
thf(fact_611_ring_Ois__root__def,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( polyno6951661231331188332t_unit @ R @ P2 @ X )
= ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
& ( ( eval_l34571156754992824t_unit @ R @ P2 @ X )
= ( zero_l4142658623432671053t_unit @ R ) )
& ( P2 != nil_list_a ) ) ) ) ).
% ring.is_root_def
thf(fact_612_ring_Ois__root__def,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,X: a] :
( ( ring_a_b @ R )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P2 @ X )
= ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( eval_a_b @ R @ P2 @ X )
= ( zero_a_b @ R ) )
& ( P2 != nil_a ) ) ) ) ).
% ring.is_root_def
thf(fact_613_monic__degree__one__root__condition,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A ) @ nil_a ) ) @ B )
= ( A = B ) ) ) ).
% monic_degree_one_root_condition
thf(fact_614_abelian__group_Oa__lcos__m__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,M2: set_a,G3: a,H: a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ G3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_l_coset_a_b @ G @ G3 @ ( a_l_coset_a_b @ G @ H @ M2 ) )
= ( a_l_coset_a_b @ G @ ( add_a_b @ G @ G3 @ H ) @ M2 ) ) ) ) ) ) ).
% abelian_group.a_lcos_m_assoc
thf(fact_615_abelian__group_Oa__lcos__m__assoc,axiom,
! [G: partia2670972154091845814t_unit,M2: set_list_a,G3: list_a,H: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( ord_le8861187494160871172list_a @ M2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ G3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ H @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( a_l_co7008843373686234386t_unit @ G @ G3 @ ( a_l_co7008843373686234386t_unit @ G @ H @ M2 ) )
= ( a_l_co7008843373686234386t_unit @ G @ ( add_li7652885771158616974t_unit @ G @ G3 @ H ) @ M2 ) ) ) ) ) ) ).
% abelian_group.a_lcos_m_assoc
thf(fact_616_normalize_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [V: a,Va: list_a] :
( X
!= ( cons_a @ V @ Va ) ) ) ).
% normalize.cases
thf(fact_617_units__of__units,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( units_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G ) )
= ( units_a_ring_ext_a_b @ G ) ) ).
% units_of_units
thf(fact_618_ring_Odense__repr_Ocases,axiom,
! [R: partia2175431115845679010xt_a_b,X: list_a] :
( ( ring_a_b @ R )
=> ( ( X != nil_a )
=> ~ ! [V: a,Va: list_a] :
( X
!= ( cons_a @ V @ Va ) ) ) ) ).
% ring.dense_repr.cases
thf(fact_619_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_620_units__of__mult,axiom,
! [G: partia2670972154091845814t_unit] :
( ( mult_l6995149843440949818t_unit @ ( units_6477118173342999439t_unit @ G ) )
= ( mult_l7073676228092353617t_unit @ G ) ) ).
% units_of_mult
thf(fact_621_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_622_units__of__carrier,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( partia6735698275553448452t_unit @ ( units_8174867845824275201xt_a_b @ G ) )
= ( units_a_ring_ext_a_b @ G ) ) ).
% units_of_carrier
thf(fact_623_monoid_Ounits__group,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( group_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G ) ) ) ).
% monoid.units_group
thf(fact_624_univ__poly__one,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ K2 ) )
= ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) ) ).
% univ_poly_one
thf(fact_625_var__def,axiom,
( var_li8453953174693405341t_unit
= ( ^ [R2: partia2670972154091845814t_unit] : ( cons_list_a @ ( one_li8328186300101108157t_unit @ R2 ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ nil_list_a ) ) ) ) ).
% var_def
thf(fact_626_var__def,axiom,
( var_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b] : ( cons_a @ ( one_a_ring_ext_a_b @ R2 ) @ ( cons_a @ ( zero_a_b @ R2 ) @ nil_a ) ) ) ) ).
% var_def
thf(fact_627_ring_Omonic__degree__one__root__condition,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( polyno6951661231331188332t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ A ) @ nil_list_a ) ) @ B )
= ( A = B ) ) ) ) ).
% ring.monic_degree_one_root_condition
thf(fact_628_ring_Omonic__degree__one__root__condition,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( polyno4133073214067823460ot_a_b @ R @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ A ) @ nil_a ) ) @ B )
= ( A = B ) ) ) ) ).
% ring.monic_degree_one_root_condition
thf(fact_629_monoid_Ounits__of__inv,axiom,
! [G: partia8223610829204095565t_unit,X: a] :
( ( monoid2746444814143937472t_unit @ G )
=> ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
=> ( ( m_inv_a_Product_unit @ ( units_7501539392726747778t_unit @ G ) @ X )
= ( m_inv_a_Product_unit @ G @ X ) ) ) ) ).
% monoid.units_of_inv
thf(fact_630_monoid_Ounits__of__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( m_inv_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G ) @ X )
= ( m_inv_a_ring_ext_a_b @ G @ X ) ) ) ) ).
% monoid.units_of_inv
thf(fact_631_subringE_I2_J,axiom,
! [H2: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H2 @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H2 ) ) ).
% subringE(2)
thf(fact_632_subringE_I2_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H2 @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H2 ) ) ).
% subringE(2)
thf(fact_633_subringE_I7_J,axiom,
! [H2: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subrin6918843898125473962t_unit @ H2 @ R )
=> ( ( member_list_a @ H12 @ H2 )
=> ( ( member_list_a @ H23 @ H2 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subringE(7)
thf(fact_634_subringE_I7_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subring_a_b @ H2 @ R )
=> ( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( add_a_b @ R @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subringE(7)
thf(fact_635_subringE_I6_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subring_a_b @ H2 @ R )
=> ( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subringE(6)
thf(fact_636_subringE_I6_J,axiom,
! [H2: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subrin6918843898125473962t_unit @ H2 @ R )
=> ( ( member_list_a @ H12 @ H2 )
=> ( ( member_list_a @ H23 @ H2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subringE(6)
thf(fact_637_subringE_I3_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H2 @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H2 ) ) ).
% subringE(3)
thf(fact_638_subringE_I5_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H: a] :
( ( subring_a_b @ H2 @ R )
=> ( ( member_a @ H @ H2 )
=> ( member_a @ ( a_inv_a_b @ R @ H ) @ H2 ) ) ) ).
% subringE(5)
thf(fact_639_subringE_I1_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H2 @ R )
=> ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subringE(1)
thf(fact_640_subringE_I1_J,axiom,
! [H2: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H2 @ R )
=> ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subringE(1)
thf(fact_641_ring_Ocarrier__is__subring,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( subring_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% ring.carrier_is_subring
thf(fact_642_ring_Ocarrier__is__subring,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).
% ring.carrier_is_subring
thf(fact_643_abelian__group_Oa__transpose__inv,axiom,
! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( abelian_group_a_b @ G )
=> ( ( ( add_a_b @ G @ X @ Y )
= Z )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ Z )
= Y ) ) ) ) ) ) ).
% abelian_group.a_transpose_inv
thf(fact_644_abelian__group_Oa__transpose__inv,axiom,
! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( ( add_li7652885771158616974t_unit @ G @ X @ Y )
= Z )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( a_inv_8944721093294617173t_unit @ G @ X ) @ Z )
= Y ) ) ) ) ) ) ).
% abelian_group.a_transpose_inv
thf(fact_645_abelian__group_Oa__l__coset__subset__G,axiom,
! [G: partia2175431115845679010xt_a_b,H2: set_a,X: a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ G @ X @ H2 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_group.a_l_coset_subset_G
thf(fact_646_abelian__group_Oa__l__coset__subset__G,axiom,
! [G: partia2670972154091845814t_unit,H2: set_list_a,X: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ G @ X @ H2 ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_group.a_l_coset_subset_G
thf(fact_647_ring_Ocarrier__polynomial__shell,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K2 ) ) )
=> ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ).
% ring.carrier_polynomial_shell
thf(fact_648_ring_Ocarrier__polynomial__shell,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) )
=> ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ).
% ring.carrier_polynomial_shell
thf(fact_649_abelian__group_Oa__lcos__mult__one,axiom,
! [G: partia2175431115845679010xt_a_b,M2: set_a] :
( ( abelian_group_a_b @ G )
=> ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_l_coset_a_b @ G @ ( zero_a_b @ G ) @ M2 )
= M2 ) ) ) ).
% abelian_group.a_lcos_mult_one
thf(fact_650_abelian__group_Oa__lcos__mult__one,axiom,
! [G: partia2670972154091845814t_unit,M2: set_list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( ord_le8861187494160871172list_a @ M2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( a_l_co7008843373686234386t_unit @ G @ ( zero_l4142658623432671053t_unit @ G ) @ M2 )
= M2 ) ) ) ).
% abelian_group.a_lcos_mult_one
thf(fact_651_factors__mult__single,axiom,
! [A: a,Fb: list_a,B: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( cons_a @ A @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% factors_mult_single
thf(fact_652_empty__subsetI,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% empty_subsetI
thf(fact_653_subset__empty,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_654_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_655_carrier__one__not__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) ) ) ).
% carrier_one_not_zero
thf(fact_656_subset__antisym,axiom,
! [A3: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ A3 )
=> ( A3 = B6 ) ) ) ).
% subset_antisym
thf(fact_657_subsetI,axiom,
! [A3: set_list_a,B6: set_list_a] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ A3 )
=> ( member_list_a @ X4 @ B6 ) )
=> ( ord_le8861187494160871172list_a @ A3 @ B6 ) ) ).
% subsetI
thf(fact_658_subsetI,axiom,
! [A3: set_a,B6: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( member_a @ X4 @ B6 ) )
=> ( ord_less_eq_set_a @ A3 @ B6 ) ) ).
% subsetI
thf(fact_659_one__zeroI,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% one_zeroI
thf(fact_660_one__zeroD,axiom,
( ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) )
=> ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% one_zeroD
thf(fact_661_carrier__one__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% carrier_one_zero
thf(fact_662_insert__subset,axiom,
! [X: list_a,A3: set_list_a,B6: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X @ A3 ) @ B6 )
= ( ( member_list_a @ X @ B6 )
& ( ord_le8861187494160871172list_a @ A3 @ B6 ) ) ) ).
% insert_subset
thf(fact_663_insert__subset,axiom,
! [X: a,A3: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A3 ) @ B6 )
= ( ( member_a @ X @ B6 )
& ( ord_less_eq_set_a @ A3 @ B6 ) ) ) ).
% insert_subset
thf(fact_664_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A3: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_665_singleton__insert__inj__eq_H,axiom,
! [A: a,A3: set_a,B: a] :
( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_666_subset__insertI2,axiom,
! [A3: set_a,B6: set_a,B: a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ B6 ) ) ) ).
% subset_insertI2
thf(fact_667_subset__insertI,axiom,
! [B6: set_a,A: a] : ( ord_less_eq_set_a @ B6 @ ( insert_a @ A @ B6 ) ) ).
% subset_insertI
thf(fact_668_subset__insert,axiom,
! [X: list_a,A3: set_list_a,B6: set_list_a] :
( ~ ( member_list_a @ X @ A3 )
=> ( ( ord_le8861187494160871172list_a @ A3 @ ( insert_list_a @ X @ B6 ) )
= ( ord_le8861187494160871172list_a @ A3 @ B6 ) ) ) ).
% subset_insert
thf(fact_669_subset__insert,axiom,
! [X: a,A3: set_a,B6: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B6 ) )
= ( ord_less_eq_set_a @ A3 @ B6 ) ) ) ).
% subset_insert
thf(fact_670_insert__mono,axiom,
! [C4: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C4 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C4 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_671_subset__singletonD,axiom,
! [A3: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A3 = bot_bot_set_a )
| ( A3
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_672_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_673_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_674_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z4: set_a] : ( Y5 = Z4 ) )
= ( ^ [A6: set_a,B7: set_a] :
( ( ord_less_eq_set_a @ A6 @ B7 )
& ( ord_less_eq_set_a @ B7 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_675_subset__trans,axiom,
! [A3: set_a,B6: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ C4 )
=> ( ord_less_eq_set_a @ A3 @ C4 ) ) ) ).
% subset_trans
thf(fact_676_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_677_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_678_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B7: set_list_a] :
! [T: list_a] :
( ( member_list_a @ T @ A6 )
=> ( member_list_a @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_679_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B7: set_a] :
! [T: a] :
( ( member_a @ T @ A6 )
=> ( member_a @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_680_equalityD2,axiom,
! [A3: set_a,B6: set_a] :
( ( A3 = B6 )
=> ( ord_less_eq_set_a @ B6 @ A3 ) ) ).
% equalityD2
thf(fact_681_equalityD1,axiom,
! [A3: set_a,B6: set_a] :
( ( A3 = B6 )
=> ( ord_less_eq_set_a @ A3 @ B6 ) ) ).
% equalityD1
thf(fact_682_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B7: set_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ A6 )
=> ( member_list_a @ X3 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_683_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B7: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A6 )
=> ( member_a @ X3 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_684_equalityE,axiom,
! [A3: set_a,B6: set_a] :
( ( A3 = B6 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B6 )
=> ~ ( ord_less_eq_set_a @ B6 @ A3 ) ) ) ).
% equalityE
thf(fact_685_subsetD,axiom,
! [A3: set_list_a,B6: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B6 )
=> ( ( member_list_a @ C @ A3 )
=> ( member_list_a @ C @ B6 ) ) ) ).
% subsetD
thf(fact_686_subsetD,axiom,
! [A3: set_a,B6: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B6 ) ) ) ).
% subsetD
thf(fact_687_in__mono,axiom,
! [A3: set_list_a,B6: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B6 )
=> ( ( member_list_a @ X @ A3 )
=> ( member_list_a @ X @ B6 ) ) ) ).
% in_mono
thf(fact_688_in__mono,axiom,
! [A3: set_a,B6: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ X @ B6 ) ) ) ).
% in_mono
thf(fact_689_ring_Ozeropideal,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( princi8786919440553033881t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R ) ) ).
% ring.zeropideal
thf(fact_690_ring_Ozeropideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( principalideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ).
% ring.zeropideal
thf(fact_691_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
!= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_692_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_693_semiring_Ocarrier__one__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_694_semiring_Ocarrier__one__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_695_semiring_Oone__zeroI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_696_semiring_Oone__zeroI,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
=> ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_697_semiring_Oone__zeroD,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) )
=> ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_698_semiring_Oone__zeroD,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_699_monoid_Ofactors__mult__single,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,Fb: list_a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( irredu6211895646901577903xt_a_b @ G @ A )
=> ( ( factor5638265376665762323xt_a_b @ G @ Fb @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( factor5638265376665762323xt_a_b @ G @ ( cons_a @ A @ Fb ) @ ( mult_a_ring_ext_a_b @ G @ A @ B ) ) ) ) ) ) ).
% monoid.factors_mult_single
thf(fact_700_monoid_Ofactors__mult__single,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,Fb: list_list_a,B: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( irredu4230924414530676029t_unit @ G @ A )
=> ( ( factor7181967632740204193t_unit @ G @ Fb @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( factor7181967632740204193t_unit @ G @ ( cons_list_a @ A @ Fb ) @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) ) ) ) ) ) ).
% monoid.factors_mult_single
thf(fact_701_genideal__one,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) )
= ( partia707051561876973205xt_a_b @ r ) ) ).
% genideal_one
thf(fact_702_Idl__subset__ideal_H,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_703_genideal__zero,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% genideal_zero
thf(fact_704_genideal__self_H,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_705_subcringI,axiom,
! [H2: set_a] :
( ( subring_a_b @ H2 @ r )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( ( mult_a_ring_ext_a_b @ r @ H1 @ H22 )
= ( mult_a_ring_ext_a_b @ r @ H22 @ H1 ) ) ) )
=> ( subcring_a_b @ H2 @ r ) ) ) ).
% subcringI
thf(fact_706_subset__Idl__subset,axiom,
! [I2: set_a,H2: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H2 @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H2 ) @ ( genideal_a_b @ r @ I2 ) ) ) ) ).
% subset_Idl_subset
thf(fact_707_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_708_subcringE_I2_J,axiom,
! [H2: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H2 @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H2 ) ) ).
% subcringE(2)
thf(fact_709_subcringE_I2_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H2 @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H2 ) ) ).
% subcringE(2)
thf(fact_710_subcringE_I7_J,axiom,
! [H2: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subcri7763218559781929323t_unit @ H2 @ R )
=> ( ( member_list_a @ H12 @ H2 )
=> ( ( member_list_a @ H23 @ H2 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subcringE(7)
thf(fact_711_subcringE_I7_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subcring_a_b @ H2 @ R )
=> ( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( add_a_b @ R @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subcringE(7)
thf(fact_712_subcring_Osub__m__comm,axiom,
! [H2: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subcri7763218559781929323t_unit @ H2 @ R )
=> ( ( member_list_a @ H12 @ H2 )
=> ( ( member_list_a @ H23 @ H2 )
=> ( ( mult_l7073676228092353617t_unit @ R @ H12 @ H23 )
= ( mult_l7073676228092353617t_unit @ R @ H23 @ H12 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_713_subcring_Osub__m__comm,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subcring_a_b @ H2 @ R )
=> ( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( ( mult_a_ring_ext_a_b @ R @ H12 @ H23 )
= ( mult_a_ring_ext_a_b @ R @ H23 @ H12 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_714_subcringE_I6_J,axiom,
! [H2: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subcri7763218559781929323t_unit @ H2 @ R )
=> ( ( member_list_a @ H12 @ H2 )
=> ( ( member_list_a @ H23 @ H2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subcringE(6)
thf(fact_715_subcringE_I6_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subcring_a_b @ H2 @ R )
=> ( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subcringE(6)
thf(fact_716_subcringE_I3_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H2 @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H2 ) ) ).
% subcringE(3)
thf(fact_717_subcringE_I5_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H: a] :
( ( subcring_a_b @ H2 @ R )
=> ( ( member_a @ H @ H2 )
=> ( member_a @ ( a_inv_a_b @ R @ H ) @ H2 ) ) ) ).
% subcringE(5)
thf(fact_718_subcringE_I1_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H2 @ R )
=> ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subcringE(1)
thf(fact_719_subcringE_I1_J,axiom,
! [H2: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H2 @ R )
=> ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subcringE(1)
thf(fact_720_ring_OsubcringI,axiom,
! [R: partia2670972154091845814t_unit,H2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ H2 @ R )
=> ( ! [H1: list_a,H22: list_a] :
( ( member_list_a @ H1 @ H2 )
=> ( ( member_list_a @ H22 @ H2 )
=> ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 )
= ( mult_l7073676228092353617t_unit @ R @ H22 @ H1 ) ) ) )
=> ( subcri7763218559781929323t_unit @ H2 @ R ) ) ) ) ).
% ring.subcringI
thf(fact_721_ring_OsubcringI,axiom,
! [R: partia2175431115845679010xt_a_b,H2: set_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ H2 @ R )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( ( mult_a_ring_ext_a_b @ R @ H1 @ H22 )
= ( mult_a_ring_ext_a_b @ R @ H22 @ H1 ) ) ) )
=> ( subcring_a_b @ H2 @ R ) ) ) ) ).
% ring.subcringI
thf(fact_722_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_723_ring_Ogenideal__self,axiom,
! [R: partia2670972154091845814t_unit,S: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ S @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ S @ ( genide3243992037924705879t_unit @ R @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_724_ring_Osubset__Idl__subset,axiom,
! [R: partia2175431115845679010xt_a_b,I2: set_a,H2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ H2 @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ R @ H2 ) @ ( genideal_a_b @ R @ I2 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_725_ring_Osubset__Idl__subset,axiom,
! [R: partia2670972154091845814t_unit,I2: set_list_a,H2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ H2 @ I2 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ R @ H2 ) @ ( genide3243992037924705879t_unit @ R @ I2 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_726_ring_Ogenideal__self_H,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ I @ ( genideal_a_b @ R @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_727_ring_Ogenideal__self_H,axiom,
! [R: partia2670972154091845814t_unit,I: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ I @ ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ I @ bot_bot_set_list_a ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_728_ring_Ogenideal__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ).
% ring.genideal_zero
thf(fact_729_ring_Ogenideal__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( genideal_a_b @ R @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ).
% ring.genideal_zero
thf(fact_730_principalideal_Ogenerate,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
& ( I2
= ( genideal_a_b @ R @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_731_principalideal_Ogenerate,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( princi8786919440553033881t_unit @ I2 @ R )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
& ( I2
= ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_732_ring_OIdl__subset__ideal_H,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ R @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ R @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ R @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_733_ring_OIdl__subset__ideal_H,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) )
= ( member_list_a @ A @ ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_734_ring_Ogenideal__one,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( genideal_a_b @ R @ ( insert_a @ ( one_a_ring_ext_a_b @ R ) @ bot_bot_set_a ) )
= ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.genideal_one
thf(fact_735_ring_Ogenideal__one,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ ( one_li8328186300101108157t_unit @ R ) @ bot_bot_set_list_a ) )
= ( partia5361259788508890537t_unit @ R ) ) ) ).
% ring.genideal_one
thf(fact_736_subdomainI,axiom,
! [H2: set_a] :
( ( subcring_a_b @ H2 @ r )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( ( ( mult_a_ring_ext_a_b @ r @ H1 @ H22 )
= ( zero_a_b @ r ) )
=> ( ( H1
= ( zero_a_b @ r ) )
| ( H22
= ( zero_a_b @ r ) ) ) ) ) )
=> ( subdomain_a_b @ H2 @ r ) ) ) ) ).
% subdomainI
thf(fact_737_add_Oint__pow__diff,axiom,
! [X: a,N: int,M: int] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ ( minus_minus_int @ N @ M ) @ X )
= ( add_a_b @ r @ ( add_pow_a_b_int @ r @ N @ X ) @ ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ M @ X ) ) ) ) ) ).
% add.int_pow_diff
thf(fact_738_subalgebra__in__carrier,axiom,
! [K2: set_a,V2: set_a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V2 @ r )
=> ( ord_less_eq_set_a @ V2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_739_carrier__is__subalgebra,axiom,
! [K2: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_740_ring_Or__right__minus__eq,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( a_minus_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
= ( A = B ) ) ) ) ) ).
% ring.r_right_minus_eq
thf(fact_741_ring_Or__right__minus__eq,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( a_minu3984020753470702548t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
= ( A = B ) ) ) ) ) ).
% ring.r_right_minus_eq
thf(fact_742_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_743_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_744_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_745_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_746_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_747_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_748_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_749_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_750_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_751_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_752_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_753_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_754_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_755_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_756_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_757_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_758_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_759_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_760_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_761_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_762_group__cancel_Osub1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_763_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_764_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_765_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_766_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_767_subdomainE_I2_J,axiom,
! [H2: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H2 @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H2 ) ) ).
% subdomainE(2)
thf(fact_768_subdomainE_I2_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H2 @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H2 ) ) ).
% subdomainE(2)
thf(fact_769_subdomainE_I7_J,axiom,
! [H2: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subdom7821232466298058046t_unit @ H2 @ R )
=> ( ( member_list_a @ H12 @ H2 )
=> ( ( member_list_a @ H23 @ H2 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subdomainE(7)
thf(fact_770_subdomainE_I7_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subdomain_a_b @ H2 @ R )
=> ( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( add_a_b @ R @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subdomainE(7)
thf(fact_771_subalgebra_Osmult__closed,axiom,
! [K2: set_list_a,V2: set_list_a,R: partia2670972154091845814t_unit,K: list_a,V3: list_a] :
( ( embedd1768981623711841426t_unit @ K2 @ V2 @ R )
=> ( ( member_list_a @ K @ K2 )
=> ( ( member_list_a @ V3 @ V2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K @ V3 ) @ V2 ) ) ) ) ).
% subalgebra.smult_closed
thf(fact_772_subalgebra_Osmult__closed,axiom,
! [K2: set_a,V2: set_a,R: partia2175431115845679010xt_a_b,K: a,V3: a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V2 @ R )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V3 @ V2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ V3 ) @ V2 ) ) ) ) ).
% subalgebra.smult_closed
thf(fact_773_subdomainE_I8_J,axiom,
! [H2: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subdom7821232466298058046t_unit @ H2 @ R )
=> ( ( member_list_a @ H12 @ H2 )
=> ( ( member_list_a @ H23 @ H2 )
=> ( ( mult_l7073676228092353617t_unit @ R @ H12 @ H23 )
= ( mult_l7073676228092353617t_unit @ R @ H23 @ H12 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_774_subdomainE_I8_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subdomain_a_b @ H2 @ R )
=> ( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( ( mult_a_ring_ext_a_b @ R @ H12 @ H23 )
= ( mult_a_ring_ext_a_b @ R @ H23 @ H12 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_775_subdomainE_I6_J,axiom,
! [H2: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subdom7821232466298058046t_unit @ H2 @ R )
=> ( ( member_list_a @ H12 @ H2 )
=> ( ( member_list_a @ H23 @ H2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subdomainE(6)
thf(fact_776_subdomainE_I6_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subdomain_a_b @ H2 @ R )
=> ( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subdomainE(6)
thf(fact_777_subdomainE_I3_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H2 @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H2 ) ) ).
% subdomainE(3)
thf(fact_778_subdomainE_I5_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H: a] :
( ( subdomain_a_b @ H2 @ R )
=> ( ( member_a @ H @ H2 )
=> ( member_a @ ( a_inv_a_b @ R @ H ) @ H2 ) ) ) ).
% subdomainE(5)
thf(fact_779_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_780_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_781_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_782_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_783_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_784_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_785_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_786_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_787_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_788_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_789_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_790_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_791_subdomainE_I1_J,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H2 @ R )
=> ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subdomainE(1)
thf(fact_792_subdomainE_I1_J,axiom,
! [H2: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H2 @ R )
=> ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subdomainE(1)
thf(fact_793_subdomain_Osubintegral,axiom,
! [H2: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subdom7821232466298058046t_unit @ H2 @ R )
=> ( ( member_list_a @ H12 @ H2 )
=> ( ( member_list_a @ H23 @ H2 )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ H12 @ H23 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( H12
= ( zero_l4142658623432671053t_unit @ R ) )
| ( H23
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_794_subdomain_Osubintegral,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subdomain_a_b @ H2 @ R )
=> ( ( member_a @ H12 @ H2 )
=> ( ( member_a @ H23 @ H2 )
=> ( ( ( mult_a_ring_ext_a_b @ R @ H12 @ H23 )
= ( zero_a_b @ R ) )
=> ( ( H12
= ( zero_a_b @ R ) )
| ( H23
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_795_subdomain_Osub__one__not__zero,axiom,
! [H2: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H2 @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_796_subdomain_Osub__one__not__zero,axiom,
! [H2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H2 @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_797_ring_Osubalgebra__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,V2: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V2 @ R )
=> ( ord_less_eq_set_a @ V2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_798_ring_Osubalgebra__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,V2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd1768981623711841426t_unit @ K2 @ V2 @ R )
=> ( ord_le8861187494160871172list_a @ V2 @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_799_ring_Ocarrier__is__subalgebra,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_800_ring_Ocarrier__is__subalgebra,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( embedd1768981623711841426t_unit @ K2 @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_801_ring_OsubdomainI,axiom,
! [R: partia2670972154091845814t_unit,H2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subcri7763218559781929323t_unit @ H2 @ R )
=> ( ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ! [H1: list_a,H22: list_a] :
( ( member_list_a @ H1 @ H2 )
=> ( ( member_list_a @ H22 @ H2 )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( H1
= ( zero_l4142658623432671053t_unit @ R ) )
| ( H22
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) )
=> ( subdom7821232466298058046t_unit @ H2 @ R ) ) ) ) ) ).
% ring.subdomainI
thf(fact_802_ring_OsubdomainI,axiom,
! [R: partia2175431115845679010xt_a_b,H2: set_a] :
( ( ring_a_b @ R )
=> ( ( subcring_a_b @ H2 @ R )
=> ( ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( ( ( mult_a_ring_ext_a_b @ R @ H1 @ H22 )
= ( zero_a_b @ R ) )
=> ( ( H1
= ( zero_a_b @ R ) )
| ( H22
= ( zero_a_b @ R ) ) ) ) ) )
=> ( subdomain_a_b @ H2 @ R ) ) ) ) ) ).
% ring.subdomainI
thf(fact_803_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_804_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_805_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_806_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_807_combine_Oelims,axiom,
! [X: list_a,Xa3: list_a,Y: a] :
( ( ( embedded_combine_a_b @ r @ X @ Xa3 )
= Y )
=> ( ! [K3: a,Ks: list_a] :
( ( X
= ( cons_a @ K3 @ Ks ) )
=> ! [U2: a,Us: list_a] :
( ( Xa3
= ( cons_a @ U2 @ Us ) )
=> ( Y
!= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K3 @ U2 ) @ ( embedded_combine_a_b @ r @ Ks @ Us ) ) ) ) )
=> ( ( ( X = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) )
=> ~ ( ( Xa3 = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% combine.elims
thf(fact_808_line__extension__smult__closed,axiom,
! [K2: set_a,E: set_a,A: a,K: a,U: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ! [K3: a,V: a] :
( ( member_a @ K3 @ K2 )
=> ( ( member_a @ V @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ V ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ U ) @ ( embedd971793762689825387on_a_b @ r @ K2 @ A @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_809_subring__props_I7_J,axiom,
! [K2: set_a,H12: a,H23: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H12 @ K2 )
=> ( ( member_a @ H23 @ K2 )
=> ( member_a @ ( add_a_b @ r @ H12 @ H23 ) @ K2 ) ) ) ) ).
% subring_props(7)
thf(fact_810_subring__props_I2_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K2 ) ) ).
% subring_props(2)
thf(fact_811_subring__props_I6_J,axiom,
! [K2: set_a,H12: a,H23: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H12 @ K2 )
=> ( ( member_a @ H23 @ K2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H23 ) @ K2 ) ) ) ) ).
% subring_props(6)
thf(fact_812_subring__props_I4_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( K2 != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_813_subring__props_I5_J,axiom,
! [K2: set_a,H: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H @ K2 )
=> ( member_a @ ( a_inv_a_b @ r @ H ) @ K2 ) ) ) ).
% subring_props(5)
thf(fact_814_subring__props_I3_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K2 ) ) ).
% subring_props(3)
thf(fact_815_subring__props_I1_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_816_combine_Osimps_I2_J,axiom,
! [Us2: list_a] :
( ( embedded_combine_a_b @ r @ nil_a @ Us2 )
= ( zero_a_b @ r ) ) ).
% combine.simps(2)
thf(fact_817_combine_Osimps_I3_J,axiom,
! [Ks2: list_a] :
( ( embedded_combine_a_b @ r @ Ks2 @ nil_a )
= ( zero_a_b @ r ) ) ).
% combine.simps(3)
thf(fact_818_combine_Osimps_I1_J,axiom,
! [K: a,Ks2: list_a,U: a,Us2: list_a] :
( ( embedded_combine_a_b @ r @ ( cons_a @ K @ Ks2 ) @ ( cons_a @ U @ Us2 ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K @ U ) @ ( embedded_combine_a_b @ r @ Ks2 @ Us2 ) ) ) ).
% combine.simps(1)
thf(fact_819_subfield__m__inv_I1_J,axiom,
! [K2: set_a,K: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ K ) @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ).
% subfield_m_inv(1)
thf(fact_820_subfield__m__inv__simprule,axiom,
! [K2: set_a,K: a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ A ) @ K2 )
=> ( member_a @ A @ K2 ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_821_Diff__eq__empty__iff,axiom,
! [A3: set_a,B6: set_a] :
( ( ( minus_minus_set_a @ A3 @ B6 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A3 @ B6 ) ) ).
% Diff_eq_empty_iff
thf(fact_822_subfield__m__inv_I3_J,axiom,
! [K2: set_a,K: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ K ) @ K )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% subfield_m_inv(3)
thf(fact_823_subfield__m__inv_I2_J,axiom,
! [K2: set_a,K: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ K @ ( m_inv_a_ring_ext_a_b @ r @ K ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% subfield_m_inv(2)
thf(fact_824_Diff__mono,axiom,
! [A3: set_a,C4: set_a,D2: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A3 @ C4 )
=> ( ( ord_less_eq_set_a @ D2 @ B6 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B6 ) @ ( minus_minus_set_a @ C4 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_825_Diff__subset,axiom,
! [A3: set_a,B6: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B6 ) @ A3 ) ).
% Diff_subset
thf(fact_826_double__diff,axiom,
! [A3: set_a,B6: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
=> ( ( ord_less_eq_set_a @ B6 @ C4 )
=> ( ( minus_minus_set_a @ B6 @ ( minus_minus_set_a @ C4 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_827_subfieldE_I4_J,axiom,
! [K2: set_list_a,R: partia2670972154091845814t_unit,K1: list_a,K22: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( member_list_a @ K1 @ K2 )
=> ( ( member_list_a @ K22 @ K2 )
=> ( ( mult_l7073676228092353617t_unit @ R @ K1 @ K22 )
= ( mult_l7073676228092353617t_unit @ R @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_828_subfieldE_I4_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K1 @ K2 )
=> ( ( member_a @ K22 @ K2 )
=> ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
= ( mult_a_ring_ext_a_b @ R @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_829_ring_Ocombine_Ocong,axiom,
embedded_combine_a_b = embedded_combine_a_b ).
% ring.combine.cong
thf(fact_830_subfieldE_I3_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K2 @ R )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subfieldE(3)
thf(fact_831_subfieldE_I3_J,axiom,
! [K2: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subfieldE(3)
thf(fact_832_subfieldE_I5_J,axiom,
! [K2: set_list_a,R: partia2670972154091845814t_unit,K1: list_a,K22: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( member_list_a @ K1 @ K2 )
=> ( ( member_list_a @ K22 @ K2 )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ K1 @ K22 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( K1
= ( zero_l4142658623432671053t_unit @ R ) )
| ( K22
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_833_subfieldE_I5_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K1 @ K2 )
=> ( ( member_a @ K22 @ K2 )
=> ( ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
= ( zero_a_b @ R ) )
=> ( ( K1
= ( zero_a_b @ R ) )
| ( K22
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_834_ring_Osubring__props_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ K2 ) ) ) ).
% ring.subring_props(2)
thf(fact_835_ring_Osubring__props_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( member_a @ ( zero_a_b @ R ) @ K2 ) ) ) ).
% ring.subring_props(2)
thf(fact_836_ring_Osubring__props_I7_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,H12: list_a,H23: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( member_list_a @ H12 @ K2 )
=> ( ( member_list_a @ H23 @ K2 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H12 @ H23 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(7)
thf(fact_837_ring_Osubring__props_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,H12: a,H23: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ H12 @ K2 )
=> ( ( member_a @ H23 @ K2 )
=> ( member_a @ ( add_a_b @ R @ H12 @ H23 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(7)
thf(fact_838_ring_Osubring__props_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( K2 != bot_bot_set_a ) ) ) ).
% ring.subring_props(4)
thf(fact_839_subfieldE_I6_J,axiom,
! [K2: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% subfieldE(6)
thf(fact_840_subfieldE_I6_J,axiom,
! [K2: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K2 @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% subfieldE(6)
thf(fact_841_ring_Osubring__props_I6_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,H12: list_a,H23: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( member_list_a @ H12 @ K2 )
=> ( ( member_list_a @ H23 @ K2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H12 @ H23 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(6)
thf(fact_842_ring_Osubring__props_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,H12: a,H23: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ H12 @ K2 )
=> ( ( member_a @ H23 @ K2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H12 @ H23 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(6)
thf(fact_843_ring_Osubring__props_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ K2 ) ) ) ).
% ring.subring_props(3)
thf(fact_844_ring_Osubring__props_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,H: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ H @ K2 )
=> ( member_a @ ( a_inv_a_b @ R @ H ) @ K2 ) ) ) ) ).
% ring.subring_props(5)
thf(fact_845_subset__Diff__insert,axiom,
! [A3: set_list_a,B6: set_list_a,X: list_a,C4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ ( minus_646659088055828811list_a @ B6 @ ( insert_list_a @ X @ C4 ) ) )
= ( ( ord_le8861187494160871172list_a @ A3 @ ( minus_646659088055828811list_a @ B6 @ C4 ) )
& ~ ( member_list_a @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_846_subset__Diff__insert,axiom,
! [A3: set_a,B6: set_a,X: a,C4: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B6 @ ( insert_a @ X @ C4 ) ) )
= ( ( ord_less_eq_set_a @ A3 @ ( minus_minus_set_a @ B6 @ C4 ) )
& ~ ( member_a @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_847_ring_Osubfield__m__inv_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,K: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( member_list_a @ K @ ( minus_646659088055828811list_a @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ R @ K ) @ ( minus_646659088055828811list_a @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ) ) ) ).
% ring.subfield_m_inv(1)
thf(fact_848_ring_Osubfield__m__inv_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,K: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ R @ K ) @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ) ) ).
% ring.subfield_m_inv(1)
thf(fact_849_ring_Osubfield__m__inv__simprule,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,K: a,A: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ A ) @ K2 )
=> ( member_a @ A @ K2 ) ) ) ) ) ) ).
% ring.subfield_m_inv_simprule
thf(fact_850_ring_Osubfield__m__inv__simprule,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,K: list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( member_list_a @ K @ ( minus_646659088055828811list_a @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K @ A ) @ K2 )
=> ( member_list_a @ A @ K2 ) ) ) ) ) ) ).
% ring.subfield_m_inv_simprule
thf(fact_851_ring_Osubring__props_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.subring_props(1)
thf(fact_852_ring_Osubring__props_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.subring_props(1)
thf(fact_853_Diff__single__insert,axiom,
! [A3: set_a,X: a,B6: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B6 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B6 ) ) ) ).
% Diff_single_insert
thf(fact_854_subset__insert__iff,axiom,
! [A3: set_list_a,X: list_a,B6: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ ( insert_list_a @ X @ B6 ) )
= ( ( ( member_list_a @ X @ A3 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B6 ) )
& ( ~ ( member_list_a @ X @ A3 )
=> ( ord_le8861187494160871172list_a @ A3 @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_855_subset__insert__iff,axiom,
! [A3: set_a,X: a,B6: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B6 ) )
= ( ( ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B6 ) )
& ( ~ ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ A3 @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_856_ring_Ocombine_Osimps_I3_J,axiom,
! [R: partia2670972154091845814t_unit,Ks2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd2435972518007585703t_unit @ R @ Ks2 @ nil_list_a )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% ring.combine.simps(3)
thf(fact_857_ring_Ocombine_Osimps_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,Ks2: list_a] :
( ( ring_a_b @ R )
=> ( ( embedded_combine_a_b @ R @ Ks2 @ nil_a )
= ( zero_a_b @ R ) ) ) ).
% ring.combine.simps(3)
thf(fact_858_ring_Ocombine_Osimps_I2_J,axiom,
! [R: partia2670972154091845814t_unit,Us2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd2435972518007585703t_unit @ R @ nil_list_a @ Us2 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% ring.combine.simps(2)
thf(fact_859_ring_Ocombine_Osimps_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,Us2: list_a] :
( ( ring_a_b @ R )
=> ( ( embedded_combine_a_b @ R @ nil_a @ Us2 )
= ( zero_a_b @ R ) ) ) ).
% ring.combine.simps(2)
thf(fact_860_ring_Osubfield__m__inv_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,K: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( member_list_a @ K @ ( minus_646659088055828811list_a @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( m_inv_2802811658206063947t_unit @ R @ K ) @ K )
= ( one_li8328186300101108157t_unit @ R ) ) ) ) ) ).
% ring.subfield_m_inv(3)
thf(fact_861_ring_Osubfield__m__inv_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,K: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( m_inv_a_ring_ext_a_b @ R @ K ) @ K )
= ( one_a_ring_ext_a_b @ R ) ) ) ) ) ).
% ring.subfield_m_inv(3)
thf(fact_862_ring_Osubfield__m__inv_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,K: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( member_list_a @ K @ ( minus_646659088055828811list_a @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ K @ ( m_inv_2802811658206063947t_unit @ R @ K ) )
= ( one_li8328186300101108157t_unit @ R ) ) ) ) ) ).
% ring.subfield_m_inv(2)
thf(fact_863_ring_Osubfield__m__inv_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,K: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ R @ K @ ( m_inv_a_ring_ext_a_b @ R @ K ) )
= ( one_a_ring_ext_a_b @ R ) ) ) ) ) ).
% ring.subfield_m_inv(2)
thf(fact_864_ring_Ocombine_Osimps_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: list_a,Ks2: list_list_a,U: list_a,Us2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd2435972518007585703t_unit @ R @ ( cons_list_a @ K @ Ks2 ) @ ( cons_list_a @ U @ Us2 ) )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ K @ U ) @ ( embedd2435972518007585703t_unit @ R @ Ks2 @ Us2 ) ) ) ) ).
% ring.combine.simps(1)
thf(fact_865_ring_Ocombine_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: a,Ks2: list_a,U: a,Us2: list_a] :
( ( ring_a_b @ R )
=> ( ( embedded_combine_a_b @ R @ ( cons_a @ K @ Ks2 ) @ ( cons_a @ U @ Us2 ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ K @ U ) @ ( embedded_combine_a_b @ R @ Ks2 @ Us2 ) ) ) ) ).
% ring.combine.simps(1)
thf(fact_866_ring_Oline__extension__smult__closed,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,E: set_list_a,A: list_a,K: list_a,U: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ! [K3: list_a,V: list_a] :
( ( member_list_a @ K3 @ K2 )
=> ( ( member_list_a @ V @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K3 @ V ) @ E ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ K @ K2 )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ R @ K2 @ A @ E ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K @ U ) @ ( embedd5150658419831591667t_unit @ R @ K2 @ A @ E ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_867_ring_Oline__extension__smult__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a,A: a,K: a,U: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ! [K3: a,V: a] :
( ( member_a @ K3 @ K2 )
=> ( ( member_a @ V @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K3 @ V ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ U ) @ ( embedd971793762689825387on_a_b @ R @ K2 @ A @ E ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_868_ring_Ocombine_Oelims,axiom,
! [R: partia2670972154091845814t_unit,X: list_list_a,Xa3: list_list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ( embedd2435972518007585703t_unit @ R @ X @ Xa3 )
= Y )
=> ( ! [K3: list_a,Ks: list_list_a] :
( ( X
= ( cons_list_a @ K3 @ Ks ) )
=> ! [U2: list_a,Us: list_list_a] :
( ( Xa3
= ( cons_list_a @ U2 @ Us ) )
=> ( Y
!= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ K3 @ U2 ) @ ( embedd2435972518007585703t_unit @ R @ Ks @ Us ) ) ) ) )
=> ( ( ( X = nil_list_a )
=> ( Y
!= ( zero_l4142658623432671053t_unit @ R ) ) )
=> ~ ( ( Xa3 = nil_list_a )
=> ( Y
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% ring.combine.elims
thf(fact_869_ring_Ocombine_Oelims,axiom,
! [R: partia2175431115845679010xt_a_b,X: list_a,Xa3: list_a,Y: a] :
( ( ring_a_b @ R )
=> ( ( ( embedded_combine_a_b @ R @ X @ Xa3 )
= Y )
=> ( ! [K3: a,Ks: list_a] :
( ( X
= ( cons_a @ K3 @ Ks ) )
=> ! [U2: a,Us: list_a] :
( ( Xa3
= ( cons_a @ U2 @ Us ) )
=> ( Y
!= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ K3 @ U2 ) @ ( embedded_combine_a_b @ R @ Ks @ Us ) ) ) ) )
=> ( ( ( X = nil_a )
=> ( Y
!= ( zero_a_b @ R ) ) )
=> ~ ( ( Xa3 = nil_a )
=> ( Y
!= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% ring.combine.elims
thf(fact_870_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_871_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_872_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_873_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_874_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_875_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_876_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_877_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_878_combine__common__factor,axiom,
! [A: int,E2: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_879_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_880_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_881_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_882_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_883_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_884_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_885_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_886_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_887_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_888_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_889_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_890_eq__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_891_eq__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_892_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_893_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_894_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_895_subalbegra__incl__imp__finite__dimension,axiom,
! [K2: set_a,E: set_a,V2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V2 @ r )
=> ( ( ord_less_eq_set_a @ V2 @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ V2 ) ) ) ) ) ).
% subalbegra_incl_imp_finite_dimension
thf(fact_896_lead__coeff__in__carrier,axiom,
! [K2: set_a,A: a,P2: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ ( cons_a @ A @ P2 ) )
=> ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ).
% lead_coeff_in_carrier
thf(fact_897_space__subgroup__props_I6_J,axiom,
! [K2: set_a,N: nat,E: set_a,K: a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ A ) @ E )
=> ( member_a @ A @ E ) ) ) ) ) ) ).
% space_subgroup_props(6)
thf(fact_898_finite__dimension__imp__subalgebra,axiom,
! [K2: set_a,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ( embedd9027525575939734154ra_a_b @ K2 @ E @ r ) ) ) ).
% finite_dimension_imp_subalgebra
thf(fact_899_telescopic__base,axiom,
! [K2: set_a,F2: set_a,N: nat,M: nat,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( subfield_a_b @ F2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ F2 )
=> ( ( embedd2795209813406577254on_a_b @ r @ M @ F2 @ E )
=> ( embedd2795209813406577254on_a_b @ r @ ( times_times_nat @ N @ M ) @ K2 @ E ) ) ) ) ) ).
% telescopic_base
thf(fact_900_dimension__is__inj,axiom,
! [K2: set_a,N: nat,E: set_a,M: nat] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M @ K2 @ E )
=> ( N = M ) ) ) ) ).
% dimension_is_inj
thf(fact_901_telescopic__base__dim_I1_J,axiom,
! [K2: set_a,F2: set_a,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( subfield_a_b @ F2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ F2 )
=> ( ( embedd8708762675212832759on_a_b @ r @ F2 @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ E ) ) ) ) ) ).
% telescopic_base_dim(1)
thf(fact_902_finite__dimension__def,axiom,
! [K2: set_a,E: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
= ( ? [N2: nat] : ( embedd2795209813406577254on_a_b @ r @ N2 @ K2 @ E ) ) ) ).
% finite_dimension_def
thf(fact_903_finite__dimensionI,axiom,
! [N: nat,K2: set_a,E: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ E ) ) ).
% finite_dimensionI
thf(fact_904_finite__dimensionE_H,axiom,
! [K2: set_a,E: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ~ ! [N3: nat] :
~ ( embedd2795209813406577254on_a_b @ r @ N3 @ K2 @ E ) ) ).
% finite_dimensionE'
thf(fact_905_space__subgroup__props_I3_J,axiom,
! [K2: set_a,N: nat,E: set_a,V1: a,V22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( member_a @ V1 @ E )
=> ( ( member_a @ V22 @ E )
=> ( member_a @ ( add_a_b @ r @ V1 @ V22 ) @ E ) ) ) ) ) ).
% space_subgroup_props(3)
thf(fact_906_space__subgroup__props_I2_J,axiom,
! [K2: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( member_a @ ( zero_a_b @ r ) @ E ) ) ) ).
% space_subgroup_props(2)
thf(fact_907_space__subgroup__props_I5_J,axiom,
! [K2: set_a,N: nat,E: set_a,K: a,V3: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ V3 ) @ E ) ) ) ) ) ).
% space_subgroup_props(5)
thf(fact_908_space__subgroup__props_I4_J,axiom,
! [K2: set_a,N: nat,E: set_a,V3: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( a_inv_a_b @ r @ V3 ) @ E ) ) ) ) ).
% space_subgroup_props(4)
thf(fact_909_unique__dimension,axiom,
! [K2: set_a,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ? [X4: nat] :
( ( embedd2795209813406577254on_a_b @ r @ X4 @ K2 @ E )
& ! [Y6: nat] :
( ( embedd2795209813406577254on_a_b @ r @ Y6 @ K2 @ E )
=> ( Y6 = X4 ) ) ) ) ) ).
% unique_dimension
thf(fact_910_space__subgroup__props_I1_J,axiom,
! [K2: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% space_subgroup_props(1)
thf(fact_911_eval__poly__in__carrier,axiom,
! [K2: set_a,P2: list_a,X: a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P2 @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% eval_poly_in_carrier
thf(fact_912_lead__coeff__not__zero,axiom,
! [K2: set_a,A: a,P2: list_a] :
( ( polynomial_a_b @ r @ K2 @ ( cons_a @ A @ P2 ) )
=> ( member_a @ A @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% lead_coeff_not_zero
thf(fact_913_zero__is__polynomial,axiom,
! [K2: set_a] : ( polynomial_a_b @ r @ K2 @ nil_a ) ).
% zero_is_polynomial
thf(fact_914_carrier__polynomial,axiom,
! [K2: set_a,P2: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P2 )
=> ( polynomial_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ P2 ) ) ) ).
% carrier_polynomial
thf(fact_915_const__is__polynomial,axiom,
! [A: a,K2: set_a] :
( ( member_a @ A @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( polynomial_a_b @ r @ K2 @ ( cons_a @ A @ nil_a ) ) ) ).
% const_is_polynomial
thf(fact_916_ring_Ounique__dimension,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E )
=> ? [X4: nat] :
( ( embedd2795209813406577254on_a_b @ R @ X4 @ K2 @ E )
& ! [Y6: nat] :
( ( embedd2795209813406577254on_a_b @ R @ Y6 @ K2 @ E )
=> ( Y6 = X4 ) ) ) ) ) ) ).
% ring.unique_dimension
thf(fact_917_ring_Ofinite__dimensionI,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( embedd8708762675212832759on_a_b @ R @ K2 @ E ) ) ) ).
% ring.finite_dimensionI
thf(fact_918_ring_Ofinite__dimensionE_H,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E )
=> ~ ! [N3: nat] :
~ ( embedd2795209813406577254on_a_b @ R @ N3 @ K2 @ E ) ) ) ).
% ring.finite_dimensionE'
thf(fact_919_ring_Ofinite__dimension__def,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E )
= ( ? [N2: nat] : ( embedd2795209813406577254on_a_b @ R @ N2 @ K2 @ E ) ) ) ) ).
% ring.finite_dimension_def
thf(fact_920_ring_Odimension_Ocong,axiom,
embedd2795209813406577254on_a_b = embedd2795209813406577254on_a_b ).
% ring.dimension.cong
thf(fact_921_ring_Ofinite__dimension_Ocong,axiom,
embedd8708762675212832759on_a_b = embedd8708762675212832759on_a_b ).
% ring.finite_dimension.cong
thf(fact_922_ring_Ozero__is__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( polynomial_a_b @ R @ K2 @ nil_a ) ) ).
% ring.zero_is_polynomial
thf(fact_923_univ__poly__carrier,axiom,
( polynomial_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,K4: set_a,P3: list_a] : ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K4 ) ) ) ) ) ).
% univ_poly_carrier
thf(fact_924_ring_Otelescopic__base,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,F2: set_a,N: nat,M: nat,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( subfield_a_b @ F2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ F2 )
=> ( ( embedd2795209813406577254on_a_b @ R @ M @ F2 @ E )
=> ( embedd2795209813406577254on_a_b @ R @ ( times_times_nat @ N @ M ) @ K2 @ E ) ) ) ) ) ) ).
% ring.telescopic_base
thf(fact_925_ring_Odimension__is__inj,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,M: nat] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ R @ M @ K2 @ E )
=> ( N = M ) ) ) ) ) ).
% ring.dimension_is_inj
thf(fact_926_ring_Otelescopic__base__dim_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,F2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( subfield_a_b @ F2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ F2 )
=> ( ( embedd8708762675212832759on_a_b @ R @ F2 @ E )
=> ( embedd8708762675212832759on_a_b @ R @ K2 @ E ) ) ) ) ) ) ).
% ring.telescopic_base_dim(1)
thf(fact_927_ring_Ocarrier__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P2 )
=> ( polynomial_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) @ P2 ) ) ) ) ).
% ring.carrier_polynomial
thf(fact_928_ring_Ocarrier__polynomial,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ P2 )
=> ( polyno1315193887021588240t_unit @ R @ ( partia5361259788508890537t_unit @ R ) @ P2 ) ) ) ) ).
% ring.carrier_polynomial
thf(fact_929_ring_Ospace__subgroup__props_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ E ) ) ) ) ).
% ring.space_subgroup_props(2)
thf(fact_930_ring_Ospace__subgroup__props_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( member_a @ ( zero_a_b @ R ) @ E ) ) ) ) ).
% ring.space_subgroup_props(2)
thf(fact_931_ring_Ospace__subgroup__props_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E: set_list_a,V1: list_a,V22: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E )
=> ( ( member_list_a @ V1 @ E )
=> ( ( member_list_a @ V22 @ E )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ V1 @ V22 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(3)
thf(fact_932_ring_Ospace__subgroup__props_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,V1: a,V22: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( member_a @ V1 @ E )
=> ( ( member_a @ V22 @ E )
=> ( member_a @ ( add_a_b @ R @ V1 @ V22 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(3)
thf(fact_933_ring_Ospace__subgroup__props_I5_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E: set_list_a,K: list_a,V3: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E )
=> ( ( member_list_a @ K @ K2 )
=> ( ( member_list_a @ V3 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K @ V3 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(5)
thf(fact_934_ring_Ospace__subgroup__props_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,K: a,V3: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ V3 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(5)
thf(fact_935_ring_Ospace__subgroup__props_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,V3: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( member_a @ V3 @ E )
=> ( member_a @ ( a_inv_a_b @ R @ V3 ) @ E ) ) ) ) ) ).
% ring.space_subgroup_props(4)
thf(fact_936_ring_Oeval__poly__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P2: list_a,X: a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( eval_a_b @ R @ P2 @ X ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% ring.eval_poly_in_carrier
thf(fact_937_ring_Oeval__poly__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P2: list_list_a,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ P2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ R @ P2 @ X ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ).
% ring.eval_poly_in_carrier
thf(fact_938_ring_Ospace__subgroup__props_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.space_subgroup_props(1)
thf(fact_939_ring_Ospace__subgroup__props_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E )
=> ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.space_subgroup_props(1)
thf(fact_940_ring_Ofinite__dimension__imp__subalgebra,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E )
=> ( embedd9027525575939734154ra_a_b @ K2 @ E @ R ) ) ) ) ).
% ring.finite_dimension_imp_subalgebra
thf(fact_941_ring_Olead__coeff__not__zero,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,A: list_a,P2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ ( cons_list_a @ A @ P2 ) )
=> ( member_list_a @ A @ ( minus_646659088055828811list_a @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ) ) ).
% ring.lead_coeff_not_zero
thf(fact_942_ring_Olead__coeff__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,A: a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( polynomial_a_b @ R @ K2 @ ( cons_a @ A @ P2 ) )
=> ( member_a @ A @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ) ).
% ring.lead_coeff_not_zero
thf(fact_943_ring_Osubalbegra__incl__imp__finite__dimension,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a,V2: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd8708762675212832759on_a_b @ R @ K2 @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V2 @ R )
=> ( ( ord_less_eq_set_a @ V2 @ E )
=> ( embedd8708762675212832759on_a_b @ R @ K2 @ V2 ) ) ) ) ) ) ).
% ring.subalbegra_incl_imp_finite_dimension
thf(fact_944_ring_Oconst__is__polynomial,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ A @ ( minus_646659088055828811list_a @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( polyno1315193887021588240t_unit @ R @ K2 @ ( cons_list_a @ A @ nil_list_a ) ) ) ) ).
% ring.const_is_polynomial
thf(fact_945_ring_Oconst__is__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( polynomial_a_b @ R @ K2 @ ( cons_a @ A @ nil_a ) ) ) ) ).
% ring.const_is_polynomial
thf(fact_946_ring_Olead__coeff__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,A: a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ ( cons_a @ A @ P2 ) )
=> ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ) ) ).
% ring.lead_coeff_in_carrier
thf(fact_947_ring_Olead__coeff__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,A: list_a,P2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ ( cons_list_a @ A @ P2 ) )
=> ( member_list_a @ A @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ) ) ) ).
% ring.lead_coeff_in_carrier
thf(fact_948_ring_Ospace__subgroup__props_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a,K: a,A: a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ R @ K @ A ) @ E )
=> ( member_a @ A @ E ) ) ) ) ) ) ) ).
% ring.space_subgroup_props(6)
thf(fact_949_ring_Ospace__subgroup__props_I6_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E: set_list_a,K: list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E )
=> ( ( member_list_a @ K @ ( minus_646659088055828811list_a @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K @ A ) @ E )
=> ( member_list_a @ A @ E ) ) ) ) ) ) ) ).
% ring.space_subgroup_props(6)
thf(fact_950_dimension__zero,axiom,
! [K2: set_a,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ zero_zero_nat @ K2 @ E )
=> ( E
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% dimension_zero
thf(fact_951_const__term__zero,axiom,
! [K2: set_a,P2: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P2 )
=> ( ( P2 != nil_a )
=> ( ( ( const_term_a_b @ r @ P2 )
= ( zero_a_b @ r ) )
=> ~ ! [P4: list_a] :
( ( polynomial_a_b @ r @ K2 @ P4 )
=> ( ( P4 != nil_a )
=> ( P2
!= ( append_a @ P4 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ) ) ).
% const_term_zero
thf(fact_952_zero__dim,axiom,
! [K2: set_a] : ( embedd2795209813406577254on_a_b @ r @ zero_zero_nat @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% zero_dim
thf(fact_953_dimension__backwards,axiom,
! [K2: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ ( suc @ N ) @ K2 @ E )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ? [E3: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E3 )
& ~ ( member_a @ X4 @ E3 )
& ( E
= ( embedd971793762689825387on_a_b @ r @ K2 @ X4 @ E3 ) ) ) ) ) ) ).
% dimension_backwards
thf(fact_954_Suc__dim,axiom,
! [V3: a,E: set_a,N: nat,K2: set_a] :
( ( member_a @ V3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ V3 @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( embedd2795209813406577254on_a_b @ r @ ( suc @ N ) @ K2 @ ( embedd971793762689825387on_a_b @ r @ K2 @ V3 @ E ) ) ) ) ) ).
% Suc_dim
thf(fact_955_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_956_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_957_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_958_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_959_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_960_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_961_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_962_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_963_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_964_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_965_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_966_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_967_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_968_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_969_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_970_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_971_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_972_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_973_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_974_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_975_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_976_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_977_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_978_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_979_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_980_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_981_dimension_Osimps,axiom,
! [A1: nat,A22: set_a,A32: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ A1 @ A22 @ A32 )
= ( ? [K4: set_a] :
( ( A1 = zero_zero_nat )
& ( A22 = K4 )
& ( A32
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
| ? [V4: a,E4: set_a,N2: nat,K4: set_a] :
( ( A1
= ( suc @ N2 ) )
& ( A22 = K4 )
& ( A32
= ( embedd971793762689825387on_a_b @ r @ K4 @ V4 @ E4 ) )
& ( member_a @ V4 @ ( partia707051561876973205xt_a_b @ r ) )
& ~ ( member_a @ V4 @ E4 )
& ( embedd2795209813406577254on_a_b @ r @ N2 @ K4 @ E4 ) ) ) ) ).
% dimension.simps
thf(fact_982_dimension_Ocases,axiom,
! [A1: nat,A22: set_a,A32: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ~ ! [V: a,E5: set_a,N3: nat] :
( ( A1
= ( suc @ N3 ) )
=> ( ( A32
= ( embedd971793762689825387on_a_b @ r @ A22 @ V @ E5 ) )
=> ( ( member_a @ V @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ V @ E5 )
=> ~ ( embedd2795209813406577254on_a_b @ r @ N3 @ A22 @ E5 ) ) ) ) ) ) ) ).
% dimension.cases
thf(fact_983_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_984_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_985_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_986_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_987_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_988_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_989_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_990_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_991_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_992_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_993_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_994_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_995_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_996_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_997_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_998_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_999_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1000_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_1001_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1002_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_1003_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1004_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_1005_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_1006_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_1007_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_1008_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_1009_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_1010_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_1011_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1012_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1013_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1014_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1015_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1016_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1017_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1018_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1019_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1020_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1021_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1022_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_1023_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_1024_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_1025_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1026_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1027_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1028_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1029_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1030_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1031_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1032_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_1033_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_1034_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1035_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1036_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1037_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1038_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1039_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1040_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1041_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1042_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_1043_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1044_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1045_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1046_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1047_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1048_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1049_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1050_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1051_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1052_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1053_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1054_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B5: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B5 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1055_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_1056_ring_Odimension_Osimps,axiom,
! [R: partia2670972154091845814t_unit,A1: nat,A22: set_list_a,A32: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ A1 @ A22 @ A32 )
= ( ? [K4: set_list_a] :
( ( A1 = zero_zero_nat )
& ( A22 = K4 )
& ( A32
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
| ? [V4: list_a,E4: set_list_a,N2: nat,K4: set_list_a] :
( ( A1
= ( suc @ N2 ) )
& ( A22 = K4 )
& ( A32
= ( embedd5150658419831591667t_unit @ R @ K4 @ V4 @ E4 ) )
& ( member_list_a @ V4 @ ( partia5361259788508890537t_unit @ R ) )
& ~ ( member_list_a @ V4 @ E4 )
& ( embedd3793949463769647726t_unit @ R @ N2 @ K4 @ E4 ) ) ) ) ) ).
% ring.dimension.simps
thf(fact_1057_ring_Odimension_Osimps,axiom,
! [R: partia2175431115845679010xt_a_b,A1: nat,A22: set_a,A32: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ A1 @ A22 @ A32 )
= ( ? [K4: set_a] :
( ( A1 = zero_zero_nat )
& ( A22 = K4 )
& ( A32
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
| ? [V4: a,E4: set_a,N2: nat,K4: set_a] :
( ( A1
= ( suc @ N2 ) )
& ( A22 = K4 )
& ( A32
= ( embedd971793762689825387on_a_b @ R @ K4 @ V4 @ E4 ) )
& ( member_a @ V4 @ ( partia707051561876973205xt_a_b @ R ) )
& ~ ( member_a @ V4 @ E4 )
& ( embedd2795209813406577254on_a_b @ R @ N2 @ K4 @ E4 ) ) ) ) ) ).
% ring.dimension.simps
thf(fact_1058_ring_Odimension_Ocases,axiom,
! [R: partia2670972154091845814t_unit,A1: nat,A22: set_list_a,A32: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ~ ! [V: list_a,E5: set_list_a,N3: nat] :
( ( A1
= ( suc @ N3 ) )
=> ( ( A32
= ( embedd5150658419831591667t_unit @ R @ A22 @ V @ E5 ) )
=> ( ( member_list_a @ V @ ( partia5361259788508890537t_unit @ R ) )
=> ( ~ ( member_list_a @ V @ E5 )
=> ~ ( embedd3793949463769647726t_unit @ R @ N3 @ A22 @ E5 ) ) ) ) ) ) ) ) ).
% ring.dimension.cases
thf(fact_1059_ring_Odimension_Ocases,axiom,
! [R: partia2175431115845679010xt_a_b,A1: nat,A22: set_a,A32: set_a] :
( ( ring_a_b @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ~ ! [V: a,E5: set_a,N3: nat] :
( ( A1
= ( suc @ N3 ) )
=> ( ( A32
= ( embedd971793762689825387on_a_b @ R @ A22 @ V @ E5 ) )
=> ( ( member_a @ V @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ~ ( member_a @ V @ E5 )
=> ~ ( embedd2795209813406577254on_a_b @ R @ N3 @ A22 @ E5 ) ) ) ) ) ) ) ) ).
% ring.dimension.cases
thf(fact_1060_ring_OSuc__dim,axiom,
! [R: partia2670972154091845814t_unit,V3: list_a,E: set_list_a,N: nat,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ V3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ~ ( member_list_a @ V3 @ E )
=> ( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E )
=> ( embedd3793949463769647726t_unit @ R @ ( suc @ N ) @ K2 @ ( embedd5150658419831591667t_unit @ R @ K2 @ V3 @ E ) ) ) ) ) ) ).
% ring.Suc_dim
thf(fact_1061_ring_OSuc__dim,axiom,
! [R: partia2175431115845679010xt_a_b,V3: a,E: set_a,N: nat,K2: set_a] :
( ( ring_a_b @ R )
=> ( ( member_a @ V3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ~ ( member_a @ V3 @ E )
=> ( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E )
=> ( embedd2795209813406577254on_a_b @ R @ ( suc @ N ) @ K2 @ ( embedd971793762689825387on_a_b @ R @ K2 @ V3 @ E ) ) ) ) ) ) ).
% ring.Suc_dim
thf(fact_1062_ring_Odimension__backwards,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,N: nat,E: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ ( suc @ N ) @ K2 @ E )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
& ? [E3: set_list_a] :
( ( embedd3793949463769647726t_unit @ R @ N @ K2 @ E3 )
& ~ ( member_list_a @ X4 @ E3 )
& ( E
= ( embedd5150658419831591667t_unit @ R @ K2 @ X4 @ E3 ) ) ) ) ) ) ) ).
% ring.dimension_backwards
thf(fact_1063_ring_Odimension__backwards,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,N: nat,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ ( suc @ N ) @ K2 @ E )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
& ? [E3: set_a] :
( ( embedd2795209813406577254on_a_b @ R @ N @ K2 @ E3 )
& ~ ( member_a @ X4 @ E3 )
& ( E
= ( embedd971793762689825387on_a_b @ R @ K2 @ X4 @ E3 ) ) ) ) ) ) ) ).
% ring.dimension_backwards
thf(fact_1064_ring_Ozero__dim,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( embedd3793949463769647726t_unit @ R @ zero_zero_nat @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ).
% ring.zero_dim
thf(fact_1065_ring_Ozero__dim,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( ring_a_b @ R )
=> ( embedd2795209813406577254on_a_b @ R @ zero_zero_nat @ K2 @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ).
% ring.zero_dim
thf(fact_1066_ring_Odimension__zero,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,E: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( embedd3793949463769647726t_unit @ R @ zero_zero_nat @ K2 @ E )
=> ( E
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ) ) ).
% ring.dimension_zero
thf(fact_1067_ring_Odimension__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,E: set_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd2795209813406577254on_a_b @ R @ zero_zero_nat @ K2 @ E )
=> ( E
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ) ).
% ring.dimension_zero
thf(fact_1068_ring_Oconst__term__zero,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ P2 )
=> ( ( P2 != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ R @ P2 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ~ ! [P4: list_list_a] :
( ( polyno1315193887021588240t_unit @ R @ K2 @ P4 )
=> ( ( P4 != nil_list_a )
=> ( P2
!= ( append_list_a @ P4 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) ) ) ) ) ) ) ) ) ).
% ring.const_term_zero
thf(fact_1069_ring_Oconst__term__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P2 )
=> ( ( P2 != nil_a )
=> ( ( ( const_term_a_b @ R @ P2 )
= ( zero_a_b @ R ) )
=> ~ ! [P4: list_a] :
( ( polynomial_a_b @ R @ K2 @ P4 )
=> ( ( P4 != nil_a )
=> ( P2
!= ( append_a @ P4 @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) ) ) ) ) ) ) ) ) ).
% ring.const_term_zero
thf(fact_1070_eval__append__aux,axiom,
! [P2: list_a,B: a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P2 @ ( cons_a @ B @ nil_a ) ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P2 @ A ) @ A ) @ B ) ) ) ) ) ).
% eval_append_aux
thf(fact_1071_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1072_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_1073_polynomial__incl,axiom,
! [K2: set_a,P2: list_a] :
( ( polynomial_a_b @ r @ K2 @ P2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ K2 ) ) ).
% polynomial_incl
thf(fact_1074_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1075_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1076_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1077_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1078_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1079_eval__in__carrier,axiom,
! [P2: list_a,X: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P2 @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier
thf(fact_1080_factors__closed,axiom,
! [Fs: list_a,A: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% factors_closed
thf(fact_1081_factors__mult,axiom,
! [Fa: list_a,A: a,Fb: list_a,B: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fa @ A )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B )
=> ( ( 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 @ A @ B ) ) ) ) ) ) ).
% factors_mult
thf(fact_1082_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1083_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1084_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1085_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1086_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1087_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1088_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1089_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_1090_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_1091_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_1092_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_1093_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_1094_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_1095_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_1096_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_1097_const__term__explicit,axiom,
! [P2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P2 != nil_a )
=> ( ( ( const_term_a_b @ r @ P2 )
= A )
=> ~ ! [P4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( P2
!= ( append_a @ P4 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ).
% const_term_explicit
thf(fact_1098_const__term__eq__last,axiom,
! [P2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( append_a @ P2 @ ( cons_a @ A @ nil_a ) ) )
= A ) ) ) ).
% const_term_eq_last
thf(fact_1099_combine__append__zero,axiom,
! [Us2: list_a,Ks2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( append_a @ Ks2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Us2 )
= ( embedded_combine_a_b @ r @ Ks2 @ Us2 ) ) ) ).
% combine_append_zero
thf(fact_1100_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_1101_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_1102_combine__in__carrier,axiom,
! [Ks2: list_a,Us2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( embedded_combine_a_b @ r @ Ks2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% combine_in_carrier
thf(fact_1103_polynomial__in__carrier,axiom,
! [K2: set_a,P2: list_a] :
( ( subring_a_b @ K2 @ r )
=> ( ( polynomial_a_b @ r @ K2 @ P2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% polynomial_in_carrier
thf(fact_1104_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1105_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1106_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1107_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1108_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1109_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1110_subset__code_I1_J,axiom,
! [Xs: list_list_a,B6: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B6 )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X3 @ B6 ) ) ) ) ).
% subset_code(1)
thf(fact_1111_subset__code_I1_J,axiom,
! [Xs: list_a,B6: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B6 )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a @ X3 @ B6 ) ) ) ) ).
% subset_code(1)
thf(fact_1112_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1113_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1114_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1115_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1116_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1117_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1118_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1119_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1120_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1121_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1122_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K5: nat] :
( N2
= ( plus_plus_nat @ M3 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1123_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_1124_set__subset__Cons,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1125_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1126_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1127_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1128_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1129_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1130_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1131_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1132_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1133_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1134_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1135_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1136_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1137_ring_Opolynomial__incl,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ K2 ) ) ) ).
% ring.polynomial_incl
thf(fact_1138_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1139_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1140_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1141_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1142_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z2: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z2 )
=> ( R @ X4 @ Z2 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1143_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1144_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N3 )
=> ( P @ M4 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1145_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1146_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1147_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1148_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_1149_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1150_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1151_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1152_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1153_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1154_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1155_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1156_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1157_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1158_ring_Ocombine__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,Ks2: list_a,Us2: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( embedded_combine_a_b @ R @ Ks2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.combine_in_carrier
thf(fact_1159_ring_Ocombine__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,Ks2: list_list_a,Us2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( embedd2435972518007585703t_unit @ R @ Ks2 @ Us2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.combine_in_carrier
thf(fact_1160_ring_Oeval__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,X: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( eval_a_b @ R @ P2 @ X ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.eval_in_carrier
thf(fact_1161_ring_Oeval__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,X: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ R @ P2 @ X ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.eval_in_carrier
thf(fact_1162_monoid_Ofactors__closed,axiom,
! [G: partia2175431115845679010xt_a_b,Fs: list_a,A: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( factor5638265376665762323xt_a_b @ G @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% monoid.factors_closed
thf(fact_1163_monoid_Ofactors__closed,axiom,
! [G: partia2670972154091845814t_unit,Fs: list_list_a,A: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( factor7181967632740204193t_unit @ G @ Fs @ A )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% monoid.factors_closed
thf(fact_1164_ring_Opolynomial__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K2 @ R )
=> ( ( polynomial_a_b @ R @ K2 @ P2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.polynomial_in_carrier
thf(fact_1165_ring_Opolynomial__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,P2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K2 @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K2 @ P2 )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.polynomial_in_carrier
thf(fact_1166_monoid_Ofactors__mult,axiom,
! [G: partia2175431115845679010xt_a_b,Fa: list_a,A: a,Fb: list_a,B: a] :
( ( monoid8385113658579753027xt_a_b @ G )
=> ( ( factor5638265376665762323xt_a_b @ G @ Fa @ A )
=> ( ( factor5638265376665762323xt_a_b @ G @ Fb @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia707051561876973205xt_a_b @ G ) )
=> ( factor5638265376665762323xt_a_b @ G @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ G @ A @ B ) ) ) ) ) ) ) ).
% monoid.factors_mult
thf(fact_1167_monoid_Ofactors__mult,axiom,
! [G: partia2670972154091845814t_unit,Fa: list_list_a,A: list_a,Fb: list_list_a,B: list_a] :
( ( monoid5589397312508706001t_unit @ G )
=> ( ( factor7181967632740204193t_unit @ G @ Fa @ A )
=> ( ( factor7181967632740204193t_unit @ G @ Fb @ B )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fa ) @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fb ) @ ( partia5361259788508890537t_unit @ G ) )
=> ( factor7181967632740204193t_unit @ G @ ( append_list_a @ Fa @ Fb ) @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) ) ) ) ) ) ) ).
% monoid.factors_mult
thf(fact_1168_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_a @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1169_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1170_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1171_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1172_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X4: a] : ( P @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1173_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X4: a,Xs2: list_a] : ( P @ ( cons_a @ X4 @ Xs2 ) @ nil_a )
=> ( ! [Y3: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y3 @ Ys2 ) )
=> ( ! [X4: a,Xs2: list_a,Y3: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X4 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1174_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y4: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1175_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X4: a] :
( X
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y3: a,Xs2: list_a] :
( X
!= ( cons_a @ X4 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1176_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X4: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X4 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_1177_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X21: a,X22: list_a] :
( Y
!= ( cons_a @ X21 @ X22 ) ) ) ).
% list.exhaust
thf(fact_1178_list_OdiscI,axiom,
! [List: list_a,X212: a,X222: list_a] :
( ( List
= ( cons_a @ X212 @ X222 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_1179_list_Odistinct_I1_J,axiom,
! [X212: a,X222: list_a] :
( nil_a
!= ( cons_a @ X212 @ X222 ) ) ).
% list.distinct(1)
thf(fact_1180_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_1181_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_1182_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_1183_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1184_ring_Oconst__term__explicit,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,A: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( P2 != nil_a )
=> ( ( ( const_term_a_b @ R @ P2 )
= A )
=> ~ ! [P4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( P2
!= ( append_a @ P4 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ) ).
% ring.const_term_explicit
thf(fact_1185_ring_Oconst__term__explicit,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( P2 != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ R @ P2 )
= A )
=> ~ ! [P4: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P4 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( P2
!= ( append_list_a @ P4 @ ( cons_list_a @ A @ nil_list_a ) ) ) ) ) ) ) ) ).
% ring.const_term_explicit
thf(fact_1186_ring_Oconst__term__eq__last,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,A: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( const_term_a_b @ R @ ( append_a @ P2 @ ( cons_a @ A @ nil_a ) ) )
= A ) ) ) ) ).
% ring.const_term_eq_last
thf(fact_1187_ring_Oconst__term__eq__last,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( append_list_a @ P2 @ ( cons_list_a @ A @ nil_list_a ) ) )
= A ) ) ) ) ).
% ring.const_term_eq_last
thf(fact_1188_ring_Ocombine__append__zero,axiom,
! [R: partia2175431115845679010xt_a_b,Us2: list_a,Ks2: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( embedded_combine_a_b @ R @ ( append_a @ Ks2 @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) @ Us2 )
= ( embedded_combine_a_b @ R @ Ks2 @ Us2 ) ) ) ) ).
% ring.combine_append_zero
thf(fact_1189_ring_Ocombine__append__zero,axiom,
! [R: partia2670972154091845814t_unit,Us2: list_list_a,Ks2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( embedd2435972518007585703t_unit @ R @ ( append_list_a @ Ks2 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) @ Us2 )
= ( embedd2435972518007585703t_unit @ R @ Ks2 @ Us2 ) ) ) ) ).
% ring.combine_append_zero
thf(fact_1190_ring_Oeval__append__aux,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,B: a,A: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( append_a @ P2 @ ( cons_a @ B @ nil_a ) ) @ A )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ ( eval_a_b @ R @ P2 @ A ) @ A ) @ B ) ) ) ) ) ) ).
% ring.eval_append_aux
thf(fact_1191_ring_Oeval__append__aux,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,B: list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( append_list_a @ P2 @ ( cons_list_a @ B @ nil_list_a ) ) @ A )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ ( eval_l34571156754992824t_unit @ R @ P2 @ A ) @ A ) @ B ) ) ) ) ) ) ).
% ring.eval_append_aux
thf(fact_1192_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X4: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X4 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_1193_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y3: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_1194_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys4: list_a] :
( ( ( cons_a @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_1195_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys4: list_a] :
( ( Ys
= ( cons_a @ X @ Ys4 ) )
& ( ( append_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_1196_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X4: a] : ( P @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X4 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_1197_exp__base__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% exp_base_closed
thf(fact_1198_Span__mem__iff,axiom,
! [K2: set_a,Us2: list_a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
& ? [Ks3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ K2 )
& ( ( embedded_combine_a_b @ r @ ( cons_a @ X3 @ Ks3 ) @ ( cons_a @ A @ Us2 ) )
= ( zero_a_b @ r ) ) ) ) ) ) ) ) ) ).
% Span_mem_iff
thf(fact_1199_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1200_Span__in__carrier,axiom,
! [K2: set_a,Us2: list_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% Span_in_carrier
thf(fact_1201_Span__subgroup__props_I1_J,axiom,
! [K2: set_a,Us2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% Span_subgroup_props(1)
thf(fact_1202_Span__base__incl,axiom,
! [K2: set_a,Us2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) ) ) ) ).
% Span_base_incl
thf(fact_1203_Span__same__set,axiom,
! [K2: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( set_a2 @ Us2 )
= ( set_a2 @ Vs ) )
=> ( ( embedded_Span_a_b @ r @ K2 @ Us2 )
= ( embedded_Span_a_b @ r @ K2 @ Vs ) ) ) ) ) ).
% Span_same_set
thf(fact_1204_mono__Span__sublist,axiom,
! [K2: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( 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 @ K2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs ) ) ) ) ) ).
% mono_Span_sublist
thf(fact_1205_mono__Span__subset,axiom,
! [K2: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs ) ) ) ) ) ).
% mono_Span_subset
thf(fact_1206_Span__subalgebraI,axiom,
! [K2: set_a,E: set_a,Us2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ E @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ E )
=> ( ! [V5: set_a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V5 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ V5 )
=> ( ord_less_eq_set_a @ E @ V5 ) ) )
=> ( E
= ( embedded_Span_a_b @ r @ K2 @ Us2 ) ) ) ) ) ) ).
% Span_subalgebraI
thf(fact_1207_subalgebra__Span__incl,axiom,
! [K2: set_a,V2: set_a,Us2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ V2 )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) @ V2 ) ) ) ) ).
% subalgebra_Span_incl
thf(fact_1208_Span__subgroup__props_I3_J,axiom,
! [K2: set_a,Us2: list_a,V1: a,V22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ V1 @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) )
=> ( ( member_a @ V22 @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) )
=> ( member_a @ ( add_a_b @ r @ V1 @ V22 ) @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) ) ) ) ) ) ).
% Span_subgroup_props(3)
thf(fact_1209_Span__subgroup__props_I2_J,axiom,
! [K2: set_a,Us2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( zero_a_b @ r ) @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) ) ) ) ).
% Span_subgroup_props(2)
thf(fact_1210_mono__Span,axiom,
! [K2: set_a,Us2: list_a,U: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K2 @ ( cons_a @ U @ Us2 ) ) ) ) ) ) ).
% mono_Span
thf(fact_1211_Span__smult__closed,axiom,
! [K2: set_a,Us2: list_a,K: a,V3: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ K2 )
=> ( ( member_a @ V3 @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ V3 ) @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) ) ) ) ) ) ).
% Span_smult_closed
thf(fact_1212_Span__subgroup__props_I4_J,axiom,
! [K2: set_a,Us2: list_a,V3: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ V3 @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) )
=> ( member_a @ ( a_inv_a_b @ r @ V3 ) @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) ) ) ) ) ).
% Span_subgroup_props(4)
thf(fact_1213_mono__Span__append_I1_J,axiom,
! [K2: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( 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 @ K2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Us2 @ Vs ) ) ) ) ) ) ).
% mono_Span_append(1)
thf(fact_1214_mono__Span__append_I2_J,axiom,
! [K2: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( 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 @ K2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Vs @ Us2 ) ) ) ) ) ) ).
% mono_Span_append(2)
thf(fact_1215_Span__finite__dimension,axiom,
! [K2: set_a,Us2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) ) ) ) ).
% Span_finite_dimension
thf(fact_1216_Span__is__subalgebra,axiom,
! [K2: set_a,Us2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) @ r ) ) ) ).
% Span_is_subalgebra
thf(fact_1217_Span__m__inv__simprule,axiom,
! [K2: set_a,Us2: list_a,K: a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ A ) @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) )
=> ( member_a @ A @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) ) ) ) ) ) ) ).
% Span_m_inv_simprule
thf(fact_1218_ring_OSpan_Ocong,axiom,
embedded_Span_a_b = embedded_Span_a_b ).
% ring.Span.cong
thf(fact_1219_ring_OSpan__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us2: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.Span_in_carrier
thf(fact_1220_ring_OSpan__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.Span_in_carrier
thf(fact_1221_ring_OSpan__subgroup__props_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.Span_subgroup_props(1)
thf(fact_1222_ring_OSpan__subgroup__props_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.Span_subgroup_props(1)
thf(fact_1223_ring_OSpan__same__set,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us2: list_a,Vs: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( set_a2 @ Us2 )
= ( set_a2 @ Vs ) )
=> ( ( embedded_Span_a_b @ R @ K2 @ Us2 )
= ( embedded_Span_a_b @ R @ K2 @ Vs ) ) ) ) ) ) ).
% ring.Span_same_set
thf(fact_1224_ring_OSpan__same__set,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( set_list_a2 @ Us2 )
= ( set_list_a2 @ Vs ) )
=> ( ( embedd4402942584324845940t_unit @ R @ K2 @ Us2 )
= ( embedd4402942584324845940t_unit @ R @ K2 @ Vs ) ) ) ) ) ) ).
% ring.Span_same_set
thf(fact_1225_ring_OSpan__base__incl,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( embedded_Span_a_b @ R @ K2 @ Us2 ) ) ) ) ) ).
% ring.Span_base_incl
thf(fact_1226_ring_OSpan__base__incl,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us2 ) ) ) ) ) ).
% ring.Span_base_incl
thf(fact_1227_ring_Omono__Span__subset,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us2: list_a,Vs: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( embedded_Span_a_b @ R @ K2 @ Vs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K2 @ Us2 ) @ ( embedded_Span_a_b @ R @ K2 @ Vs ) ) ) ) ) ) ).
% ring.mono_Span_subset
thf(fact_1228_ring_Omono__Span__subset,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ Vs ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us2 ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ Vs ) ) ) ) ) ) ).
% ring.mono_Span_subset
thf(fact_1229_ring_Omono__Span__sublist,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,Us2: list_a,Vs: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( 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 @ K2 @ Us2 ) @ ( embedded_Span_a_b @ R @ K2 @ Vs ) ) ) ) ) ) ).
% ring.mono_Span_sublist
thf(fact_1230_ring_Omono__Span__sublist,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K2 @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( set_list_a2 @ Vs ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ R @ K2 @ Us2 ) @ ( embedd4402942584324845940t_unit @ R @ K2 @ Vs ) ) ) ) ) ) ).
% ring.mono_Span_sublist
thf(fact_1231_ring_Osubalgebra__Span__incl,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a,V2: set_a,Us2: list_a] :
( ( ring_a_b @ R )
=> ( ( subfield_a_b @ K2 @ R )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V2 @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ V2 )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ R @ K2 @ Us2 ) @ V2 ) ) ) ) ) ).
% ring.subalgebra_Span_incl
thf(fact_1232_Span__mem__imp__non__trivial__combine,axiom,
! [K2: set_a,Us2: list_a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) )
=> ~ ! [K3: a] :
( ( member_a @ K3 @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ! [Ks: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ K2 )
=> ( ( ( size_size_list_a @ Ks )
= ( size_size_list_a @ Us2 ) )
=> ( ( embedded_combine_a_b @ r @ ( cons_a @ K3 @ Ks ) @ ( cons_a @ A @ Us2 ) )
!= ( zero_a_b @ r ) ) ) ) ) ) ) ) ).
% Span_mem_imp_non_trivial_combine
thf(fact_1233_Span__append__eq__set__add,axiom,
! [K2: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_Span_a_b @ r @ K2 @ ( append_a @ Us2 @ Vs ) )
= ( set_add_a_b @ r @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K2 @ Vs ) ) ) ) ) ) ).
% Span_append_eq_set_add
thf(fact_1234_set__add__closed,axiom,
! [A3: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A3 @ B6 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_add_closed
thf(fact_1235_set__add__comm,axiom,
! [I2: set_a,J2: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ J2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ I2 @ J2 )
= ( set_add_a_b @ r @ J2 @ I2 ) ) ) ) ).
% set_add_comm
thf(fact_1236_setadd__subset__G,axiom,
! [H2: set_a,K2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H2 @ K2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% setadd_subset_G
thf(fact_1237_sum__space__dim_I1_J,axiom,
! [K2: set_a,E: set_a,F2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ E )
=> ( ( embedd8708762675212832759on_a_b @ r @ K2 @ F2 )
=> ( embedd8708762675212832759on_a_b @ r @ K2 @ ( set_add_a_b @ r @ E @ F2 ) ) ) ) ) ).
% sum_space_dim(1)
thf(fact_1238_combine__eq__eval,axiom,
! [Ks2: list_a,X: a] :
( ( embedded_combine_a_b @ r @ Ks2 @ ( polyno2922411391617481336se_a_b @ r @ X @ ( size_size_list_a @ Ks2 ) ) )
= ( eval_a_b @ r @ Ks2 @ X ) ) ).
% combine_eq_eval
thf(fact_1239_combine__append,axiom,
! [Ks2: list_a,Us2: list_a,Ks4: list_a,Vs: list_a] :
( ( ( size_size_list_a @ Ks2 )
= ( size_size_list_a @ Us2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( 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 @ Ks2 @ Us2 ) @ ( embedded_combine_a_b @ r @ Ks4 @ Vs ) )
= ( embedded_combine_a_b @ r @ ( append_a @ Ks2 @ Ks4 ) @ ( append_a @ Us2 @ Vs ) ) ) ) ) ) ) ) ).
% combine_append
thf(fact_1240_Span__mem__iff__length__version,axiom,
! [K2: set_a,Us2: list_a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K2 @ Us2 ) )
= ( ? [Ks3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ K2 )
& ( ( size_size_list_a @ Ks3 )
= ( size_size_list_a @ Us2 ) )
& ( A
= ( embedded_combine_a_b @ r @ Ks3 @ Us2 ) ) ) ) ) ) ) ).
% Span_mem_iff_length_version
thf(fact_1241_degree__oneE,axiom,
! [P2: list_a,K2: set_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A2: a] :
( ( member_a @ A2 @ K2 )
=> ( ( A2
!= ( zero_a_b @ r ) )
=> ! [B2: a] :
( ( member_a @ B2 @ K2 )
=> ( P2
!= ( cons_a @ A2 @ ( cons_a @ B2 @ nil_a ) ) ) ) ) ) ) ) ).
% degree_oneE
thf(fact_1242_dimension__direct__sum__space,axiom,
! [K2: set_a,N: nat,E: set_a,M: nat,F2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M @ K2 @ F2 )
=> ( ( ( inf_inf_set_a @ E @ F2 )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( embedd2795209813406577254on_a_b @ r @ ( plus_plus_nat @ N @ M ) @ K2 @ ( set_add_a_b @ r @ E @ F2 ) ) ) ) ) ) ).
% dimension_direct_sum_space
thf(fact_1243_subring__inter,axiom,
! [I2: set_a,J2: set_a] :
( ( subring_a_b @ I2 @ r )
=> ( ( subring_a_b @ J2 @ r )
=> ( subring_a_b @ ( inf_inf_set_a @ I2 @ J2 ) @ r ) ) ) ).
% subring_inter
thf(fact_1244_subalgebra__inter,axiom,
! [K2: set_a,V2: set_a,V6: set_a] :
( ( embedd9027525575939734154ra_a_b @ K2 @ V2 @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K2 @ V6 @ r )
=> ( embedd9027525575939734154ra_a_b @ K2 @ ( inf_inf_set_a @ V2 @ V6 ) @ r ) ) ) ).
% subalgebra_inter
thf(fact_1245_subcring__inter,axiom,
! [I2: set_a,J2: set_a] :
( ( subcring_a_b @ I2 @ r )
=> ( ( subcring_a_b @ J2 @ r )
=> ( subcring_a_b @ ( inf_inf_set_a @ I2 @ J2 ) @ r ) ) ) ).
% subcring_inter
thf(fact_1246_telescopic__base__aux,axiom,
! [K2: set_a,F2: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( subfield_a_b @ F2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ F2 )
=> ( ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ F2 @ E )
=> ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E ) ) ) ) ) ).
% telescopic_base_aux
thf(fact_1247_dimension__sum__space,axiom,
! [K2: set_a,N: nat,E: set_a,M: nat,F2: set_a,K: nat] :
( ( subfield_a_b @ K2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M @ K2 @ F2 )
=> ( ( embedd2795209813406577254on_a_b @ r @ K @ K2 @ ( inf_inf_set_a @ E @ F2 ) )
=> ( embedd2795209813406577254on_a_b @ r @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) @ K2 @ ( set_add_a_b @ r @ E @ F2 ) ) ) ) ) ) ).
% dimension_sum_space
thf(fact_1248_dimension__one,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ K2 @ K2 ) ) ).
% dimension_one
thf(fact_1249_pirreducible__degree,axiom,
! [K2: set_a,P2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).
% pirreducible_degree
thf(fact_1250_subfield__long__division__theorem__shell,axiom,
! [K2: set_a,P2: list_a,B: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( B
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ? [Q2: list_a,R5: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
& ( member_list_a @ R5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
& ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ B @ Q2 ) @ R5 ) )
& ( ( R5
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R5 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% subfield_long_division_theorem_shell
thf(fact_1251_nunit__factors,axiom,
! [A: a,As: list_a] :
( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( factor5638265376665762323xt_a_b @ r @ As @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ As ) ) ) ) ).
% nunit_factors
thf(fact_1252_add_Oint__pow__1,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_int @ r @ one_one_int @ X )
= X ) ) ).
% add.int_pow_1
thf(fact_1253_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1254_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M6: nat,N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ( ord_less_nat @ ( F @ M6 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1255_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1256_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1257_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1258_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1259_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1260_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1261_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1262_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1263_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1264_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1265_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1266_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1267_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1268_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1269_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1270_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1271_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1272_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1273_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1274_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1275_boundD__carrier,axiom,
! [N: nat,F: nat > a,M: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_a @ ( F @ M ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
% Conjectures (1)
thf(conj_0,conjecture,
( ( eval_a_b @ r @ ( poly_of_const_a_b @ r @ x ) @ y )
= x ) ).
%------------------------------------------------------------------------------