TPTP Problem File: SLH0350^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Interpolation_Polynomials_HOL_Algebra/0001_Lagrange_Interpolation/prob_00258_010033__17369498_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1580 ( 521 unt; 296 typ; 0 def)
% Number of atoms : 3732 (1215 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 16134 ( 231 ~; 30 |; 154 &;13870 @)
% ( 0 <=>;1849 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 31 ( 30 usr)
% Number of type conns : 1075 (1075 >; 0 *; 0 +; 0 <<)
% Number of symbols : 269 ( 266 usr; 13 con; 0-4 aty)
% Number of variables : 3374 ( 471 ^;2837 !; 66 ?;3374 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:38:23.335
%------------------------------------------------------------------------------
% Could-be-implicit typings (30)
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member6714375691612171394list_a: ( list_a > list_list_a ) > set_li6773872926390105121list_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_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member5910328476188217884list_a: ( set_list_a > list_a ) > set_se5067313844698916539list_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
member_set_list_a_a: ( set_list_a > a ) > set_set_list_a_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member_set_a_list_a: ( set_a > list_a ) > set_set_a_list_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
member_set_a_set_a: ( set_a > set_a ) > set_set_a_set_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mtf__a_J,type,
member_set_a_a: ( set_a > a ) > set_set_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member_a_list_list_a: ( a > list_list_a ) > set_a_list_list_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_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member_a_set_list_a: ( a > set_list_a ) > set_a_set_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Set__Oset_Itf__a_J_J,type,
member_a_set_a: ( a > set_a ) > set_a_set_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__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_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_S,type,
s: set_a ).
thf(sy_v_p____,type,
p: list_a ).
thf(sy_v_s____,type,
s2: a ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1278)
thf(fact_0_assms_I1_J,axiom,
finite_finite_a @ s ).
% assms(1)
thf(fact_1__092_060open_062s_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
member_a @ s2 @ ( partia707051561876973205xt_a_b @ r ) ).
% \<open>s \<in> carrier R\<close>
thf(fact_2_assms_I2_J,axiom,
ord_less_eq_set_a @ s @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(2)
thf(fact_3_d,axiom,
member_a @ s2 @ s ).
% d
thf(fact_4_assms_I3_J,axiom,
member_a @ x @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ s ) ).
% assms(3)
thf(fact_5_factorial__domain__axioms,axiom,
ring_f5272581269873410839in_a_b @ r ).
% factorial_domain_axioms
thf(fact_6_local_Ofield__axioms,axiom,
field_a_b @ r ).
% local.field_axioms
thf(fact_7_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_8__092_060open_062_092_060And_062y_O_Ay_A_092_060in_062_AS_A_092_060Longrightarrow_062_Ax_A_092_060ominus_062_Ay_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
! [Y: a] :
( ( member_a @ Y @ s )
=> ( member_a @ ( a_minus_a_b @ r @ x @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% \<open>\<And>y. y \<in> S \<Longrightarrow> x \<ominus> y \<in> carrier R\<close>
thf(fact_9_p__def,axiom,
( p
= ( lagran9092808442999052491ux_a_b @ r @ s ) ) ).
% p_def
thf(fact_10_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_11_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_12_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_13_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_14_c,axiom,
member_a @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) @ ( units_a_ring_ext_a_b @ r ) ).
% c
thf(fact_15_finprod__one__eqI,axiom,
! [A: set_list_a,F: list_a > a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( F @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_16_finprod__one__eqI,axiom,
! [A: set_list_list_a,F: list_list_a > a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( ( F @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro5500967685102550467list_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_17_finprod__one__eqI,axiom,
! [A: set_a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( F @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_18_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_19_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_20_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_21_a,axiom,
( ( eval_a_b @ r @ p @ x )
= ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) ) ).
% a
thf(fact_22_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_23_lagrange__aux__eval,axiom,
! [S: set_a,X: a] :
( ( finite_finite_a @ S )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( lagran9092808442999052491ux_a_b @ r @ S ) @ X )
= ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ X ) @ S ) ) ) ) ) ).
% lagrange_aux_eval
thf(fact_24_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_25_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_26_finprod__one,axiom,
! [A: set_a] :
( ( finpro205304725090349623_a_b_a @ r
@ ^ [I: a] : ( one_a_ring_ext_a_b @ r )
@ A )
= ( one_a_ring_ext_a_b @ r ) ) ).
% finprod_one
thf(fact_27_finprod__infinite,axiom,
! [A: set_list_a,F: list_a > a] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finpro6052973074229812797list_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_28_finprod__infinite,axiom,
! [A: set_a,F: a > a] :
( ~ ( finite_finite_a @ A )
=> ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_29_finite__ring__finite__units,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( finite_finite_a @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% finite_ring_finite_units
thf(fact_30__092_060open_062local_Oeval_A_Ilagrange__basis__polynomial_AS_Ax_J_Ax_A_061_A_092_060one_062_092_060close_062,axiom,
( ( eval_a_b @ r @ ( lagran2649660974587678107al_a_b @ r @ s @ x ) @ x )
= ( one_a_ring_ext_a_b @ r ) ) ).
% \<open>local.eval (lagrange_basis_polynomial S x) x = \<one>\<close>
thf(fact_31_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
lagran9092808442999052491ux_a_b = lagran9092808442999052491ux_a_b ).
% ring.lagrange_basis_polynomial_aux.cong
thf(fact_32_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
lagran3534788790333317459t_unit = lagran3534788790333317459t_unit ).
% ring.lagrange_basis_polynomial_aux.cong
thf(fact_33_ring__irreducibleE_I4_J,axiom,
! [R: a] :
( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R )
=> ~ ( member_a @ R @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_34_finite__Collect__subsets,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B: set_a] : ( ord_less_eq_set_a @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_35_finite__Collect__subsets,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( finite5282473924520328461list_a
@ ( collect_set_list_a
@ ^ [B: set_list_a] : ( ord_le8861187494160871172list_a @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_36_finite__Diff,axiom,
! [A: set_a,B2: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( minus_minus_set_a @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_37_finite__Diff,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_38_finite__Diff2,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A @ B2 ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_Diff2
thf(fact_39_finite__Diff2,axiom,
! [B2: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B2 ) )
= ( finite_finite_list_a @ A ) ) ) ).
% finite_Diff2
thf(fact_40_finprod__mono__neutral__cong__right,axiom,
! [B2: set_list_list_a,A: set_list_list_a,G: list_list_a > a,H: list_list_a > a] :
( ( finite1660835950917165235list_a @ B2 )
=> ( ( ord_le8488217952732425610list_a @ A @ B2 )
=> ( ! [I2: list_list_a] :
( ( member_list_list_a @ I2 @ ( minus_5335179877275218001list_a @ B2 @ A ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_list_a_a @ G
@ ( pi_list_list_a_a @ B2
@ ^ [Uu: list_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5500967685102550467list_a @ r @ G @ B2 )
= ( finpro5500967685102550467list_a @ r @ H @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_41_finprod__mono__neutral__cong__right,axiom,
! [B2: set_a,A: set_a,G: a > a,H: a > a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ B2 @ A ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ G @ B2 )
= ( finpro205304725090349623_a_b_a @ r @ H @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_42_finprod__mono__neutral__cong__right,axiom,
! [B2: set_list_a,A: set_list_a,G: list_a > a,H: list_a > a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ ( minus_646659088055828811list_a @ B2 @ A ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r @ G @ B2 )
= ( finpro6052973074229812797list_a @ r @ H @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_43_finprod__mono__neutral__cong__left,axiom,
! [B2: set_list_list_a,A: set_list_list_a,H: list_list_a > a,G: list_list_a > a] :
( ( finite1660835950917165235list_a @ B2 )
=> ( ( ord_le8488217952732425610list_a @ A @ B2 )
=> ( ! [I2: list_list_a] :
( ( member_list_list_a @ I2 @ ( minus_5335179877275218001list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_list_a_a @ H
@ ( pi_list_list_a_a @ B2
@ ^ [Uu: list_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5500967685102550467list_a @ r @ G @ A )
= ( finpro5500967685102550467list_a @ r @ H @ B2 ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_44_finprod__mono__neutral__cong__left,axiom,
! [B2: set_a,A: set_a,H: a > a,G: a > a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_a @ H
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ G @ A )
= ( finpro205304725090349623_a_b_a @ r @ H @ B2 ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_45_finprod__mono__neutral__cong__left,axiom,
! [B2: set_list_a,A: set_list_a,H: list_a > a,G: list_a > a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ ( minus_646659088055828811list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_a_a @ H
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r @ G @ A )
= ( finpro6052973074229812797list_a @ r @ H @ B2 ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_46_subalgebra__in__carrier,axiom,
! [K: set_a,V: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_47_carrier__is__subalgebra,axiom,
! [K: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_48_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_49_subset__Idl__subset,axiom,
! [I3: set_a,H2: set_a] :
( ( ord_less_eq_set_a @ I3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H2 @ I3 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H2 ) @ ( genideal_a_b @ r @ I3 ) ) ) ) ).
% subset_Idl_subset
thf(fact_50_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_51_finprod__closed,axiom,
! [F: a > a,A: set_a] :
( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_a @ ( finpro205304725090349623_a_b_a @ r @ F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% finprod_closed
thf(fact_52_finprod__cong_H,axiom,
! [A: set_list_a,B2: set_list_a,G: list_a > a,F: list_a > a] :
( ( A = B2 )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finpro6052973074229812797list_a @ r @ F @ A )
= ( finpro6052973074229812797list_a @ r @ G @ B2 ) ) ) ) ) ).
% finprod_cong'
thf(fact_53_finprod__cong_H,axiom,
! [A: set_list_list_a,B2: set_list_list_a,G: list_list_a > a,F: list_list_a > a] :
( ( A = B2 )
=> ( ( member_list_list_a_a @ G
@ ( pi_list_list_a_a @ B2
@ ^ [Uu: list_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: list_list_a] :
( ( member_list_list_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finpro5500967685102550467list_a @ r @ F @ A )
= ( finpro5500967685102550467list_a @ r @ G @ B2 ) ) ) ) ) ).
% finprod_cong'
thf(fact_54_finprod__cong_H,axiom,
! [A: set_a,B2: set_a,G: a > a,F: a > a] :
( ( A = B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: a] :
( ( member_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( finpro205304725090349623_a_b_a @ r @ G @ B2 ) ) ) ) ) ).
% finprod_cong'
thf(fact_55_finite__Collect__disjI,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_set_a @ ( collect_set_a @ P ) )
& ( finite_finite_set_a @ ( collect_set_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_56_finite__Collect__disjI,axiom,
! [P: set_list_a > $o,Q: set_list_a > $o] :
( ( finite5282473924520328461list_a
@ ( collect_set_list_a
@ ^ [X3: set_list_a] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite5282473924520328461list_a @ ( collect_set_list_a @ P ) )
& ( finite5282473924520328461list_a @ ( collect_set_list_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_57_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_58_finite__Collect__disjI,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X3: list_a] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
& ( finite_finite_list_a @ ( collect_list_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_59_mem__Collect__eq,axiom,
! [A2: list_list_a,P: list_list_a > $o] :
( ( member_list_list_a @ A2 @ ( collect_list_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
! [A2: list_a,P: list_a > $o] :
( ( member_list_a @ A2 @ ( collect_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A2: set_list_a,P: set_list_a > $o] :
( ( member_set_list_a @ A2 @ ( collect_set_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_65_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A: set_set_list_a] :
( ( collect_set_list_a
@ ^ [X3: set_list_a] : ( member_set_list_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_69_Collect__cong,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X2: list_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_list_a @ P )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_70_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_71_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_72_Collect__cong,axiom,
! [P: set_list_a > $o,Q: set_list_a > $o] :
( ! [X2: set_list_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_list_a @ P )
= ( collect_set_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_73_finite__Collect__conjI,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ( finite_finite_set_a @ ( collect_set_a @ P ) )
| ( finite_finite_set_a @ ( collect_set_a @ Q ) ) )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_74_finite__Collect__conjI,axiom,
! [P: set_list_a > $o,Q: set_list_a > $o] :
( ( ( finite5282473924520328461list_a @ ( collect_set_list_a @ P ) )
| ( finite5282473924520328461list_a @ ( collect_set_list_a @ Q ) ) )
=> ( finite5282473924520328461list_a
@ ( collect_set_list_a
@ ^ [X3: set_list_a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_75_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_76_finite__Collect__conjI,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
| ( finite_finite_list_a @ ( collect_list_a @ Q ) ) )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X3: list_a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_77_finprod__singleton,axiom,
! [I4: list_list_a,A: set_list_list_a,F: list_list_a > a] :
( ( member_list_list_a @ I4 @ A )
=> ( ( finite1660835950917165235list_a @ A )
=> ( ( member_list_list_a_a @ F
@ ( pi_list_list_a_a @ A
@ ^ [Uu: list_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5500967685102550467list_a @ r
@ ^ [J: list_list_a] : ( if_a @ ( I4 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton
thf(fact_78_finprod__singleton,axiom,
! [I4: list_a,A: set_list_a,F: list_a > a] :
( ( member_list_a @ I4 @ A )
=> ( ( finite_finite_list_a @ A )
=> ( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r
@ ^ [J: list_a] : ( if_a @ ( I4 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton
thf(fact_79_finprod__singleton,axiom,
! [I4: a,A: set_a,F: a > a] :
( ( member_a @ I4 @ A )
=> ( ( finite_finite_a @ A )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r
@ ^ [J: a] : ( if_a @ ( I4 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton
thf(fact_80_finprod__singleton__swap,axiom,
! [I4: list_list_a,A: set_list_list_a,F: list_list_a > a] :
( ( member_list_list_a @ I4 @ A )
=> ( ( finite1660835950917165235list_a @ A )
=> ( ( member_list_list_a_a @ F
@ ( pi_list_list_a_a @ A
@ ^ [Uu: list_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5500967685102550467list_a @ r
@ ^ [J: list_list_a] : ( if_a @ ( J = I4 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_81_finprod__singleton__swap,axiom,
! [I4: list_a,A: set_list_a,F: list_a > a] :
( ( member_list_a @ I4 @ A )
=> ( ( finite_finite_list_a @ A )
=> ( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r
@ ^ [J: list_a] : ( if_a @ ( J = I4 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_82_finprod__singleton__swap,axiom,
! [I4: a,A: set_a,F: a > a] :
( ( member_a @ I4 @ A )
=> ( ( finite_finite_a @ A )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r
@ ^ [J: a] : ( if_a @ ( J = I4 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_83_ring_Olagrange__basis__polynomial_Ocong,axiom,
lagran2649660974587678107al_a_b = lagran2649660974587678107al_a_b ).
% ring.lagrange_basis_polynomial.cong
thf(fact_84_ring_Olagrange__basis__polynomial_Ocong,axiom,
lagran6985349428869127715t_unit = lagran6985349428869127715t_unit ).
% ring.lagrange_basis_polynomial.cong
thf(fact_85_pigeonhole__infinite__rel,axiom,
! [A: set_list_list_a,B2: set_a,R2: list_list_a > a > $o] :
( ~ ( finite1660835950917165235list_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B2 )
& ~ ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [A3: list_list_a] :
( ( member_list_list_a @ A3 @ A )
& ( R2 @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_86_pigeonhole__infinite__rel,axiom,
! [A: set_set_a,B2: set_a,R2: set_a > a > $o] :
( ~ ( finite_finite_set_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B2 )
& ~ ( finite_finite_set_a
@ ( collect_set_a
@ ^ [A3: set_a] :
( ( member_set_a @ A3 @ A )
& ( R2 @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_87_pigeonhole__infinite__rel,axiom,
! [A: set_set_list_a,B2: set_a,R2: set_list_a > a > $o] :
( ~ ( finite5282473924520328461list_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B2 )
& ~ ( finite5282473924520328461list_a
@ ( collect_set_list_a
@ ^ [A3: set_list_a] :
( ( member_set_list_a @ A3 @ A )
& ( R2 @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_88_pigeonhole__infinite__rel,axiom,
! [A: set_list_list_a,B2: set_list_a,R2: list_list_a > list_a > $o] :
( ~ ( finite1660835950917165235list_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ B2 )
& ~ ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [A3: list_list_a] :
( ( member_list_list_a @ A3 @ A )
& ( R2 @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_89_pigeonhole__infinite__rel,axiom,
! [A: set_set_a,B2: set_list_a,R2: set_a > list_a > $o] :
( ~ ( finite_finite_set_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ B2 )
& ~ ( finite_finite_set_a
@ ( collect_set_a
@ ^ [A3: set_a] :
( ( member_set_a @ A3 @ A )
& ( R2 @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_90_pigeonhole__infinite__rel,axiom,
! [A: set_set_list_a,B2: set_list_a,R2: set_list_a > list_a > $o] :
( ~ ( finite5282473924520328461list_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ B2 )
& ~ ( finite5282473924520328461list_a
@ ( collect_set_list_a
@ ^ [A3: set_list_a] :
( ( member_set_list_a @ A3 @ A )
& ( R2 @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_91_pigeonhole__infinite__rel,axiom,
! [A: set_a,B2: set_a,R2: a > a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A )
& ( R2 @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_92_pigeonhole__infinite__rel,axiom,
! [A: set_a,B2: set_list_a,R2: a > list_a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A )
& ( R2 @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_93_pigeonhole__infinite__rel,axiom,
! [A: set_list_a,B2: set_a,R2: list_a > a > $o] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B2 )
& ~ ( finite_finite_list_a
@ ( collect_list_a
@ ^ [A3: list_a] :
( ( member_list_a @ A3 @ A )
& ( R2 @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_94_pigeonhole__infinite__rel,axiom,
! [A: set_list_a,B2: set_list_a,R2: list_a > list_a > $o] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ B2 )
& ( R2 @ X2 @ Xa ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ B2 )
& ~ ( finite_finite_list_a
@ ( collect_list_a
@ ^ [A3: list_a] :
( ( member_list_a @ A3 @ A )
& ( R2 @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_95_not__finite__existsD,axiom,
! [P: set_a > $o] :
( ~ ( finite_finite_set_a @ ( collect_set_a @ P ) )
=> ? [X_1: set_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_96_not__finite__existsD,axiom,
! [P: set_list_a > $o] :
( ~ ( finite5282473924520328461list_a @ ( collect_set_list_a @ P ) )
=> ? [X_1: set_list_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_97_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_98_not__finite__existsD,axiom,
! [P: list_a > $o] :
( ~ ( finite_finite_list_a @ ( collect_list_a @ P ) )
=> ? [X_1: list_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_99_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_100_finite__has__minimal2,axiom,
! [A: set_set_list_a,A2: set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a @ A2 @ A )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
& ( ord_le8861187494160871172list_a @ X2 @ A2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_101_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ A2 @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_102_finite__has__maximal2,axiom,
! [A: set_set_list_a,A2: set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a @ A2 @ A )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
& ( ord_le8861187494160871172list_a @ A2 @ X2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_103_rev__finite__subset,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_104_rev__finite__subset,axiom,
! [B2: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( finite_finite_list_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_105_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_106_infinite__super,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_super
thf(fact_107_finite__subset,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_108_finite__subset,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( finite_finite_list_a @ B2 )
=> ( finite_finite_list_a @ A ) ) ) ).
% finite_subset
thf(fact_109_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_110_Diff__infinite__finite,axiom,
! [T: set_list_a,S: set_list_a] :
( ( finite_finite_list_a @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_111_primeness__condition,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P2 )
= ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% primeness_condition
thf(fact_112_finprod__mono__neutral__cong,axiom,
! [B2: set_list_list_a,A: set_list_list_a,H: list_list_a > a,G: list_list_a > a] :
( ( finite1660835950917165235list_a @ B2 )
=> ( ( finite1660835950917165235list_a @ A )
=> ( ! [I2: list_list_a] :
( ( member_list_list_a @ I2 @ ( minus_5335179877275218001list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I2: list_list_a] :
( ( member_list_list_a @ I2 @ ( minus_5335179877275218001list_a @ A @ B2 ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( inf_in7423150557312423384list_a @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_list_a_a @ G
@ ( pi_list_list_a_a @ A
@ ^ [Uu: list_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a_a @ H
@ ( pi_list_list_a_a @ B2
@ ^ [Uu: list_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5500967685102550467list_a @ r @ G @ A )
= ( finpro5500967685102550467list_a @ r @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_113_finprod__mono__neutral__cong,axiom,
! [B2: set_a,A: set_a,H: a > a,G: a > a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ A )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ A @ B2 ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( inf_inf_set_a @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a @ H
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ G @ A )
= ( finpro205304725090349623_a_b_a @ r @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_114_finprod__mono__neutral__cong,axiom,
! [B2: set_list_a,A: set_list_a,H: list_a > a,G: list_a > a] :
( ( finite_finite_list_a @ B2 )
=> ( ( finite_finite_list_a @ A )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ ( minus_646659088055828811list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ ( minus_646659088055828811list_a @ A @ B2 ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( inf_inf_set_list_a @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a_a @ H
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r @ G @ A )
= ( finpro6052973074229812797list_a @ r @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_115_noetherian__ringI,axiom,
( ! [I5: set_a] :
( ( ideal_a_b @ I5 @ r )
=> ? [A4: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( finite_finite_a @ A4 )
& ( I5
= ( genideal_a_b @ r @ A4 ) ) ) )
=> ( ring_n3639167112692572309ng_a_b @ r ) ) ).
% noetherian_ringI
thf(fact_116_a__lcos__mult__one,axiom,
! [M: set_a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M )
= M ) ) ).
% a_lcos_mult_one
thf(fact_117_ring__irreducibleE_I5_J,axiom,
! [R: a,A2: a,B3: a] :
( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R
= ( mult_a_ring_ext_a_b @ r @ A2 @ B3 ) )
=> ( ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_118_a__lcos__m__assoc,axiom,
! [M: set_a,G: a,H: a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H @ M ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H ) @ M ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_119_cgenideal__is__principalideal,axiom,
! [I4: a] :
( ( member_a @ I4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I4 ) @ r ) ) ).
% cgenideal_is_principalideal
thf(fact_120_finetely__gen,axiom,
! [I3: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ? [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( finite_finite_a @ A5 )
& ( I3
= ( genideal_a_b @ r @ A5 ) ) ) ) ).
% finetely_gen
thf(fact_121_finprod__zero__iff,axiom,
! [A: set_list_list_a,F: list_list_a > a] :
( ( finite1660835950917165235list_a @ A )
=> ( ! [A6: list_list_a] :
( ( member_list_list_a @ A6 @ A )
=> ( member_a @ ( F @ A6 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro5500967685102550467list_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A )
& ( ( F @ X3 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_122_finprod__zero__iff,axiom,
! [A: set_list_a,F: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ! [A6: list_a] :
( ( member_list_a @ A6 @ A )
=> ( member_a @ ( F @ A6 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro6052973074229812797list_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ A )
& ( ( F @ X3 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_123_finprod__zero__iff,axiom,
! [A: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A )
=> ( member_a @ ( F @ A6 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A )
& ( ( F @ X3 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_124_Units__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X2 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_125_Units__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_126_i__intersect,axiom,
! [I3: set_a,J2: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ( ( ideal_a_b @ J2 @ r )
=> ( ideal_a_b @ ( inf_inf_set_a @ I3 @ J2 ) @ r ) ) ) ).
% i_intersect
thf(fact_127_subalgebra__inter,axiom,
! [K: set_a,V: set_a,V2: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V2 @ r )
=> ( embedd9027525575939734154ra_a_b @ K @ ( inf_inf_set_a @ V @ V2 ) @ r ) ) ) ).
% subalgebra_inter
thf(fact_128_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_129_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_130_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_131_add_Or__cancel,axiom,
! [A2: a,C: a,B3: a] :
( ( ( add_a_b @ r @ A2 @ C )
= ( add_a_b @ r @ B3 @ C ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B3 ) ) ) ) ) ).
% add.r_cancel
thf(fact_132_add_Ol__cancel,axiom,
! [C: a,A2: a,B3: a] :
( ( ( add_a_b @ r @ C @ A2 )
= ( add_a_b @ r @ C @ B3 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B3 ) ) ) ) ) ).
% add.l_cancel
thf(fact_133_m__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_134_m__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% m_comm
thf(fact_135_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_136_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_137_oneideal,axiom,
ideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% oneideal
thf(fact_138_cgenideal__self,axiom,
! [I4: a] :
( ( member_a @ I4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I4 @ ( cgenid547466209912283029xt_a_b @ r @ I4 ) ) ) ).
% cgenideal_self
thf(fact_139_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_140_add_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X2 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_141_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_142_add_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X2 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_143_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_144_m__rcancel,axiom,
! [A2: a,B3: a,C: a] :
( ( A2
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B3 @ A2 )
= ( mult_a_ring_ext_a_b @ r @ C @ A2 ) )
= ( B3 = C ) ) ) ) ) ) ).
% m_rcancel
thf(fact_145_m__lcancel,axiom,
! [A2: a,B3: a,C: a] :
( ( A2
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A2 @ B3 )
= ( mult_a_ring_ext_a_b @ r @ A2 @ C ) )
= ( B3 = C ) ) ) ) ) ) ).
% m_lcancel
thf(fact_146_integral__iff,axiom,
! [A2: a,B3: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A2 @ B3 )
= ( zero_a_b @ r ) )
= ( ( A2
= ( zero_a_b @ r ) )
| ( B3
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_147_local_Ointegral,axiom,
! [A2: a,B3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A2 @ B3 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A2
= ( zero_a_b @ r ) )
| ( B3
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_148_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_149_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_150_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_151_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_152_unit__factor,axiom,
! [A2: a,B3: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A2 @ B3 ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_153_prod__unit__r,axiom,
! [A2: a,B3: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A2 @ B3 ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_154_prod__unit__l,axiom,
! [A2: a,B3: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A2 @ B3 ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_155_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_156_ideal__eq__carrier__iff,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( partia707051561876973205xt_a_b @ r )
= ( cgenid547466209912283029xt_a_b @ r @ A2 ) )
= ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ideal_eq_carrier_iff
thf(fact_157_ring__irreducibleE_I1_J,axiom,
! [R: a] :
( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R )
=> ( R
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_158_exists__gen,axiom,
! [I3: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( I3
= ( cgenid547466209912283029xt_a_b @ r @ X2 ) ) ) ) ).
% exists_gen
thf(fact_159_cgenideal__ideal,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ A2 ) @ r ) ) ).
% cgenideal_ideal
thf(fact_160_ring__primeE_I1_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( P2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_161_cgenideal__minimal,axiom,
! [J2: set_a,A2: a] :
( ( ideal_a_b @ J2 @ r )
=> ( ( member_a @ A2 @ J2 )
=> ( ord_less_eq_set_a @ ( cgenid547466209912283029xt_a_b @ r @ A2 ) @ J2 ) ) ) ).
% cgenideal_minimal
thf(fact_162_genideal__minimal,axiom,
! [I3: set_a,S: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ( ( ord_less_eq_set_a @ S @ I3 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ S ) @ I3 ) ) ) ).
% genideal_minimal
thf(fact_163_genideal__ideal,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ideal_a_b @ ( genideal_a_b @ r @ S ) @ r ) ) ).
% genideal_ideal
thf(fact_164_Idl__subset__ideal,axiom,
! [I3: set_a,H2: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H2 ) @ I3 )
= ( ord_less_eq_set_a @ H2 @ I3 ) ) ) ) ).
% Idl_subset_ideal
thf(fact_165_ideal__is__subalgebra,axiom,
! [K: set_a,I3: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ideal_a_b @ I3 @ r )
=> ( embedd9027525575939734154ra_a_b @ K @ I3 @ r ) ) ) ).
% ideal_is_subalgebra
thf(fact_166_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A6: a] :
( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A6
!= ( zero_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A6 @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_167_finite__Int,axiom,
! [F2: set_a,G2: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G2 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_168_finite__Int,axiom,
! [F2: set_list_a,G2: set_list_a] :
( ( ( finite_finite_list_a @ F2 )
| ( finite_finite_list_a @ G2 ) )
=> ( finite_finite_list_a @ ( inf_inf_set_list_a @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_169_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_170_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_171_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_172_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_173_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_174_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_175_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_176_add_Or__cancel__one_H,axiom,
! [X: a,A2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ A2 @ X ) )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_177_add_Or__cancel__one,axiom,
! [X: a,A2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A2 @ X )
= X )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_178_add_Ol__cancel__one_H,axiom,
! [X: a,A2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ X @ A2 ) )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_179_add_Ol__cancel__one,axiom,
! [X: a,A2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ A2 )
= X )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_180_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_181_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_182_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_183_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_184_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_185_r__right__minus__eq,axiom,
! [A2: a,B3: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A2 @ B3 )
= ( zero_a_b @ r ) )
= ( A2 = B3 ) ) ) ) ).
% r_right_minus_eq
thf(fact_186_finprod__multf,axiom,
! [F: a > a,A: set_a,G: a > a] :
( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r
@ ^ [X3: a] : ( mult_a_ring_ext_a_b @ r @ ( F @ X3 ) @ ( G @ X3 ) )
@ A )
= ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ F @ A ) @ ( finpro205304725090349623_a_b_a @ r @ G @ A ) ) ) ) ) ).
% finprod_multf
thf(fact_187_irreducible__imp__maximalideal,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P2 )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P2 ) @ r ) ) ) ).
% irreducible_imp_maximalideal
thf(fact_188_ring__iso__imp__img__field,axiom,
! [H: a > a,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ r @ S ) )
=> ( field_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_field
thf(fact_189_ring__iso__imp__img__field,axiom,
! [H: a > list_a,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ r @ S ) )
=> ( field_6388047844668329575t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_field
thf(fact_190_ring__iso__imp__img__field,axiom,
! [H: a > set_a,S: partia6043505979758434576t_unit] :
( ( member_a_set_a @ H @ ( ring_i7849008455817099456t_unit @ r @ S ) )
=> ( field_6045675692312731021t_unit
@ ( zero_u8960205505688764764t_unit
@ ^ [Uu: set_a] : ( H @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_field
thf(fact_191_ring__iso__imp__img__field,axiom,
! [H: a > set_list_a,S: partia7496981018696276118t_unit] :
( ( member_a_set_list_a @ H @ ( ring_i5325512697602418746t_unit @ r @ S ) )
=> ( field_26233345952514695t_unit
@ ( zero_u3422801854947504854t_unit
@ ^ [Uu: set_list_a] : ( H @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_field
thf(fact_192_ring__primeI,axiom,
! [P2: a] :
( ( P2
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P2 )
=> ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% ring_primeI
thf(fact_193_ring__primeE_I3_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( prime_a_ring_ext_a_b @ r @ P2 ) ) ) ).
% ring_primeE(3)
thf(fact_194_noetherian__ring_Ofinetely__gen,axiom,
! [R2: partia2956882679547061052t_unit,I3: set_list_list_a] :
( ( ring_n1719824158142654231t_unit @ R2 )
=> ( ( ideal_7391923968229085103t_unit @ I3 @ R2 )
=> ? [A5: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ A5 @ ( partia2464479390973590831t_unit @ R2 ) )
& ( finite1660835950917165235list_a @ A5 )
& ( I3
= ( genide2671672708880404049t_unit @ R2 @ A5 ) ) ) ) ) ).
% noetherian_ring.finetely_gen
thf(fact_195_noetherian__ring_Ofinetely__gen,axiom,
! [R2: partia2175431115845679010xt_a_b,I3: set_a] :
( ( ring_n3639167112692572309ng_a_b @ R2 )
=> ( ( ideal_a_b @ I3 @ R2 )
=> ? [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ ( partia707051561876973205xt_a_b @ R2 ) )
& ( finite_finite_a @ A5 )
& ( I3
= ( genideal_a_b @ R2 @ A5 ) ) ) ) ) ).
% noetherian_ring.finetely_gen
thf(fact_196_noetherian__ring_Ofinetely__gen,axiom,
! [R2: partia2670972154091845814t_unit,I3: set_list_a] :
( ( ring_n5188127996776581661t_unit @ R2 )
=> ( ( ideal_8896367198367571637t_unit @ I3 @ R2 )
=> ? [A5: set_list_a] :
( ( ord_le8861187494160871172list_a @ A5 @ ( partia5361259788508890537t_unit @ R2 ) )
& ( finite_finite_list_a @ A5 )
& ( I3
= ( genide3243992037924705879t_unit @ R2 @ A5 ) ) ) ) ) ).
% noetherian_ring.finetely_gen
thf(fact_197_finprod__Un__Int,axiom,
! [A: set_a,B2: set_a,G: a > a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ G @ ( sup_sup_set_a @ A @ B2 ) ) @ ( finpro205304725090349623_a_b_a @ r @ G @ ( inf_inf_set_a @ A @ B2 ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ G @ A ) @ ( finpro205304725090349623_a_b_a @ r @ G @ B2 ) ) ) ) ) ) ) ).
% finprod_Un_Int
thf(fact_198_finprod__Un__Int,axiom,
! [A: set_list_a,B2: set_list_a,G: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( finpro6052973074229812797list_a @ r @ G @ ( sup_sup_set_list_a @ A @ B2 ) ) @ ( finpro6052973074229812797list_a @ r @ G @ ( inf_inf_set_list_a @ A @ B2 ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro6052973074229812797list_a @ r @ G @ A ) @ ( finpro6052973074229812797list_a @ r @ G @ B2 ) ) ) ) ) ) ) ).
% finprod_Un_Int
thf(fact_199_monoid__cancelI,axiom,
( ! [A6: a,B4: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A6 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B4 ) )
=> ( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A6 = B4 ) ) ) ) )
=> ( ! [A6: a,B4: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A6 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B4 @ C2 ) )
=> ( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A6 = B4 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_200_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_201_Int__subset__iff,axiom,
! [C3: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A @ B2 ) )
= ( ( ord_less_eq_set_a @ C3 @ A )
& ( ord_less_eq_set_a @ C3 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_202_Int__subset__iff,axiom,
! [C3: set_list_a,A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C3 @ ( inf_inf_set_list_a @ A @ B2 ) )
= ( ( ord_le8861187494160871172list_a @ C3 @ A )
& ( ord_le8861187494160871172list_a @ C3 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_203_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_204_le__inf__iff,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) )
= ( ( ord_le8861187494160871172list_a @ X @ Y )
& ( ord_le8861187494160871172list_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_205_subset__antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_206_subset__antisym,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_207_subsetI,axiom,
! [A: set_list_list_a,B2: set_list_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( member_list_list_a @ X2 @ B2 ) )
=> ( ord_le8488217952732425610list_a @ A @ B2 ) ) ).
% subsetI
thf(fact_208_subsetI,axiom,
! [A: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% subsetI
thf(fact_209_subsetI,axiom,
! [A: set_list_a,B2: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_list_a @ X2 @ B2 ) )
=> ( ord_le8861187494160871172list_a @ A @ B2 ) ) ).
% subsetI
thf(fact_210_inf__right__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_211_inf__right__idem,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ Y )
= ( inf_inf_set_list_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_212_inf_Oright__idem,axiom,
! [A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ B3 )
= ( inf_inf_set_a @ A2 @ B3 ) ) ).
% inf.right_idem
thf(fact_213_inf_Oright__idem,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ B3 )
= ( inf_inf_set_list_a @ A2 @ B3 ) ) ).
% inf.right_idem
thf(fact_214_inf__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_215_inf__left__idem,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ X @ Y ) )
= ( inf_inf_set_list_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_216_inf_Oleft__idem,axiom,
! [A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( inf_inf_set_a @ A2 @ B3 ) ) ).
% inf.left_idem
thf(fact_217_inf_Oleft__idem,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ ( inf_inf_set_list_a @ A2 @ B3 ) )
= ( inf_inf_set_list_a @ A2 @ B3 ) ) ).
% inf.left_idem
thf(fact_218_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_219_inf__idem,axiom,
! [X: set_list_a] :
( ( inf_inf_set_list_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_220_inf_Oidem,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_221_inf_Oidem,axiom,
! [A2: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_222_Int__iff,axiom,
! [C: list_list_a,A: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( inf_in7423150557312423384list_a @ A @ B2 ) )
= ( ( member_list_list_a @ C @ A )
& ( member_list_list_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_223_Int__iff,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B2 ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_224_Int__iff,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B2 ) )
= ( ( member_list_a @ C @ A )
& ( member_list_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_225_IntI,axiom,
! [C: list_list_a,A: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ A )
=> ( ( member_list_list_a @ C @ B2 )
=> ( member_list_list_a @ C @ ( inf_in7423150557312423384list_a @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_226_IntI,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_227_IntI,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A )
=> ( ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_228_Diff__idemp,axiom,
! [A: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B2 ) @ B2 )
= ( minus_minus_set_a @ A @ B2 ) ) ).
% Diff_idemp
thf(fact_229_Diff__idemp,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A @ B2 ) @ B2 )
= ( minus_646659088055828811list_a @ A @ B2 ) ) ).
% Diff_idemp
thf(fact_230_Diff__iff,axiom,
! [C: list_list_a,A: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( minus_5335179877275218001list_a @ A @ B2 ) )
= ( ( member_list_list_a @ C @ A )
& ~ ( member_list_list_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_231_Diff__iff,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_232_Diff__iff,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) )
= ( ( member_list_a @ C @ A )
& ~ ( member_list_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_233_DiffI,axiom,
! [C: list_list_a,A: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ A )
=> ( ~ ( member_list_list_a @ C @ B2 )
=> ( member_list_list_a @ C @ ( minus_5335179877275218001list_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_234_DiffI,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_235_DiffI,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A )
=> ( ~ ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_236_Un__iff,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B2 ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_237_Un__iff,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B2 ) )
= ( ( member_list_a @ C @ A )
| ( member_list_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_238_Un__iff,axiom,
! [C: list_list_a,A: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( sup_su1566401739067690942list_a @ A @ B2 ) )
= ( ( member_list_list_a @ C @ A )
| ( member_list_list_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_239_UnCI,axiom,
! [C: a,B2: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% UnCI
thf(fact_240_UnCI,axiom,
! [C: list_a,B2: set_list_a,A: set_list_a] :
( ( ~ ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ A ) )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B2 ) ) ) ).
% UnCI
thf(fact_241_UnCI,axiom,
! [C: list_list_a,B2: set_list_list_a,A: set_list_list_a] :
( ( ~ ( member_list_list_a @ C @ B2 )
=> ( member_list_list_a @ C @ A ) )
=> ( member_list_list_a @ C @ ( sup_su1566401739067690942list_a @ A @ B2 ) ) ) ).
% UnCI
thf(fact_242_inf_Obounded__iff,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C ) )
= ( ( ord_less_eq_set_a @ A2 @ B3 )
& ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_243_inf_Obounded__iff,axiom,
! [A2: set_list_a,B3: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( inf_inf_set_list_a @ B3 @ C ) )
= ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
& ( ord_le8861187494160871172list_a @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_244_sup_Obounded__iff,axiom,
! [B3: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C ) @ A2 )
= ( ( ord_less_eq_set_a @ B3 @ A2 )
& ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_245_sup_Obounded__iff,axiom,
! [B3: set_list_a,C: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ B3 @ C ) @ A2 )
= ( ( ord_le8861187494160871172list_a @ B3 @ A2 )
& ( ord_le8861187494160871172list_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_246_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_247_le__sup__iff,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ X @ Y ) @ Z )
= ( ( ord_le8861187494160871172list_a @ X @ Z )
& ( ord_le8861187494160871172list_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_248_Un__subset__iff,axiom,
! [A: set_a,B2: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ C3 )
= ( ( ord_less_eq_set_a @ A @ C3 )
& ( ord_less_eq_set_a @ B2 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_249_Un__subset__iff,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B2 ) @ C3 )
= ( ( ord_le8861187494160871172list_a @ A @ C3 )
& ( ord_le8861187494160871172list_a @ B2 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_250_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_251_sup__inf__absorb,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( sup_sup_set_list_a @ X @ ( inf_inf_set_list_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_252_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_253_inf__sup__absorb,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( sup_sup_set_list_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_254_finite__Un,axiom,
! [F2: set_a,G2: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G2 ) )
= ( ( finite_finite_a @ F2 )
& ( finite_finite_a @ G2 ) ) ) ).
% finite_Un
thf(fact_255_finite__Un,axiom,
! [F2: set_list_a,G2: set_list_a] :
( ( finite_finite_list_a @ ( sup_sup_set_list_a @ F2 @ G2 ) )
= ( ( finite_finite_list_a @ F2 )
& ( finite_finite_list_a @ G2 ) ) ) ).
% finite_Un
thf(fact_256_Un__Int__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_257_Un__Int__eq_I1_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_258_Un__Int__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_259_Un__Int__eq_I2_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_260_Un__Int__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_261_Un__Int__eq_I3_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( inf_inf_set_list_a @ S @ ( sup_sup_set_list_a @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_262_Un__Int__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( inf_inf_set_a @ T @ ( sup_sup_set_a @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_263_Un__Int__eq_I4_J,axiom,
! [T: set_list_a,S: set_list_a] :
( ( inf_inf_set_list_a @ T @ ( sup_sup_set_list_a @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_264_Int__Un__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_265_Int__Un__eq_I1_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_266_Int__Un__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_267_Int__Un__eq_I2_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_268_Int__Un__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_269_Int__Un__eq_I3_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( sup_sup_set_list_a @ S @ ( inf_inf_set_list_a @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_270_Int__Un__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( sup_sup_set_a @ T @ ( inf_inf_set_a @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_271_Int__Un__eq_I4_J,axiom,
! [T: set_list_a,S: set_list_a] :
( ( sup_sup_set_list_a @ T @ ( inf_inf_set_list_a @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_272_Un__Diff__cancel2,axiom,
! [B2: set_a,A: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B2 @ A ) @ A )
= ( sup_sup_set_a @ B2 @ A ) ) ).
% Un_Diff_cancel2
thf(fact_273_Un__Diff__cancel2,axiom,
! [B2: set_list_a,A: set_list_a] :
( ( sup_sup_set_list_a @ ( minus_646659088055828811list_a @ B2 @ A ) @ A )
= ( sup_sup_set_list_a @ B2 @ A ) ) ).
% Un_Diff_cancel2
thf(fact_274_Un__Diff__cancel,axiom,
! [A: set_a,B2: set_a] :
( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B2 @ A ) )
= ( sup_sup_set_a @ A @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_275_Un__Diff__cancel,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( sup_sup_set_list_a @ A @ ( minus_646659088055828811list_a @ B2 @ A ) )
= ( sup_sup_set_list_a @ A @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_276_Un__def,axiom,
( sup_su1566401739067690942list_a
= ( ^ [A7: set_list_list_a,B: set_list_list_a] :
( collect_list_list_a
@ ^ [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A7 )
| ( member_list_list_a @ X3 @ B ) ) ) ) ) ).
% Un_def
thf(fact_277_Un__def,axiom,
( sup_sup_set_list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
( collect_list_a
@ ^ [X3: list_a] :
( ( member_list_a @ X3 @ A7 )
| ( member_list_a @ X3 @ B ) ) ) ) ) ).
% Un_def
thf(fact_278_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A7: set_a,B: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A7 )
| ( member_a @ X3 @ B ) ) ) ) ) ).
% Un_def
thf(fact_279_Un__def,axiom,
( sup_sup_set_set_a
= ( ^ [A7: set_set_a,B: set_set_a] :
( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ A7 )
| ( member_set_a @ X3 @ B ) ) ) ) ) ).
% Un_def
thf(fact_280_Un__def,axiom,
( sup_su4537662296134749976list_a
= ( ^ [A7: set_set_list_a,B: set_set_list_a] :
( collect_set_list_a
@ ^ [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A7 )
| ( member_set_list_a @ X3 @ B ) ) ) ) ) ).
% Un_def
thf(fact_281_Collect__disj__eq,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( collect_list_a
@ ^ [X3: list_a] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_282_Collect__disj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_283_Collect__disj__eq,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( collect_set_a
@ ^ [X3: set_a] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_284_Collect__disj__eq,axiom,
! [P: set_list_a > $o,Q: set_list_a > $o] :
( ( collect_set_list_a
@ ^ [X3: set_list_a] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_su4537662296134749976list_a @ ( collect_set_list_a @ P ) @ ( collect_set_list_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_285_UnI2,axiom,
! [C: a,B2: set_a,A: set_a] :
( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% UnI2
thf(fact_286_UnI2,axiom,
! [C: list_a,B2: set_list_a,A: set_list_a] :
( ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B2 ) ) ) ).
% UnI2
thf(fact_287_UnI2,axiom,
! [C: list_list_a,B2: set_list_list_a,A: set_list_list_a] :
( ( member_list_list_a @ C @ B2 )
=> ( member_list_list_a @ C @ ( sup_su1566401739067690942list_a @ A @ B2 ) ) ) ).
% UnI2
thf(fact_288_UnI1,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% UnI1
thf(fact_289_UnI1,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B2 ) ) ) ).
% UnI1
thf(fact_290_UnI1,axiom,
! [C: list_list_a,A: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ A )
=> ( member_list_list_a @ C @ ( sup_su1566401739067690942list_a @ A @ B2 ) ) ) ).
% UnI1
thf(fact_291_UnE,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B2 ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_292_UnE,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B2 ) )
=> ( ~ ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_293_UnE,axiom,
! [C: list_list_a,A: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( sup_su1566401739067690942list_a @ A @ B2 ) )
=> ( ~ ( member_list_list_a @ C @ A )
=> ( member_list_list_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_294_sup_OcoboundedI2,axiom,
! [C: set_a,B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C @ B3 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_295_sup_OcoboundedI2,axiom,
! [C: set_list_a,B3: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C @ B3 )
=> ( ord_le8861187494160871172list_a @ C @ ( sup_sup_set_list_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_296_sup_OcoboundedI1,axiom,
! [C: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_297_sup_OcoboundedI1,axiom,
! [C: set_list_a,A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C @ A2 )
=> ( ord_le8861187494160871172list_a @ C @ ( sup_sup_set_list_a @ A2 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_298_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B5: set_a] :
( ( sup_sup_set_a @ A3 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_299_sup_Oabsorb__iff2,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B5: set_list_a] :
( ( sup_sup_set_list_a @ A3 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_300_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A3: set_a] :
( ( sup_sup_set_a @ A3 @ B5 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_301_sup_Oabsorb__iff1,axiom,
( ord_le8861187494160871172list_a
= ( ^ [B5: set_list_a,A3: set_list_a] :
( ( sup_sup_set_list_a @ A3 @ B5 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_302_sup_Ocobounded2,axiom,
! [B3: set_a,A2: set_a] : ( ord_less_eq_set_a @ B3 @ ( sup_sup_set_a @ A2 @ B3 ) ) ).
% sup.cobounded2
thf(fact_303_sup_Ocobounded2,axiom,
! [B3: set_list_a,A2: set_list_a] : ( ord_le8861187494160871172list_a @ B3 @ ( sup_sup_set_list_a @ A2 @ B3 ) ) ).
% sup.cobounded2
thf(fact_304_sup_Ocobounded1,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B3 ) ) ).
% sup.cobounded1
thf(fact_305_sup_Ocobounded1,axiom,
! [A2: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ ( sup_sup_set_list_a @ A2 @ B3 ) ) ).
% sup.cobounded1
thf(fact_306_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A3: set_a] :
( A3
= ( sup_sup_set_a @ A3 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_307_sup_Oorder__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [B5: set_list_a,A3: set_list_a] :
( A3
= ( sup_sup_set_list_a @ A3 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_308_sup_OboundedI,axiom,
! [B3: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_309_sup_OboundedI,axiom,
! [B3: set_list_a,A2: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ C @ A2 )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ B3 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_310_sup_OboundedE,axiom,
! [B3: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ~ ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_311_sup_OboundedE,axiom,
! [B3: set_list_a,C: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ B3 @ C ) @ A2 )
=> ~ ( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ~ ( ord_le8861187494160871172list_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_312_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_313_sup__absorb2,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( sup_sup_set_list_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_314_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_315_sup__absorb1,axiom,
! [Y: set_list_a,X: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X )
=> ( ( sup_sup_set_list_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_316_sup_Oabsorb2,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( sup_sup_set_a @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_317_sup_Oabsorb2,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( sup_sup_set_list_a @ A2 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_318_sup_Oabsorb1,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb1
thf(fact_319_sup_Oabsorb1,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( ( sup_sup_set_list_a @ A2 @ B3 )
= A2 ) ) ).
% sup.absorb1
thf(fact_320_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( ord_less_eq_set_a @ Z2 @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y3 @ Z2 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_321_sup__unique,axiom,
! [F: set_list_a > set_list_a > set_list_a,X: set_list_a,Y: set_list_a] :
( ! [X2: set_list_a,Y3: set_list_a] : ( ord_le8861187494160871172list_a @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_list_a,Y3: set_list_a] : ( ord_le8861187494160871172list_a @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_list_a,Y3: set_list_a,Z2: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y3 @ X2 )
=> ( ( ord_le8861187494160871172list_a @ Z2 @ X2 )
=> ( ord_le8861187494160871172list_a @ ( F @ Y3 @ Z2 ) @ X2 ) ) )
=> ( ( sup_sup_set_list_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_322_sup_OorderI,axiom,
! [A2: set_a,B3: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B3 ) )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% sup.orderI
thf(fact_323_sup_OorderI,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( A2
= ( sup_sup_set_list_a @ A2 @ B3 ) )
=> ( ord_le8861187494160871172list_a @ B3 @ A2 ) ) ).
% sup.orderI
thf(fact_324_sup_OorderE,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% sup.orderE
thf(fact_325_sup_OorderE,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( A2
= ( sup_sup_set_list_a @ A2 @ B3 ) ) ) ).
% sup.orderE
thf(fact_326_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_327_le__iff__sup,axiom,
( ord_le8861187494160871172list_a
= ( ^ [X3: set_list_a,Y4: set_list_a] :
( ( sup_sup_set_list_a @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_328_sup__least,axiom,
! [Y: set_a,X: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_329_sup__least,axiom,
! [Y: set_list_a,X: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X )
=> ( ( ord_le8861187494160871172list_a @ Z @ X )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_330_sup__mono,axiom,
! [A2: set_a,C: set_a,B3: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B3 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_331_sup__mono,axiom,
! [A2: set_list_a,C: set_list_a,B3: set_list_a,D: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ C )
=> ( ( ord_le8861187494160871172list_a @ B3 @ D )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A2 @ B3 ) @ ( sup_sup_set_list_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_332_sup_Omono,axiom,
! [C: set_a,A2: set_a,D: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ D @ B3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_333_sup_Omono,axiom,
! [C: set_list_a,A2: set_list_a,D: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C @ A2 )
=> ( ( ord_le8861187494160871172list_a @ D @ B3 )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ C @ D ) @ ( sup_sup_set_list_a @ A2 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_334_le__supI2,axiom,
! [X: set_a,B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X @ B3 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_335_le__supI2,axiom,
! [X: set_list_a,B3: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ B3 )
=> ( ord_le8861187494160871172list_a @ X @ ( sup_sup_set_list_a @ A2 @ B3 ) ) ) ).
% le_supI2
thf(fact_336_le__supI1,axiom,
! [X: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_337_le__supI1,axiom,
! [X: set_list_a,A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ A2 )
=> ( ord_le8861187494160871172list_a @ X @ ( sup_sup_set_list_a @ A2 @ B3 ) ) ) ).
% le_supI1
thf(fact_338_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_339_sup__ge2,axiom,
! [Y: set_list_a,X: set_list_a] : ( ord_le8861187494160871172list_a @ Y @ ( sup_sup_set_list_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_340_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_341_sup__ge1,axiom,
! [X: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ X @ ( sup_sup_set_list_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_342_le__supI,axiom,
! [A2: set_a,X: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ( ord_less_eq_set_a @ B3 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ X ) ) ) ).
% le_supI
thf(fact_343_le__supI,axiom,
! [A2: set_list_a,X: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ X )
=> ( ( ord_le8861187494160871172list_a @ B3 @ X )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A2 @ B3 ) @ X ) ) ) ).
% le_supI
thf(fact_344_le__supE,axiom,
! [A2: set_a,B3: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B3 ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X )
=> ~ ( ord_less_eq_set_a @ B3 @ X ) ) ) ).
% le_supE
thf(fact_345_le__supE,axiom,
! [A2: set_list_a,B3: set_list_a,X: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A2 @ B3 ) @ X )
=> ~ ( ( ord_le8861187494160871172list_a @ A2 @ X )
=> ~ ( ord_le8861187494160871172list_a @ B3 @ X ) ) ) ).
% le_supE
thf(fact_346_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_347_inf__sup__ord_I3_J,axiom,
! [X: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ X @ ( sup_sup_set_list_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_348_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_349_inf__sup__ord_I4_J,axiom,
! [Y: set_list_a,X: set_list_a] : ( ord_le8861187494160871172list_a @ Y @ ( sup_sup_set_list_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_350_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B: set_a] :
( ( sup_sup_set_a @ A7 @ B )
= B ) ) ) ).
% subset_Un_eq
thf(fact_351_subset__Un__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
( ( sup_sup_set_list_a @ A7 @ B )
= B ) ) ) ).
% subset_Un_eq
thf(fact_352_subset__UnE,axiom,
! [C3: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( sup_sup_set_a @ A @ B2 ) )
=> ~ ! [A8: set_a] :
( ( ord_less_eq_set_a @ A8 @ A )
=> ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ B2 )
=> ( C3
!= ( sup_sup_set_a @ A8 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_353_subset__UnE,axiom,
! [C3: set_list_a,A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C3 @ ( sup_sup_set_list_a @ A @ B2 ) )
=> ~ ! [A8: set_list_a] :
( ( ord_le8861187494160871172list_a @ A8 @ A )
=> ! [B6: set_list_a] :
( ( ord_le8861187494160871172list_a @ B6 @ B2 )
=> ( C3
!= ( sup_sup_set_list_a @ A8 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_354_Un__absorb2,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( sup_sup_set_a @ A @ B2 )
= A ) ) ).
% Un_absorb2
thf(fact_355_Un__absorb2,axiom,
! [B2: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B2 @ A )
=> ( ( sup_sup_set_list_a @ A @ B2 )
= A ) ) ).
% Un_absorb2
thf(fact_356_Un__absorb1,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( sup_sup_set_a @ A @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_357_Un__absorb1,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( sup_sup_set_list_a @ A @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_358_Un__upper2,axiom,
! [B2: set_a,A: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A @ B2 ) ) ).
% Un_upper2
thf(fact_359_Un__upper2,axiom,
! [B2: set_list_a,A: set_list_a] : ( ord_le8861187494160871172list_a @ B2 @ ( sup_sup_set_list_a @ A @ B2 ) ) ).
% Un_upper2
thf(fact_360_Un__upper1,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B2 ) ) ).
% Un_upper1
thf(fact_361_Un__upper1,axiom,
! [A: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ A @ ( sup_sup_set_list_a @ A @ B2 ) ) ).
% Un_upper1
thf(fact_362_Un__least,axiom,
! [A: set_a,C3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ C3 ) ) ) ).
% Un_least
thf(fact_363_Un__least,axiom,
! [A: set_list_a,C3: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C3 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C3 )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B2 ) @ C3 ) ) ) ).
% Un_least
thf(fact_364_Un__mono,axiom,
! [A: set_a,C3: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ ( sup_sup_set_a @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_365_Un__mono,axiom,
! [A: set_list_a,C3: set_list_a,B2: set_list_a,D2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C3 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ D2 )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B2 ) @ ( sup_sup_set_list_a @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_366_sup__inf__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_367_sup__inf__distrib2,axiom,
! [Y: set_list_a,Z: set_list_a,X: set_list_a] :
( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ Y @ Z ) @ X )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ Y @ X ) @ ( sup_sup_set_list_a @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_368_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_369_sup__inf__distrib1,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( sup_sup_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ X @ Y ) @ ( sup_sup_set_list_a @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_370_inf__sup__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_371_inf__sup__distrib2,axiom,
! [Y: set_list_a,Z: set_list_a,X: set_list_a] :
( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ Y @ Z ) @ X )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ Y @ X ) @ ( inf_inf_set_list_a @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_372_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_373_inf__sup__distrib1,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( sup_sup_set_list_a @ Y @ Z ) )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ ( inf_inf_set_list_a @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_374_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y3 @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y3 ) @ ( sup_sup_set_a @ X2 @ Z2 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_375_distrib__imp2,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ! [X2: set_list_a,Y3: set_list_a,Z2: set_list_a] :
( ( sup_sup_set_list_a @ X2 @ ( inf_inf_set_list_a @ Y3 @ Z2 ) )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ X2 @ Y3 ) @ ( sup_sup_set_list_a @ X2 @ Z2 ) ) )
=> ( ( inf_inf_set_list_a @ X @ ( sup_sup_set_list_a @ Y @ Z ) )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ ( inf_inf_set_list_a @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_376_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y3 @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y3 ) @ ( inf_inf_set_a @ X2 @ Z2 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_377_distrib__imp1,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ! [X2: set_list_a,Y3: set_list_a,Z2: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( sup_sup_set_list_a @ Y3 @ Z2 ) )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ X2 @ Y3 ) @ ( inf_inf_set_list_a @ X2 @ Z2 ) ) )
=> ( ( sup_sup_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ X @ Y ) @ ( sup_sup_set_list_a @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_378_Un__Int__distrib2,axiom,
! [B2: set_a,C3: set_a,A: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B2 @ C3 ) @ A )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B2 @ A ) @ ( sup_sup_set_a @ C3 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_379_Un__Int__distrib2,axiom,
! [B2: set_list_a,C3: set_list_a,A: set_list_a] :
( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ B2 @ C3 ) @ A )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ B2 @ A ) @ ( sup_sup_set_list_a @ C3 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_380_Int__Un__distrib2,axiom,
! [B2: set_a,C3: set_a,A: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B2 @ C3 ) @ A )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B2 @ A ) @ ( inf_inf_set_a @ C3 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_381_Int__Un__distrib2,axiom,
! [B2: set_list_a,C3: set_list_a,A: set_list_a] :
( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ B2 @ C3 ) @ A )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ B2 @ A ) @ ( inf_inf_set_list_a @ C3 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_382_Un__Int__distrib,axiom,
! [A: set_a,B2: set_a,C3: set_a] :
( ( sup_sup_set_a @ A @ ( inf_inf_set_a @ B2 @ C3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B2 ) @ ( sup_sup_set_a @ A @ C3 ) ) ) ).
% Un_Int_distrib
thf(fact_383_Un__Int__distrib,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( sup_sup_set_list_a @ A @ ( inf_inf_set_list_a @ B2 @ C3 ) )
= ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ A @ B2 ) @ ( sup_sup_set_list_a @ A @ C3 ) ) ) ).
% Un_Int_distrib
thf(fact_384_Int__Un__distrib,axiom,
! [A: set_a,B2: set_a,C3: set_a] :
( ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B2 @ C3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_inf_set_a @ A @ C3 ) ) ) ).
% Int_Un_distrib
thf(fact_385_Int__Un__distrib,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( sup_sup_set_list_a @ B2 @ C3 ) )
= ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ ( inf_inf_set_list_a @ A @ C3 ) ) ) ).
% Int_Un_distrib
thf(fact_386_Un__Int__crazy,axiom,
! [A: set_a,B2: set_a,C3: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_inf_set_a @ B2 @ C3 ) ) @ ( inf_inf_set_a @ C3 @ A ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A @ B2 ) @ ( sup_sup_set_a @ B2 @ C3 ) ) @ ( sup_sup_set_a @ C3 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_387_Un__Int__crazy,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( sup_sup_set_list_a @ ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ ( inf_inf_set_list_a @ B2 @ C3 ) ) @ ( inf_inf_set_list_a @ C3 @ A ) )
= ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ A @ B2 ) @ ( sup_sup_set_list_a @ B2 @ C3 ) ) @ ( sup_sup_set_list_a @ C3 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_388_Un__Diff,axiom,
! [A: set_a,B2: set_a,C3: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A @ B2 ) @ C3 )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A @ C3 ) @ ( minus_minus_set_a @ B2 @ C3 ) ) ) ).
% Un_Diff
thf(fact_389_Un__Diff,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( minus_646659088055828811list_a @ ( sup_sup_set_list_a @ A @ B2 ) @ C3 )
= ( sup_sup_set_list_a @ ( minus_646659088055828811list_a @ A @ C3 ) @ ( minus_646659088055828811list_a @ B2 @ C3 ) ) ) ).
% Un_Diff
thf(fact_390_inf__set__def,axiom,
( inf_in7423150557312423384list_a
= ( ^ [A7: set_list_list_a,B: set_list_list_a] :
( collect_list_list_a
@ ( inf_in2769718188528944901st_a_o
@ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ A7 )
@ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ B ) ) ) ) ) ).
% inf_set_def
thf(fact_391_inf__set__def,axiom,
( inf_inf_set_set_a
= ( ^ [A7: set_set_a,B: set_set_a] :
( collect_set_a
@ ( inf_inf_set_a_o
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A7 )
@ ^ [X3: set_a] : ( member_set_a @ X3 @ B ) ) ) ) ) ).
% inf_set_def
thf(fact_392_inf__set__def,axiom,
( inf_in4657809108759609906list_a
= ( ^ [A7: set_set_list_a,B: set_set_list_a] :
( collect_set_list_a
@ ( inf_inf_set_list_a_o
@ ^ [X3: set_list_a] : ( member_set_list_a @ X3 @ A7 )
@ ^ [X3: set_list_a] : ( member_set_list_a @ X3 @ B ) ) ) ) ) ).
% inf_set_def
thf(fact_393_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A7: set_a,B: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A7 )
@ ^ [X3: a] : ( member_a @ X3 @ B ) ) ) ) ) ).
% inf_set_def
thf(fact_394_inf__set__def,axiom,
( inf_inf_set_list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
( collect_list_a
@ ( inf_inf_list_a_o
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A7 )
@ ^ [X3: list_a] : ( member_list_a @ X3 @ B ) ) ) ) ) ).
% inf_set_def
thf(fact_395_infinite__Un,axiom,
! [S: set_a,T: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T ) ) ) ).
% infinite_Un
thf(fact_396_infinite__Un,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S @ T ) ) )
= ( ~ ( finite_finite_list_a @ S )
| ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_Un
thf(fact_397_Un__infinite,axiom,
! [S: set_a,T: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_398_Un__infinite,axiom,
! [S: set_list_a,T: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_399_finite__UnI,axiom,
! [F2: set_a,G2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( finite_finite_a @ G2 )
=> ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_400_finite__UnI,axiom,
! [F2: set_list_a,G2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ( finite_finite_list_a @ G2 )
=> ( finite_finite_list_a @ ( sup_sup_set_list_a @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_401_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_402_distrib__sup__le,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] : ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) ) @ ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ X @ Y ) @ ( sup_sup_set_list_a @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_403_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_404_distrib__inf__le,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] : ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ ( inf_inf_set_list_a @ X @ Z ) ) @ ( inf_inf_set_list_a @ X @ ( sup_sup_set_list_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_405_Un__Int__assoc__eq,axiom,
! [A: set_a,B2: set_a,C3: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C3 )
= ( inf_inf_set_a @ A @ ( sup_sup_set_a @ B2 @ C3 ) ) )
= ( ord_less_eq_set_a @ C3 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_406_Un__Int__assoc__eq,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ C3 )
= ( inf_inf_set_list_a @ A @ ( sup_sup_set_list_a @ B2 @ C3 ) ) )
= ( ord_le8861187494160871172list_a @ C3 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_407_Diff__subset__conv,axiom,
! [A: set_a,B2: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B2 ) @ C3 )
= ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ B2 @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_408_Diff__subset__conv,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A @ B2 ) @ C3 )
= ( ord_le8861187494160871172list_a @ A @ ( sup_sup_set_list_a @ B2 @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_409_Diff__partition,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B2 @ A ) )
= B2 ) ) ).
% Diff_partition
thf(fact_410_Diff__partition,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( sup_sup_set_list_a @ A @ ( minus_646659088055828811list_a @ B2 @ A ) )
= B2 ) ) ).
% Diff_partition
thf(fact_411_Un__Diff__Int,axiom,
! [A: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ A @ B2 ) @ ( inf_inf_set_a @ A @ B2 ) )
= A ) ).
% Un_Diff_Int
thf(fact_412_Un__Diff__Int,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( sup_sup_set_list_a @ ( minus_646659088055828811list_a @ A @ B2 ) @ ( inf_inf_set_list_a @ A @ B2 ) )
= A ) ).
% Un_Diff_Int
thf(fact_413_Int__Diff__Un,axiom,
! [A: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( minus_minus_set_a @ A @ B2 ) )
= A ) ).
% Int_Diff_Un
thf(fact_414_Int__Diff__Un,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ ( minus_646659088055828811list_a @ A @ B2 ) )
= A ) ).
% Int_Diff_Un
thf(fact_415_Diff__Int,axiom,
! [A: set_a,B2: set_a,C3: set_a] :
( ( minus_minus_set_a @ A @ ( inf_inf_set_a @ B2 @ C3 ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A @ B2 ) @ ( minus_minus_set_a @ A @ C3 ) ) ) ).
% Diff_Int
thf(fact_416_Diff__Int,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( minus_646659088055828811list_a @ A @ ( inf_inf_set_list_a @ B2 @ C3 ) )
= ( sup_sup_set_list_a @ ( minus_646659088055828811list_a @ A @ B2 ) @ ( minus_646659088055828811list_a @ A @ C3 ) ) ) ).
% Diff_Int
thf(fact_417_Diff__Un,axiom,
! [A: set_a,B2: set_a,C3: set_a] :
( ( minus_minus_set_a @ A @ ( sup_sup_set_a @ B2 @ C3 ) )
= ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B2 ) @ ( minus_minus_set_a @ A @ C3 ) ) ) ).
% Diff_Un
thf(fact_418_Diff__Un,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( minus_646659088055828811list_a @ A @ ( sup_sup_set_list_a @ B2 @ C3 ) )
= ( inf_inf_set_list_a @ ( minus_646659088055828811list_a @ A @ B2 ) @ ( minus_646659088055828811list_a @ A @ C3 ) ) ) ).
% Diff_Un
thf(fact_419_ring__prime__def,axiom,
( ring_ring_prime_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,A3: a] :
( ( A3
!= ( zero_a_b @ R3 ) )
& ( prime_a_ring_ext_a_b @ R3 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_420_ring__prime__def,axiom,
( ring_r6430282645014804837t_unit
= ( ^ [R3: partia2670972154091845814t_unit,A3: list_a] :
( ( A3
!= ( zero_l4142658623432671053t_unit @ R3 ) )
& ( prime_2011924034616061926t_unit @ R3 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_421_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_422_Collect__mono__iff,axiom,
! [P: set_list_a > $o,Q: set_list_a > $o] :
( ( ord_le8877086941679407844list_a @ ( collect_set_list_a @ P ) @ ( collect_set_list_a @ Q ) )
= ( ! [X3: set_list_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_423_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_424_Collect__mono__iff,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) )
= ( ! [X3: list_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_425_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A7: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A7 @ B )
& ( ord_less_eq_set_a @ B @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_426_set__eq__subset,axiom,
( ( ^ [Y5: set_list_a,Z3: set_list_a] : ( Y5 = Z3 ) )
= ( ^ [A7: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A7 @ B )
& ( ord_le8861187494160871172list_a @ B @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_427_subset__trans,axiom,
! [A: set_a,B2: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ord_less_eq_set_a @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_428_subset__trans,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C3 )
=> ( ord_le8861187494160871172list_a @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_429_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_430_Collect__mono,axiom,
! [P: set_list_a > $o,Q: set_list_a > $o] :
( ! [X2: set_list_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le8877086941679407844list_a @ ( collect_set_list_a @ P ) @ ( collect_set_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_431_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_432_Collect__mono,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X2: list_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_433_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_434_subset__refl,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).
% subset_refl
thf(fact_435_subset__iff,axiom,
( ord_le8488217952732425610list_a
= ( ^ [A7: set_list_list_a,B: set_list_list_a] :
! [T2: list_list_a] :
( ( member_list_list_a @ T2 @ A7 )
=> ( member_list_list_a @ T2 @ B ) ) ) ) ).
% subset_iff
thf(fact_436_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A7 )
=> ( member_a @ T2 @ B ) ) ) ) ).
% subset_iff
thf(fact_437_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
! [T2: list_a] :
( ( member_list_a @ T2 @ A7 )
=> ( member_list_a @ T2 @ B ) ) ) ) ).
% subset_iff
thf(fact_438_equalityD2,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ).
% equalityD2
thf(fact_439_equalityD2,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( A = B2 )
=> ( ord_le8861187494160871172list_a @ B2 @ A ) ) ).
% equalityD2
thf(fact_440_equalityD1,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% equalityD1
thf(fact_441_equalityD1,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( A = B2 )
=> ( ord_le8861187494160871172list_a @ A @ B2 ) ) ).
% equalityD1
thf(fact_442_subset__eq,axiom,
( ord_le8488217952732425610list_a
= ( ^ [A7: set_list_list_a,B: set_list_list_a] :
! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A7 )
=> ( member_list_list_a @ X3 @ B ) ) ) ) ).
% subset_eq
thf(fact_443_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A7 )
=> ( member_a @ X3 @ B ) ) ) ) ).
% subset_eq
thf(fact_444_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ A7 )
=> ( member_list_a @ X3 @ B ) ) ) ) ).
% subset_eq
thf(fact_445_equalityE,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ~ ( ( ord_less_eq_set_a @ A @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A ) ) ) ).
% equalityE
thf(fact_446_equalityE,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( A = B2 )
=> ~ ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ~ ( ord_le8861187494160871172list_a @ B2 @ A ) ) ) ).
% equalityE
thf(fact_447_subsetD,axiom,
! [A: set_list_list_a,B2: set_list_list_a,C: list_list_a] :
( ( ord_le8488217952732425610list_a @ A @ B2 )
=> ( ( member_list_list_a @ C @ A )
=> ( member_list_list_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_448_subsetD,axiom,
! [A: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_449_subsetD,axiom,
! [A: set_list_a,B2: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_450_in__mono,axiom,
! [A: set_list_list_a,B2: set_list_list_a,X: list_list_a] :
( ( ord_le8488217952732425610list_a @ A @ B2 )
=> ( ( member_list_list_a @ X @ A )
=> ( member_list_list_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_451_in__mono,axiom,
! [A: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_452_in__mono,axiom,
! [A: set_list_a,B2: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( member_list_a @ X @ A )
=> ( member_list_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_453_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_454_inf__left__commute,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) )
= ( inf_inf_set_list_a @ Y @ ( inf_inf_set_list_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_455_inf_Oleft__commute,axiom,
! [B3: set_a,A2: set_a,C: set_a] :
( ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A2 @ C ) )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C ) ) ) ).
% inf.left_commute
thf(fact_456_inf_Oleft__commute,axiom,
! [B3: set_list_a,A2: set_list_a,C: set_list_a] :
( ( inf_inf_set_list_a @ B3 @ ( inf_inf_set_list_a @ A2 @ C ) )
= ( inf_inf_set_list_a @ A2 @ ( inf_inf_set_list_a @ B3 @ C ) ) ) ).
% inf.left_commute
thf(fact_457_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X3 ) ) ) ).
% inf_commute
thf(fact_458_inf__commute,axiom,
( inf_inf_set_list_a
= ( ^ [X3: set_list_a,Y4: set_list_a] : ( inf_inf_set_list_a @ Y4 @ X3 ) ) ) ).
% inf_commute
thf(fact_459_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A3 ) ) ) ).
% inf.commute
thf(fact_460_inf_Ocommute,axiom,
( inf_inf_set_list_a
= ( ^ [A3: set_list_a,B5: set_list_a] : ( inf_inf_set_list_a @ B5 @ A3 ) ) ) ).
% inf.commute
thf(fact_461_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_462_inf__assoc,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ Z )
= ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_463_inf_Oassoc,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C ) ) ) ).
% inf.assoc
thf(fact_464_inf_Oassoc,axiom,
! [A2: set_list_a,B3: set_list_a,C: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ C )
= ( inf_inf_set_list_a @ A2 @ ( inf_inf_set_list_a @ B3 @ C ) ) ) ).
% inf.assoc
thf(fact_465_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_466_inf__sup__aci_I1_J,axiom,
( inf_inf_set_list_a
= ( ^ [X3: set_list_a,Y4: set_list_a] : ( inf_inf_set_list_a @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_467_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_468_inf__sup__aci_I2_J,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X @ Y ) @ Z )
= ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_469_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_470_inf__sup__aci_I3_J,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) )
= ( inf_inf_set_list_a @ Y @ ( inf_inf_set_list_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_471_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_472_inf__sup__aci_I4_J,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ X @ ( inf_inf_set_list_a @ X @ Y ) )
= ( inf_inf_set_list_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_473_Int__left__commute,axiom,
! [A: set_a,B2: set_a,C3: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B2 @ C3 ) )
= ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A @ C3 ) ) ) ).
% Int_left_commute
thf(fact_474_Int__left__commute,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B2 @ C3 ) )
= ( inf_inf_set_list_a @ B2 @ ( inf_inf_set_list_a @ A @ C3 ) ) ) ).
% Int_left_commute
thf(fact_475_Int__left__absorb,axiom,
! [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B2 ) )
= ( inf_inf_set_a @ A @ B2 ) ) ).
% Int_left_absorb
thf(fact_476_Int__left__absorb,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ A @ B2 ) )
= ( inf_inf_set_list_a @ A @ B2 ) ) ).
% Int_left_absorb
thf(fact_477_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A7: set_a,B: set_a] : ( inf_inf_set_a @ B @ A7 ) ) ) ).
% Int_commute
thf(fact_478_Int__commute,axiom,
( inf_inf_set_list_a
= ( ^ [A7: set_list_a,B: set_list_a] : ( inf_inf_set_list_a @ B @ A7 ) ) ) ).
% Int_commute
thf(fact_479_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_480_Int__absorb,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_481_Int__assoc,axiom,
! [A: set_a,B2: set_a,C3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C3 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B2 @ C3 ) ) ) ).
% Int_assoc
thf(fact_482_Int__assoc,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ C3 )
= ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B2 @ C3 ) ) ) ).
% Int_assoc
thf(fact_483_IntD2,axiom,
! [C: list_list_a,A: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( inf_in7423150557312423384list_a @ A @ B2 ) )
=> ( member_list_list_a @ C @ B2 ) ) ).
% IntD2
thf(fact_484_IntD2,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B2 ) )
=> ( member_a @ C @ B2 ) ) ).
% IntD2
thf(fact_485_IntD2,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B2 ) )
=> ( member_list_a @ C @ B2 ) ) ).
% IntD2
thf(fact_486_IntD1,axiom,
! [C: list_list_a,A: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( inf_in7423150557312423384list_a @ A @ B2 ) )
=> ( member_list_list_a @ C @ A ) ) ).
% IntD1
thf(fact_487_IntD1,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B2 ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_488_IntD1,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B2 ) )
=> ( member_list_a @ C @ A ) ) ).
% IntD1
thf(fact_489_IntE,axiom,
! [C: list_list_a,A: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( inf_in7423150557312423384list_a @ A @ B2 ) )
=> ~ ( ( member_list_list_a @ C @ A )
=> ~ ( member_list_list_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_490_IntE,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B2 ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_491_IntE,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B2 ) )
=> ~ ( ( member_list_a @ C @ A )
=> ~ ( member_list_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_492_DiffD2,axiom,
! [C: list_list_a,A: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( minus_5335179877275218001list_a @ A @ B2 ) )
=> ~ ( member_list_list_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_493_DiffD2,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_494_DiffD2,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) )
=> ~ ( member_list_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_495_DiffD1,axiom,
! [C: list_list_a,A: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( minus_5335179877275218001list_a @ A @ B2 ) )
=> ( member_list_list_a @ C @ A ) ) ).
% DiffD1
thf(fact_496_DiffD1,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_497_DiffD1,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) )
=> ( member_list_a @ C @ A ) ) ).
% DiffD1
thf(fact_498_DiffE,axiom,
! [C: list_list_a,A: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( minus_5335179877275218001list_a @ A @ B2 ) )
=> ~ ( ( member_list_list_a @ C @ A )
=> ( member_list_list_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_499_DiffE,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_500_DiffE,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) )
=> ~ ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_501_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R2: partia2956882679547061052t_unit,P2: list_list_a] :
( ( ring_p715737262848045090t_unit @ R2 )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( ring_r360171070648044744t_unit @ R2 @ P2 )
=> ( maxima7552488817642790894t_unit @ ( cgenid24865672677839267t_unit @ R2 @ P2 ) @ R2 ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_502_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R2: partia2175431115845679010xt_a_b,P2: a] :
( ( ring_p8803135361686045600in_a_b @ R2 )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_r999134135267193926le_a_b @ R2 @ P2 )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ R2 @ P2 ) @ R2 ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_503_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R2: partia2670972154091845814t_unit,P2: list_a] :
( ( ring_p8098905331641078952t_unit @ R2 )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ring_r932985474545269838t_unit @ R2 @ P2 )
=> ( maxima6585700282301356660t_unit @ ( cgenid9131348535277946915t_unit @ R2 @ P2 ) @ R2 ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_504_Collect__subset,axiom,
! [A: set_list_list_a,P: list_list_a > $o] :
( ord_le8488217952732425610list_a
@ ( collect_list_list_a
@ ^ [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_505_Collect__subset,axiom,
! [A: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_506_Collect__subset,axiom,
! [A: set_set_list_a,P: set_list_a > $o] :
( ord_le8877086941679407844list_a
@ ( collect_set_list_a
@ ^ [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_507_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_508_Collect__subset,axiom,
! [A: set_list_a,P: list_a > $o] :
( ord_le8861187494160871172list_a
@ ( collect_list_a
@ ^ [X3: list_a] :
( ( member_list_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_509_less__eq__set__def,axiom,
( ord_le8488217952732425610list_a
= ( ^ [A7: set_list_list_a,B: set_list_list_a] :
( ord_le1801313680655002067st_a_o
@ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ A7 )
@ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ B ) ) ) ) ).
% less_eq_set_def
thf(fact_510_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B: set_a] :
( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A7 )
@ ^ [X3: a] : ( member_a @ X3 @ B ) ) ) ) ).
% less_eq_set_def
thf(fact_511_less__eq__set__def,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
( ord_less_eq_list_a_o
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A7 )
@ ^ [X3: list_a] : ( member_list_a @ X3 @ B ) ) ) ) ).
% less_eq_set_def
thf(fact_512_Int__def,axiom,
( inf_in7423150557312423384list_a
= ( ^ [A7: set_list_list_a,B: set_list_list_a] :
( collect_list_list_a
@ ^ [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A7 )
& ( member_list_list_a @ X3 @ B ) ) ) ) ) ).
% Int_def
thf(fact_513_Int__def,axiom,
( inf_inf_set_set_a
= ( ^ [A7: set_set_a,B: set_set_a] :
( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ A7 )
& ( member_set_a @ X3 @ B ) ) ) ) ) ).
% Int_def
thf(fact_514_Int__def,axiom,
( inf_in4657809108759609906list_a
= ( ^ [A7: set_set_list_a,B: set_set_list_a] :
( collect_set_list_a
@ ^ [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A7 )
& ( member_set_list_a @ X3 @ B ) ) ) ) ) ).
% Int_def
thf(fact_515_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A7: set_a,B: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A7 )
& ( member_a @ X3 @ B ) ) ) ) ) ).
% Int_def
thf(fact_516_Int__def,axiom,
( inf_inf_set_list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
( collect_list_a
@ ^ [X3: list_a] :
( ( member_list_a @ X3 @ A7 )
& ( member_list_a @ X3 @ B ) ) ) ) ) ).
% Int_def
thf(fact_517_Int__Collect,axiom,
! [X: list_list_a,A: set_list_list_a,P: list_list_a > $o] :
( ( member_list_list_a @ X @ ( inf_in7423150557312423384list_a @ A @ ( collect_list_list_a @ P ) ) )
= ( ( member_list_list_a @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_518_Int__Collect,axiom,
! [X: set_a,A: set_set_a,P: set_a > $o] :
( ( member_set_a @ X @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) )
= ( ( member_set_a @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_519_Int__Collect,axiom,
! [X: set_list_a,A: set_set_list_a,P: set_list_a > $o] :
( ( member_set_list_a @ X @ ( inf_in4657809108759609906list_a @ A @ ( collect_set_list_a @ P ) ) )
= ( ( member_set_list_a @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_520_Int__Collect,axiom,
! [X: a,A: set_a,P: a > $o] :
( ( member_a @ X @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) )
= ( ( member_a @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_521_Int__Collect,axiom,
! [X: list_a,A: set_list_a,P: list_a > $o] :
( ( member_list_a @ X @ ( inf_inf_set_list_a @ A @ ( collect_list_a @ P ) ) )
= ( ( member_list_a @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_522_Collect__conj__eq,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( collect_set_a
@ ^ [X3: set_a] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_523_Collect__conj__eq,axiom,
! [P: set_list_a > $o,Q: set_list_a > $o] :
( ( collect_set_list_a
@ ^ [X3: set_list_a] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_in4657809108759609906list_a @ ( collect_set_list_a @ P ) @ ( collect_set_list_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_524_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_525_Collect__conj__eq,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( collect_list_a
@ ^ [X3: list_a] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_526_set__diff__eq,axiom,
( minus_5335179877275218001list_a
= ( ^ [A7: set_list_list_a,B: set_list_list_a] :
( collect_list_list_a
@ ^ [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A7 )
& ~ ( member_list_list_a @ X3 @ B ) ) ) ) ) ).
% set_diff_eq
thf(fact_527_set__diff__eq,axiom,
( minus_5736297505244876581_set_a
= ( ^ [A7: set_set_a,B: set_set_a] :
( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ A7 )
& ~ ( member_set_a @ X3 @ B ) ) ) ) ) ).
% set_diff_eq
thf(fact_528_set__diff__eq,axiom,
( minus_4782336368215558443list_a
= ( ^ [A7: set_set_list_a,B: set_set_list_a] :
( collect_set_list_a
@ ^ [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A7 )
& ~ ( member_set_list_a @ X3 @ B ) ) ) ) ) ).
% set_diff_eq
thf(fact_529_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A7: set_a,B: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A7 )
& ~ ( member_a @ X3 @ B ) ) ) ) ) ).
% set_diff_eq
thf(fact_530_set__diff__eq,axiom,
( minus_646659088055828811list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
( collect_list_a
@ ^ [X3: list_a] :
( ( member_list_a @ X3 @ A7 )
& ~ ( member_list_a @ X3 @ B ) ) ) ) ) ).
% set_diff_eq
thf(fact_531_inf_OcoboundedI2,axiom,
! [B3: set_a,C: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_532_inf_OcoboundedI2,axiom,
! [B3: set_list_a,C: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ C )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_533_inf_OcoboundedI1,axiom,
! [A2: set_a,C: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_534_inf_OcoboundedI1,axiom,
! [A2: set_list_a,C: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ C )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_535_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A3: set_a] :
( ( inf_inf_set_a @ A3 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_536_inf_Oabsorb__iff2,axiom,
( ord_le8861187494160871172list_a
= ( ^ [B5: set_list_a,A3: set_list_a] :
( ( inf_inf_set_list_a @ A3 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_537_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B5: set_a] :
( ( inf_inf_set_a @ A3 @ B5 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_538_inf_Oabsorb__iff1,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B5: set_list_a] :
( ( inf_inf_set_list_a @ A3 @ B5 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_539_inf_Ocobounded2,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_540_inf_Ocobounded2,axiom,
! [A2: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_541_inf_Ocobounded1,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ A2 ) ).
% inf.cobounded1
thf(fact_542_inf_Ocobounded1,axiom,
! [A2: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ A2 ) ).
% inf.cobounded1
thf(fact_543_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B5: set_a] :
( A3
= ( inf_inf_set_a @ A3 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_544_inf_Oorder__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B5: set_list_a] :
( A3
= ( inf_inf_set_list_a @ A3 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_545_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_546_inf__greatest,axiom,
! [X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( ord_le8861187494160871172list_a @ X @ Z )
=> ( ord_le8861187494160871172list_a @ X @ ( inf_inf_set_list_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_547_inf_OboundedI,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ C )
=> ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_548_inf_OboundedI,axiom,
! [A2: set_list_a,B3: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ C )
=> ( ord_le8861187494160871172list_a @ A2 @ ( inf_inf_set_list_a @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_549_inf_OboundedE,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ~ ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_550_inf_OboundedE,axiom,
! [A2: set_list_a,B3: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( inf_inf_set_list_a @ B3 @ C ) )
=> ~ ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ~ ( ord_le8861187494160871172list_a @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_551_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_552_inf__absorb2,axiom,
! [Y: set_list_a,X: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X )
=> ( ( inf_inf_set_list_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_553_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_554_inf__absorb1,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
=> ( ( inf_inf_set_list_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_555_inf_Oabsorb2,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_556_inf_Oabsorb2,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_557_inf_Oabsorb1,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= A2 ) ) ).
% inf.absorb1
thf(fact_558_inf_Oabsorb1,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( inf_inf_set_list_a @ A2 @ B3 )
= A2 ) ) ).
% inf.absorb1
thf(fact_559_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_560_le__iff__inf,axiom,
( ord_le8861187494160871172list_a
= ( ^ [X3: set_list_a,Y4: set_list_a] :
( ( inf_inf_set_list_a @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_561_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ X2 @ Z2 )
=> ( ord_less_eq_set_a @ X2 @ ( F @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_562_inf__unique,axiom,
! [F: set_list_a > set_list_a > set_list_a,X: set_list_a,Y: set_list_a] :
( ! [X2: set_list_a,Y3: set_list_a] : ( ord_le8861187494160871172list_a @ ( F @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: set_list_a,Y3: set_list_a] : ( ord_le8861187494160871172list_a @ ( F @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: set_list_a,Y3: set_list_a,Z2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y3 )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Z2 )
=> ( ord_le8861187494160871172list_a @ X2 @ ( F @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_set_list_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_563_inf_OorderI,axiom,
! [A2: set_a,B3: set_a] :
( ( A2
= ( inf_inf_set_a @ A2 @ B3 ) )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% inf.orderI
thf(fact_564_inf_OorderI,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( A2
= ( inf_inf_set_list_a @ A2 @ B3 ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ).
% inf.orderI
thf(fact_565_inf_OorderE,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( A2
= ( inf_inf_set_a @ A2 @ B3 ) ) ) ).
% inf.orderE
thf(fact_566_inf_OorderE,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( A2
= ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ).
% inf.orderE
thf(fact_567_le__infI2,axiom,
! [B3: set_a,X: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ X ) ) ).
% le_infI2
thf(fact_568_le__infI2,axiom,
! [B3: set_list_a,X: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ X )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ X ) ) ).
% le_infI2
thf(fact_569_le__infI1,axiom,
! [A2: set_a,X: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ X ) ) ).
% le_infI1
thf(fact_570_le__infI1,axiom,
! [A2: set_list_a,X: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ X )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ X ) ) ).
% le_infI1
thf(fact_571_inf__mono,axiom,
! [A2: set_a,C: set_a,B3: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B3 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_572_inf__mono,axiom,
! [A2: set_list_a,C: set_list_a,B3: set_list_a,D: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ C )
=> ( ( ord_le8861187494160871172list_a @ B3 @ D )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ ( inf_inf_set_list_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_573_le__infI,axiom,
! [X: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X @ B3 )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% le_infI
thf(fact_574_le__infI,axiom,
! [X: set_list_a,A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ A2 )
=> ( ( ord_le8861187494160871172list_a @ X @ B3 )
=> ( ord_le8861187494160871172list_a @ X @ ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ) ).
% le_infI
thf(fact_575_le__infE,axiom,
! [X: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A2 )
=> ~ ( ord_less_eq_set_a @ X @ B3 ) ) ) ).
% le_infE
thf(fact_576_le__infE,axiom,
! [X: set_list_a,A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ ( inf_inf_set_list_a @ A2 @ B3 ) )
=> ~ ( ( ord_le8861187494160871172list_a @ X @ A2 )
=> ~ ( ord_le8861187494160871172list_a @ X @ B3 ) ) ) ).
% le_infE
thf(fact_577_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_578_inf__le2,axiom,
! [X: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_579_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_580_inf__le1,axiom,
! [X: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_581_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_582_inf__sup__ord_I1_J,axiom,
! [X: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_583_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_584_inf__sup__ord_I2_J,axiom,
! [X: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_585_Int__Collect__mono,axiom,
! [A: set_list_list_a,B2: set_list_list_a,P: list_list_a > $o,Q: list_list_a > $o] :
( ( ord_le8488217952732425610list_a @ A @ B2 )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le8488217952732425610list_a @ ( inf_in7423150557312423384list_a @ A @ ( collect_list_list_a @ P ) ) @ ( inf_in7423150557312423384list_a @ B2 @ ( collect_list_list_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_586_Int__Collect__mono,axiom,
! [A: set_set_a,B2: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B2 @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_587_Int__Collect__mono,axiom,
! [A: set_set_list_a,B2: set_set_list_a,P: set_list_a > $o,Q: set_list_a > $o] :
( ( ord_le8877086941679407844list_a @ A @ B2 )
=> ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ A @ ( collect_set_list_a @ P ) ) @ ( inf_in4657809108759609906list_a @ B2 @ ( collect_set_list_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_588_Int__Collect__mono,axiom,
! [A: set_a,B2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_589_Int__Collect__mono,axiom,
! [A: set_list_a,B2: set_list_a,P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ ( collect_list_a @ P ) ) @ ( inf_inf_set_list_a @ B2 @ ( collect_list_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_590_Int__greatest,axiom,
! [C3: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( ( ord_less_eq_set_a @ C3 @ B2 )
=> ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_591_Int__greatest,axiom,
! [C3: set_list_a,A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C3 @ A )
=> ( ( ord_le8861187494160871172list_a @ C3 @ B2 )
=> ( ord_le8861187494160871172list_a @ C3 @ ( inf_inf_set_list_a @ A @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_592_Int__absorb2,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( inf_inf_set_a @ A @ B2 )
= A ) ) ).
% Int_absorb2
thf(fact_593_Int__absorb2,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( inf_inf_set_list_a @ A @ B2 )
= A ) ) ).
% Int_absorb2
thf(fact_594_Int__absorb1,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( inf_inf_set_a @ A @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_595_Int__absorb1,axiom,
! [B2: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B2 @ A )
=> ( ( inf_inf_set_list_a @ A @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_596_Int__lower2,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_597_Int__lower2,axiom,
! [A: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_598_Int__lower1,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ A ) ).
% Int_lower1
thf(fact_599_Int__lower1,axiom,
! [A: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ A ) ).
% Int_lower1
thf(fact_600_Int__mono,axiom,
! [A: set_a,C3: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_inf_set_a @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_601_Int__mono,axiom,
! [A: set_list_a,C3: set_list_a,B2: set_list_a,D2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C3 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ D2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ ( inf_inf_set_list_a @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_602_double__diff,axiom,
! [A: set_a,B2: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C3 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_603_double__diff,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C3 )
=> ( ( minus_646659088055828811list_a @ B2 @ ( minus_646659088055828811list_a @ C3 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_604_Diff__subset,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B2 ) @ A ) ).
% Diff_subset
thf(fact_605_Diff__subset,axiom,
! [A: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A @ B2 ) @ A ) ).
% Diff_subset
thf(fact_606_Diff__mono,axiom,
! [A: set_a,C3: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B2 ) @ ( minus_minus_set_a @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_607_Diff__mono,axiom,
! [A: set_list_a,C3: set_list_a,D2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C3 )
=> ( ( ord_le8861187494160871172list_a @ D2 @ B2 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A @ B2 ) @ ( minus_646659088055828811list_a @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_608_Diff__Int__distrib2,axiom,
! [A: set_a,B2: set_a,C3: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B2 ) @ C3 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C3 ) @ ( inf_inf_set_a @ B2 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_609_Diff__Int__distrib2,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( inf_inf_set_list_a @ ( minus_646659088055828811list_a @ A @ B2 ) @ C3 )
= ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ A @ C3 ) @ ( inf_inf_set_list_a @ B2 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_610_Diff__Int__distrib,axiom,
! [C3: set_a,A: set_a,B2: set_a] :
( ( inf_inf_set_a @ C3 @ ( minus_minus_set_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C3 @ A ) @ ( inf_inf_set_a @ C3 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_611_Diff__Int__distrib,axiom,
! [C3: set_list_a,A: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ C3 @ ( minus_646659088055828811list_a @ A @ B2 ) )
= ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ C3 @ A ) @ ( inf_inf_set_list_a @ C3 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_612_Diff__Diff__Int,axiom,
! [A: set_a,B2: set_a] :
( ( minus_minus_set_a @ A @ ( minus_minus_set_a @ A @ B2 ) )
= ( inf_inf_set_a @ A @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_613_Diff__Diff__Int,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ A @ ( minus_646659088055828811list_a @ A @ B2 ) )
= ( inf_inf_set_list_a @ A @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_614_Diff__Int2,axiom,
! [A: set_a,C3: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C3 ) @ ( inf_inf_set_a @ B2 @ C3 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C3 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_615_Diff__Int2,axiom,
! [A: set_list_a,C3: set_list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ A @ C3 ) @ ( inf_inf_set_list_a @ B2 @ C3 ) )
= ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ A @ C3 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_616_Int__Diff,axiom,
! [A: set_a,B2: set_a,C3: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C3 )
= ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B2 @ C3 ) ) ) ).
% Int_Diff
thf(fact_617_Int__Diff,axiom,
! [A: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ C3 )
= ( inf_inf_set_list_a @ A @ ( minus_646659088055828811list_a @ B2 @ C3 ) ) ) ).
% Int_Diff
thf(fact_618_noetherian__domain_Oaxioms_I1_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( ring_n4705423059119889713t_unit @ R2 )
=> ( ring_n5188127996776581661t_unit @ R2 ) ) ).
% noetherian_domain.axioms(1)
thf(fact_619_noetherian__domain_Oaxioms_I1_J,axiom,
! [R2: partia2175431115845679010xt_a_b] :
( ( ring_n4045954140777738665in_a_b @ R2 )
=> ( ring_n3639167112692572309ng_a_b @ R2 ) ) ).
% noetherian_domain.axioms(1)
thf(fact_620_principal__domain_Oexists__gen,axiom,
! [R2: partia2956882679547061052t_unit,I3: set_list_list_a] :
( ( ring_p715737262848045090t_unit @ R2 )
=> ( ( ideal_7391923968229085103t_unit @ I3 @ R2 )
=> ? [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R2 ) )
& ( I3
= ( cgenid24865672677839267t_unit @ R2 @ X2 ) ) ) ) ) ).
% principal_domain.exists_gen
thf(fact_621_principal__domain_Oexists__gen,axiom,
! [R2: partia2175431115845679010xt_a_b,I3: set_a] :
( ( ring_p8803135361686045600in_a_b @ R2 )
=> ( ( ideal_a_b @ I3 @ R2 )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R2 ) )
& ( I3
= ( cgenid547466209912283029xt_a_b @ R2 @ X2 ) ) ) ) ) ).
% principal_domain.exists_gen
thf(fact_622_principal__domain_Oexists__gen,axiom,
! [R2: partia2670972154091845814t_unit,I3: set_list_a] :
( ( ring_p8098905331641078952t_unit @ R2 )
=> ( ( ideal_8896367198367571637t_unit @ I3 @ R2 )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R2 ) )
& ( I3
= ( cgenid9131348535277946915t_unit @ R2 @ X2 ) ) ) ) ) ).
% principal_domain.exists_gen
thf(fact_623_principal__domain_Oprimeness__condition,axiom,
! [R2: partia2956882679547061052t_unit,P2: list_list_a] :
( ( ring_p715737262848045090t_unit @ R2 )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( ring_r360171070648044744t_unit @ R2 @ P2 )
= ( ring_r5437400583859147359t_unit @ R2 @ P2 ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_624_principal__domain_Oprimeness__condition,axiom,
! [R2: partia2175431115845679010xt_a_b,P2: a] :
( ( ring_p8803135361686045600in_a_b @ R2 )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( ring_r999134135267193926le_a_b @ R2 @ P2 )
= ( ring_ring_prime_a_b @ R2 @ P2 ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_625_principal__domain_Oprimeness__condition,axiom,
! [R2: partia2670972154091845814t_unit,P2: list_a] :
( ( ring_p8098905331641078952t_unit @ R2 )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ring_r932985474545269838t_unit @ R2 @ P2 )
= ( ring_r6430282645014804837t_unit @ R2 @ P2 ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_626_finprod__Un__disjoint,axiom,
! [A: set_a,B2: set_a,G: a > a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( ( inf_inf_set_a @ A @ B2 )
= bot_bot_set_a )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ G @ ( sup_sup_set_a @ A @ B2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ G @ A ) @ ( finpro205304725090349623_a_b_a @ r @ G @ B2 ) ) ) ) ) ) ) ) ).
% finprod_Un_disjoint
thf(fact_627_finprod__Un__disjoint,axiom,
! [A: set_list_a,B2: set_list_a,G: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r @ G @ ( sup_sup_set_list_a @ A @ B2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro6052973074229812797list_a @ r @ G @ A ) @ ( finpro6052973074229812797list_a @ r @ G @ B2 ) ) ) ) ) ) ) ) ).
% finprod_Un_disjoint
thf(fact_628_finprod__Un__disjoint,axiom,
! [A: set_set_a,B2: set_set_a,G: set_a > a] :
( ( finite_finite_set_a @ A )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( ( inf_inf_set_set_a @ A @ B2 )
= bot_bot_set_set_a )
=> ( ( member_set_a_a @ G
@ ( pi_set_a_a @ A
@ ^ [Uu: set_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_set_a_a @ G
@ ( pi_set_a_a @ B2
@ ^ [Uu: set_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro934595834566309783_set_a @ r @ G @ ( sup_sup_set_set_a @ A @ B2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro934595834566309783_set_a @ r @ G @ A ) @ ( finpro934595834566309783_set_a @ r @ G @ B2 ) ) ) ) ) ) ) ) ).
% finprod_Un_disjoint
thf(fact_629_finprod__Un__disjoint,axiom,
! [A: set_set_list_a,B2: set_set_list_a,G: set_list_a > a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( finite5282473924520328461list_a @ B2 )
=> ( ( ( inf_in4657809108759609906list_a @ A @ B2 )
= bot_bo3186585308812441520list_a )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ B2
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3826550488720007709list_a @ r @ G @ ( sup_su4537662296134749976list_a @ A @ B2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro3826550488720007709list_a @ r @ G @ A ) @ ( finpro3826550488720007709list_a @ r @ G @ B2 ) ) ) ) ) ) ) ) ).
% finprod_Un_disjoint
thf(fact_630_s_Oring_Oideal__vimage,axiom,
! [I3: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ( ideal_8896367198367571637t_unit
@ ( collect_list_a
@ ^ [R4: list_a] :
( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( member_a @ ( eval_a_b @ r @ R4 @ s2 ) @ I3 ) ) )
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% s.ring.ideal_vimage
thf(fact_631_x_Oring_Oideal__vimage,axiom,
! [I3: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ( ideal_8896367198367571637t_unit
@ ( collect_list_a
@ ^ [R4: list_a] :
( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( member_a @ ( eval_a_b @ r @ R4 @ x ) @ I3 ) ) )
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.ring.ideal_vimage
thf(fact_632_lagrange__aux__poly,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( lagran9092808442999052491ux_a_b @ r @ S ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% lagrange_aux_poly
thf(fact_633_field_Oring__iso__imp__img__field,axiom,
! [R2: partia2175431115845679010xt_a_b,H: a > a,S: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R2 )
=> ( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R2 @ S ) )
=> ( field_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H @ ( zero_a_b @ R2 ) )
@ S ) ) ) ) ).
% field.ring_iso_imp_img_field
thf(fact_634_field_Oring__iso__imp__img__field,axiom,
! [R2: partia6043505979758434576t_unit,H: set_a > a,S: partia2175431115845679010xt_a_b] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( ( member_set_a_a @ H @ ( ring_i4751279688611836608it_a_b @ R2 @ S ) )
=> ( field_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H @ ( zero_s2174465271003423091t_unit @ R2 ) )
@ S ) ) ) ) ).
% field.ring_iso_imp_img_field
thf(fact_635_field_Oring__iso__imp__img__field,axiom,
! [R2: partia2175431115845679010xt_a_b,H: a > list_a,S: partia2670972154091845814t_unit] :
( ( field_a_b @ R2 )
=> ( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R2 @ S ) )
=> ( field_6388047844668329575t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H @ ( zero_a_b @ R2 ) )
@ S ) ) ) ) ).
% field.ring_iso_imp_img_field
thf(fact_636_field_Oring__iso__imp__img__field,axiom,
! [R2: partia2175431115845679010xt_a_b,H: a > set_a,S: partia6043505979758434576t_unit] :
( ( field_a_b @ R2 )
=> ( ( member_a_set_a @ H @ ( ring_i7849008455817099456t_unit @ R2 @ S ) )
=> ( field_6045675692312731021t_unit
@ ( zero_u8960205505688764764t_unit
@ ^ [Uu: set_a] : ( H @ ( zero_a_b @ R2 ) )
@ S ) ) ) ) ).
% field.ring_iso_imp_img_field
thf(fact_637_field_Oring__iso__imp__img__field,axiom,
! [R2: partia2670972154091845814t_unit,H: list_a > a,S: partia2175431115845679010xt_a_b] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R2 @ S ) )
=> ( field_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H @ ( zero_l4142658623432671053t_unit @ R2 ) )
@ S ) ) ) ) ).
% field.ring_iso_imp_img_field
thf(fact_638_field_Oring__iso__imp__img__field,axiom,
! [R2: partia6043505979758434576t_unit,H: set_a > list_a,S: partia2670972154091845814t_unit] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( ( member_set_a_list_a @ H @ ( ring_i2253880877871059912t_unit @ R2 @ S ) )
=> ( field_6388047844668329575t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H @ ( zero_s2174465271003423091t_unit @ R2 ) )
@ S ) ) ) ) ).
% field.ring_iso_imp_img_field
thf(fact_639_field_Oring__iso__imp__img__field,axiom,
! [R2: partia6043505979758434576t_unit,H: set_a > set_a,S: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R2 )
=> ( ( member_set_a_set_a @ H @ ( ring_i7821126338072249198t_unit @ R2 @ S ) )
=> ( field_6045675692312731021t_unit
@ ( zero_u8960205505688764764t_unit
@ ^ [Uu: set_a] : ( H @ ( zero_s2174465271003423091t_unit @ R2 ) )
@ S ) ) ) ) ).
% field.ring_iso_imp_img_field
thf(fact_640_field_Oring__iso__imp__img__field,axiom,
! [R2: partia7496981018696276118t_unit,H: set_list_a > a,S: partia2175431115845679010xt_a_b] :
( ( field_26233345952514695t_unit @ R2 )
=> ( ( member_set_list_a_a @ H @ ( ring_i8122894263081988538it_a_b @ R2 @ S ) )
=> ( field_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H @ ( zero_s2910681146719230829t_unit @ R2 ) )
@ S ) ) ) ) ).
% field.ring_iso_imp_img_field
thf(fact_641_field_Oring__iso__imp__img__field,axiom,
! [R2: partia2175431115845679010xt_a_b,H: a > set_list_a,S: partia7496981018696276118t_unit] :
( ( field_a_b @ R2 )
=> ( ( member_a_set_list_a @ H @ ( ring_i5325512697602418746t_unit @ R2 @ S ) )
=> ( field_26233345952514695t_unit
@ ( zero_u3422801854947504854t_unit
@ ^ [Uu: set_list_a] : ( H @ ( zero_a_b @ R2 ) )
@ S ) ) ) ) ).
% field.ring_iso_imp_img_field
thf(fact_642_field_Oring__iso__imp__img__field,axiom,
! [R2: partia2670972154091845814t_unit,H: list_a > list_a,S: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R2 )
=> ( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R2 @ S ) )
=> ( field_6388047844668329575t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H @ ( zero_l4142658623432671053t_unit @ R2 ) )
@ S ) ) ) ) ).
% field.ring_iso_imp_img_field
thf(fact_643_finsum__Un__Int,axiom,
! [A: set_a,B2: set_a,G: a > a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( add_a_b @ r @ ( finsum_a_b_a @ r @ G @ ( sup_sup_set_a @ A @ B2 ) ) @ ( finsum_a_b_a @ r @ G @ ( inf_inf_set_a @ A @ B2 ) ) )
= ( add_a_b @ r @ ( finsum_a_b_a @ r @ G @ A ) @ ( finsum_a_b_a @ r @ G @ B2 ) ) ) ) ) ) ) ).
% finsum_Un_Int
thf(fact_644_finsum__Un__Int,axiom,
! [A: set_list_a,B2: set_list_a,G: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( add_a_b @ r @ ( finsum_a_b_list_a @ r @ G @ ( sup_sup_set_list_a @ A @ B2 ) ) @ ( finsum_a_b_list_a @ r @ G @ ( inf_inf_set_list_a @ A @ B2 ) ) )
= ( add_a_b @ r @ ( finsum_a_b_list_a @ r @ G @ A ) @ ( finsum_a_b_list_a @ r @ G @ B2 ) ) ) ) ) ) ) ).
% finsum_Un_Int
thf(fact_645_add_Ofinprod__mono__neutral__cong,axiom,
! [B2: set_list_list_a,A: set_list_list_a,H: list_list_a > a,G: list_list_a > a] :
( ( finite1660835950917165235list_a @ B2 )
=> ( ( finite1660835950917165235list_a @ A )
=> ( ! [I2: list_list_a] :
( ( member_list_list_a @ I2 @ ( minus_5335179877275218001list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [I2: list_list_a] :
( ( member_list_list_a @ I2 @ ( minus_5335179877275218001list_a @ A @ B2 ) )
=> ( ( G @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( inf_in7423150557312423384list_a @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_list_a_a @ G
@ ( pi_list_list_a_a @ A
@ ^ [Uu: list_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a_a @ H
@ ( pi_list_list_a_a @ B2
@ ^ [Uu: list_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum1795837918752241516list_a @ r @ G @ A )
= ( finsum1795837918752241516list_a @ r @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong
thf(fact_646_add_Ofinprod__mono__neutral__cong,axiom,
! [B2: set_a,A: set_a,H: a > a,G: a > a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ A )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ A @ B2 ) )
=> ( ( G @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( inf_inf_set_a @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a @ H
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a @ r @ G @ A )
= ( finsum_a_b_a @ r @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong
thf(fact_647_add_Ofinprod__mono__neutral__cong,axiom,
! [B2: set_list_a,A: set_list_a,H: list_a > a,G: list_a > a] :
( ( finite_finite_list_a @ B2 )
=> ( ( finite_finite_list_a @ A )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ ( minus_646659088055828811list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ ( minus_646659088055828811list_a @ A @ B2 ) )
=> ( ( G @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( inf_inf_set_list_a @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a_a @ H
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_list_a @ r @ G @ A )
= ( finsum_a_b_list_a @ r @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong
thf(fact_648_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B2: set_list_list_a,A: set_list_list_a,G: list_list_a > a,H: list_list_a > a] :
( ( finite1660835950917165235list_a @ B2 )
=> ( ( ord_le8488217952732425610list_a @ A @ B2 )
=> ( ! [I2: list_list_a] :
( ( member_list_list_a @ I2 @ ( minus_5335179877275218001list_a @ B2 @ A ) )
=> ( ( G @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_list_a_a @ G
@ ( pi_list_list_a_a @ B2
@ ^ [Uu: list_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum1795837918752241516list_a @ r @ G @ B2 )
= ( finsum1795837918752241516list_a @ r @ H @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_649_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B2: set_a,A: set_a,G: a > a,H: a > a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ B2 @ A ) )
=> ( ( G @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a @ r @ G @ B2 )
= ( finsum_a_b_a @ r @ H @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_650_add_Ofinprod__mono__neutral__cong__right,axiom,
! [B2: set_list_a,A: set_list_a,G: list_a > a,H: list_a > a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ ( minus_646659088055828811list_a @ B2 @ A ) )
=> ( ( G @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_list_a @ r @ G @ B2 )
= ( finsum_a_b_list_a @ r @ H @ A ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_right
thf(fact_651_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_652_x_Oring__primeI,axiom,
! [P2: list_a] :
( ( P2
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ).
% x.ring_primeI
thf(fact_653_add_Ofinprod__one__eqI,axiom,
! [A: set_a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( F @ X2 )
= ( zero_a_b @ r ) ) )
=> ( ( finsum_a_b_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_654_add_Ofinprod__one__eqI,axiom,
! [A: set_list_a,F: list_a > a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( F @ X2 )
= ( zero_a_b @ r ) ) )
=> ( ( finsum_a_b_list_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_655_add_Ofinprod__one__eqI,axiom,
! [A: set_list_list_a,F: list_list_a > a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( ( F @ X2 )
= ( zero_a_b @ r ) ) )
=> ( ( finsum1795837918752241516list_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_656_x_Ocgenideal__self,axiom,
! [I4: list_a] :
( ( member_list_a @ I4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ I4 @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 ) ) ) ).
% x.cgenideal_self
thf(fact_657_x_Ocarrier__not__empty,axiom,
( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= bot_bot_set_list_a ) ).
% x.carrier_not_empty
thf(fact_658_x_Ogenideal__self,axiom,
! [S: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ S @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S ) ) ) ).
% x.genideal_self
thf(fact_659_x_Osubset__Idl__subset,axiom,
! [I3: set_list_a,H2: set_list_a] :
( ( ord_le8861187494160871172list_a @ I3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ H2 @ I3 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H2 ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I3 ) ) ) ) ).
% x.subset_Idl_subset
thf(fact_660_x_Ocgenideal__minimal,axiom,
! [J2: set_list_a,A2: list_a] :
( ( ideal_8896367198367571637t_unit @ J2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ A2 @ J2 )
=> ( ord_le8861187494160871172list_a @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 ) @ J2 ) ) ) ).
% x.cgenideal_minimal
thf(fact_661_x_Oi__intersect,axiom,
! [I3: set_list_a,J2: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ideal_8896367198367571637t_unit @ J2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ideal_8896367198367571637t_unit @ ( inf_inf_set_list_a @ I3 @ J2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.i_intersect
thf(fact_662_x_Ogenideal__minimal,axiom,
! [I3: set_list_a,S: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ S @ I3 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S ) @ I3 ) ) ) ).
% x.genideal_minimal
thf(fact_663_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_664_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_665_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X3: set_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_666_empty__Collect__eq,axiom,
! [P: set_list_a > $o] :
( ( bot_bo3186585308812441520list_a
= ( collect_set_list_a @ P ) )
= ( ! [X3: set_list_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_667_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_668_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_669_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_670_Collect__empty__eq,axiom,
! [P: set_list_a > $o] :
( ( ( collect_set_list_a @ P )
= bot_bo3186585308812441520list_a )
= ( ! [X3: set_list_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_671_all__not__in__conv,axiom,
! [A: set_list_list_a] :
( ( ! [X3: list_list_a] :
~ ( member_list_list_a @ X3 @ A ) )
= ( A = bot_bo1875519244922727510list_a ) ) ).
% all_not_in_conv
thf(fact_672_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_673_all__not__in__conv,axiom,
! [A: set_list_a] :
( ( ! [X3: list_a] :
~ ( member_list_a @ X3 @ A ) )
= ( A = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_674_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_675_all__not__in__conv,axiom,
! [A: set_set_list_a] :
( ( ! [X3: set_list_a] :
~ ( member_set_list_a @ X3 @ A ) )
= ( A = bot_bo3186585308812441520list_a ) ) ).
% all_not_in_conv
thf(fact_676_empty__iff,axiom,
! [C: list_list_a] :
~ ( member_list_list_a @ C @ bot_bo1875519244922727510list_a ) ).
% empty_iff
thf(fact_677_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_678_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_679_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_680_empty__iff,axiom,
! [C: set_list_a] :
~ ( member_set_list_a @ C @ bot_bo3186585308812441520list_a ) ).
% empty_iff
thf(fact_681_Pi__I,axiom,
! [A: set_a,F: a > a,B2: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_a @ F @ ( pi_a_a @ A @ B2 ) ) ) ).
% Pi_I
thf(fact_682_Pi__I,axiom,
! [A: set_a,F: a > list_a,B2: a > set_list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_list_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_list_a @ F @ ( pi_a_list_a @ A @ B2 ) ) ) ).
% Pi_I
thf(fact_683_Pi__I,axiom,
! [A: set_a,F: a > list_list_a,B2: a > set_list_list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_list_list_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_list_list_a @ F @ ( pi_a_list_list_a @ A @ B2 ) ) ) ).
% Pi_I
thf(fact_684_Pi__I,axiom,
! [A: set_list_a,F: list_a > a,B2: list_a > set_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_list_a_a @ F @ ( pi_list_a_a @ A @ B2 ) ) ) ).
% Pi_I
thf(fact_685_Pi__I,axiom,
! [A: set_list_a,F: list_a > list_a,B2: list_a > set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_list_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_list_a_list_a @ F @ ( pi_list_a_list_a @ A @ B2 ) ) ) ).
% Pi_I
thf(fact_686_Pi__I,axiom,
! [A: set_list_a,F: list_a > list_list_a,B2: list_a > set_list_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_list_list_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member6714375691612171394list_a @ F @ ( pi_lis3067418140807155665list_a @ A @ B2 ) ) ) ).
% Pi_I
thf(fact_687_Pi__I,axiom,
! [A: set_list_list_a,F: list_list_a > a,B2: list_list_a > set_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_list_list_a_a @ F @ ( pi_list_list_a_a @ A @ B2 ) ) ) ).
% Pi_I
thf(fact_688_Pi__I,axiom,
! [A: set_list_list_a,F: list_list_a > list_a,B2: list_list_a > set_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( member_list_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member7168557129179038582list_a @ F @ ( pi_lis8207908228422549957list_a @ A @ B2 ) ) ) ).
% Pi_I
thf(fact_689_Pi__I,axiom,
! [A: set_list_list_a,F: list_list_a > list_list_a,B2: list_list_a > set_list_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( member_list_list_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member8231385768148312316list_a @ F @ ( pi_lis7180132755996294475list_a @ A @ B2 ) ) ) ).
% Pi_I
thf(fact_690_finsum__cong_H,axiom,
! [A: set_a,B2: set_a,G: a > a,F: a > a] :
( ( A = B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: a] :
( ( member_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finsum_a_b_a @ r @ F @ A )
= ( finsum_a_b_a @ r @ G @ B2 ) ) ) ) ) ).
% finsum_cong'
thf(fact_691_finsum__cong_H,axiom,
! [A: set_list_a,B2: set_list_a,G: list_a > a,F: list_a > a] :
( ( A = B2 )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finsum_a_b_list_a @ r @ F @ A )
= ( finsum_a_b_list_a @ r @ G @ B2 ) ) ) ) ) ).
% finsum_cong'
thf(fact_692_finsum__cong_H,axiom,
! [A: set_list_list_a,B2: set_list_list_a,G: list_list_a > a,F: list_list_a > a] :
( ( A = B2 )
=> ( ( member_list_list_a_a @ G
@ ( pi_list_list_a_a @ B2
@ ^ [Uu: list_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: list_list_a] :
( ( member_list_list_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finsum1795837918752241516list_a @ r @ F @ A )
= ( finsum1795837918752241516list_a @ r @ G @ B2 ) ) ) ) ) ).
% finsum_cong'
thf(fact_693_eval__in__carrier__2,axiom,
! [X: list_a,Y: a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier_2
thf(fact_694_x_Ocgenideal__ideal,axiom,
! [A2: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ideal_8896367198367571637t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.cgenideal_ideal
thf(fact_695_x_Ooneideal,axiom,
ideal_8896367198367571637t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.oneideal
thf(fact_696_x_OIdl__subset__ideal,axiom,
! [I3: set_list_a,H2: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H2 ) @ I3 )
= ( ord_le8861187494160871172list_a @ H2 @ I3 ) ) ) ) ).
% x.Idl_subset_ideal
thf(fact_697_x_Ogenideal__ideal,axiom,
! [S: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ideal_8896367198367571637t_unit @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.genideal_ideal
thf(fact_698_x_Onoetherian__ringI,axiom,
( ! [I5: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I5 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ? [A4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( finite_finite_list_a @ A4 )
& ( I5
= ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A4 ) ) ) )
=> ( ring_n5188127996776581661t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.noetherian_ringI
thf(fact_699_x_Oring_Ozero__closed,axiom,
member_a @ ( eval_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x ) @ ( partia707051561876973205xt_a_b @ r ) ).
% x.ring.zero_closed
thf(fact_700_s_Oring_Ozero__closed,axiom,
member_a @ ( eval_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ s2 ) @ ( partia707051561876973205xt_a_b @ r ) ).
% s.ring.zero_closed
thf(fact_701_b,axiom,
member_list_a @ p @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% b
thf(fact_702_add_Ofinprod__singleton__swap,axiom,
! [I4: list_list_a,A: set_list_list_a,F: list_list_a > a] :
( ( member_list_list_a @ I4 @ A )
=> ( ( finite1660835950917165235list_a @ A )
=> ( ( member_list_list_a_a @ F
@ ( pi_list_list_a_a @ A
@ ^ [Uu: list_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum1795837918752241516list_a @ r
@ ^ [J: list_list_a] : ( if_a @ ( J = I4 ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% add.finprod_singleton_swap
thf(fact_703_add_Ofinprod__singleton__swap,axiom,
! [I4: a,A: set_a,F: a > a] :
( ( member_a @ I4 @ A )
=> ( ( finite_finite_a @ A )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a @ r
@ ^ [J: a] : ( if_a @ ( J = I4 ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% add.finprod_singleton_swap
thf(fact_704_add_Ofinprod__singleton__swap,axiom,
! [I4: list_a,A: set_list_a,F: list_a > a] :
( ( member_list_a @ I4 @ A )
=> ( ( finite_finite_list_a @ A )
=> ( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_list_a @ r
@ ^ [J: list_a] : ( if_a @ ( J = I4 ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% add.finprod_singleton_swap
thf(fact_705_finsum__singleton,axiom,
! [I4: list_list_a,A: set_list_list_a,F: list_list_a > a] :
( ( member_list_list_a @ I4 @ A )
=> ( ( finite1660835950917165235list_a @ A )
=> ( ( member_list_list_a_a @ F
@ ( pi_list_list_a_a @ A
@ ^ [Uu: list_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum1795837918752241516list_a @ r
@ ^ [J: list_list_a] : ( if_a @ ( I4 = J ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finsum_singleton
thf(fact_706_finsum__singleton,axiom,
! [I4: a,A: set_a,F: a > a] :
( ( member_a @ I4 @ A )
=> ( ( finite_finite_a @ A )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a @ r
@ ^ [J: a] : ( if_a @ ( I4 = J ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finsum_singleton
thf(fact_707_finsum__singleton,axiom,
! [I4: list_a,A: set_list_a,F: list_a > a] :
( ( member_list_a @ I4 @ A )
=> ( ( finite_finite_list_a @ A )
=> ( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_list_a @ r
@ ^ [J: list_a] : ( if_a @ ( I4 = J ) @ ( F @ J ) @ ( zero_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finsum_singleton
thf(fact_708_finsum__ldistr,axiom,
! [A: set_a,A2: a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( finsum_a_b_a @ r @ F @ A ) @ A2 )
= ( finsum_a_b_a @ r
@ ^ [I: a] : ( mult_a_ring_ext_a_b @ r @ ( F @ I ) @ A2 )
@ A ) ) ) ) ) ).
% finsum_ldistr
thf(fact_709_finsum__ldistr,axiom,
! [A: set_list_a,A2: a,F: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( finsum_a_b_list_a @ r @ F @ A ) @ A2 )
= ( finsum_a_b_list_a @ r
@ ^ [I: list_a] : ( mult_a_ring_ext_a_b @ r @ ( F @ I ) @ A2 )
@ A ) ) ) ) ) ).
% finsum_ldistr
thf(fact_710_finsum__rdistr,axiom,
! [A: set_a,A2: a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A2 @ ( finsum_a_b_a @ r @ F @ A ) )
= ( finsum_a_b_a @ r
@ ^ [I: a] : ( mult_a_ring_ext_a_b @ r @ A2 @ ( F @ I ) )
@ A ) ) ) ) ) ).
% finsum_rdistr
thf(fact_711_finsum__rdistr,axiom,
! [A: set_list_a,A2: a,F: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A2 @ ( finsum_a_b_list_a @ r @ F @ A ) )
= ( finsum_a_b_list_a @ r
@ ^ [I: list_a] : ( mult_a_ring_ext_a_b @ r @ A2 @ ( F @ I ) )
@ A ) ) ) ) ) ).
% finsum_rdistr
thf(fact_712_subset__empty,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_713_subset__empty,axiom,
! [A: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A @ bot_bo3186585308812441520list_a )
= ( A = bot_bo3186585308812441520list_a ) ) ).
% subset_empty
thf(fact_714_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_715_subset__empty,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
= ( A = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_716_empty__subsetI,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% empty_subsetI
thf(fact_717_empty__subsetI,axiom,
! [A: set_set_list_a] : ( ord_le8877086941679407844list_a @ bot_bo3186585308812441520list_a @ A ) ).
% empty_subsetI
thf(fact_718_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_719_empty__subsetI,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A ) ).
% empty_subsetI
thf(fact_720_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_721_inf__bot__left,axiom,
! [X: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X )
= bot_bot_set_list_a ) ).
% inf_bot_left
thf(fact_722_inf__bot__left,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X )
= bot_bot_set_set_a ) ).
% inf_bot_left
thf(fact_723_inf__bot__left,axiom,
! [X: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ bot_bo3186585308812441520list_a @ X )
= bot_bo3186585308812441520list_a ) ).
% inf_bot_left
thf(fact_724_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_725_inf__bot__right,axiom,
! [X: set_list_a] :
( ( inf_inf_set_list_a @ X @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% inf_bot_right
thf(fact_726_inf__bot__right,axiom,
! [X: set_set_a] :
( ( inf_inf_set_set_a @ X @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% inf_bot_right
thf(fact_727_inf__bot__right,axiom,
! [X: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ X @ bot_bo3186585308812441520list_a )
= bot_bo3186585308812441520list_a ) ).
% inf_bot_right
thf(fact_728_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_729_sup__bot__left,axiom,
! [X: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ X )
= X ) ).
% sup_bot_left
thf(fact_730_sup__bot__left,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_731_sup__bot__left,axiom,
! [X: set_set_list_a] :
( ( sup_su4537662296134749976list_a @ bot_bo3186585308812441520list_a @ X )
= X ) ).
% sup_bot_left
thf(fact_732_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_733_sup__bot__right,axiom,
! [X: set_list_a] :
( ( sup_sup_set_list_a @ X @ bot_bot_set_list_a )
= X ) ).
% sup_bot_right
thf(fact_734_sup__bot__right,axiom,
! [X: set_set_a] :
( ( sup_sup_set_set_a @ X @ bot_bot_set_set_a )
= X ) ).
% sup_bot_right
thf(fact_735_sup__bot__right,axiom,
! [X: set_set_list_a] :
( ( sup_su4537662296134749976list_a @ X @ bot_bo3186585308812441520list_a )
= X ) ).
% sup_bot_right
thf(fact_736_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_737_bot__eq__sup__iff,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( bot_bot_set_list_a
= ( sup_sup_set_list_a @ X @ Y ) )
= ( ( X = bot_bot_set_list_a )
& ( Y = bot_bot_set_list_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_738_bot__eq__sup__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_set_a )
& ( Y = bot_bot_set_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_739_bot__eq__sup__iff,axiom,
! [X: set_set_list_a,Y: set_set_list_a] :
( ( bot_bo3186585308812441520list_a
= ( sup_su4537662296134749976list_a @ X @ Y ) )
= ( ( X = bot_bo3186585308812441520list_a )
& ( Y = bot_bo3186585308812441520list_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_740_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_741_sup__eq__bot__iff,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ( sup_sup_set_list_a @ X @ Y )
= bot_bot_set_list_a )
= ( ( X = bot_bot_set_list_a )
& ( Y = bot_bot_set_list_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_742_sup__eq__bot__iff,axiom,
! [X: set_set_a,Y: set_set_a] :
( ( ( sup_sup_set_set_a @ X @ Y )
= bot_bot_set_set_a )
= ( ( X = bot_bot_set_set_a )
& ( Y = bot_bot_set_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_743_sup__eq__bot__iff,axiom,
! [X: set_set_list_a,Y: set_set_list_a] :
( ( ( sup_su4537662296134749976list_a @ X @ Y )
= bot_bo3186585308812441520list_a )
= ( ( X = bot_bo3186585308812441520list_a )
& ( Y = bot_bo3186585308812441520list_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_744_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ( sup_sup_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B3 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_745_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ( sup_sup_set_list_a @ A2 @ B3 )
= bot_bot_set_list_a )
= ( ( A2 = bot_bot_set_list_a )
& ( B3 = bot_bot_set_list_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_746_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ( sup_sup_set_set_a @ A2 @ B3 )
= bot_bot_set_set_a )
= ( ( A2 = bot_bot_set_set_a )
& ( B3 = bot_bot_set_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_747_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_list_a,B3: set_set_list_a] :
( ( ( sup_su4537662296134749976list_a @ A2 @ B3 )
= bot_bo3186585308812441520list_a )
= ( ( A2 = bot_bo3186585308812441520list_a )
& ( B3 = bot_bo3186585308812441520list_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_748_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_749_sup__bot_Oleft__neutral,axiom,
! [A2: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_750_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_751_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_list_a] :
( ( sup_su4537662296134749976list_a @ bot_bo3186585308812441520list_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_752_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B3 ) )
= ( ( A2 = bot_bot_set_a )
& ( B3 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_753_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( bot_bot_set_list_a
= ( sup_sup_set_list_a @ A2 @ B3 ) )
= ( ( A2 = bot_bot_set_list_a )
& ( B3 = bot_bot_set_list_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_754_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( bot_bot_set_set_a
= ( sup_sup_set_set_a @ A2 @ B3 ) )
= ( ( A2 = bot_bot_set_set_a )
& ( B3 = bot_bot_set_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_755_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_list_a,B3: set_set_list_a] :
( ( bot_bo3186585308812441520list_a
= ( sup_su4537662296134749976list_a @ A2 @ B3 ) )
= ( ( A2 = bot_bo3186585308812441520list_a )
& ( B3 = bot_bo3186585308812441520list_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_756_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_757_sup__bot_Oright__neutral,axiom,
! [A2: set_list_a] :
( ( sup_sup_set_list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_758_sup__bot_Oright__neutral,axiom,
! [A2: set_set_a] :
( ( sup_sup_set_set_a @ A2 @ bot_bot_set_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_759_sup__bot_Oright__neutral,axiom,
! [A2: set_set_list_a] :
( ( sup_su4537662296134749976list_a @ A2 @ bot_bo3186585308812441520list_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_760_Diff__empty,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% Diff_empty
thf(fact_761_Diff__empty,axiom,
! [A: set_set_list_a] :
( ( minus_4782336368215558443list_a @ A @ bot_bo3186585308812441520list_a )
= A ) ).
% Diff_empty
thf(fact_762_Diff__empty,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ) ).
% Diff_empty
thf(fact_763_Diff__empty,axiom,
! [A: set_list_a] :
( ( minus_646659088055828811list_a @ A @ bot_bot_set_list_a )
= A ) ).
% Diff_empty
thf(fact_764_empty__Diff,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ bot_bot_set_set_a @ A )
= bot_bot_set_set_a ) ).
% empty_Diff
thf(fact_765_empty__Diff,axiom,
! [A: set_set_list_a] :
( ( minus_4782336368215558443list_a @ bot_bo3186585308812441520list_a @ A )
= bot_bo3186585308812441520list_a ) ).
% empty_Diff
thf(fact_766_empty__Diff,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_767_empty__Diff,axiom,
! [A: set_list_a] :
( ( minus_646659088055828811list_a @ bot_bot_set_list_a @ A )
= bot_bot_set_list_a ) ).
% empty_Diff
thf(fact_768_Diff__cancel,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ A )
= bot_bot_set_set_a ) ).
% Diff_cancel
thf(fact_769_Diff__cancel,axiom,
! [A: set_set_list_a] :
( ( minus_4782336368215558443list_a @ A @ A )
= bot_bo3186585308812441520list_a ) ).
% Diff_cancel
thf(fact_770_Diff__cancel,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_771_Diff__cancel,axiom,
! [A: set_list_a] :
( ( minus_646659088055828811list_a @ A @ A )
= bot_bot_set_list_a ) ).
% Diff_cancel
thf(fact_772_Un__empty,axiom,
! [A: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_773_Un__empty,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ( sup_sup_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
= ( ( A = bot_bot_set_list_a )
& ( B2 = bot_bot_set_list_a ) ) ) ).
% Un_empty
thf(fact_774_Un__empty,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ( sup_sup_set_set_a @ A @ B2 )
= bot_bot_set_set_a )
= ( ( A = bot_bot_set_set_a )
& ( B2 = bot_bot_set_set_a ) ) ) ).
% Un_empty
thf(fact_775_Un__empty,axiom,
! [A: set_set_list_a,B2: set_set_list_a] :
( ( ( sup_su4537662296134749976list_a @ A @ B2 )
= bot_bo3186585308812441520list_a )
= ( ( A = bot_bo3186585308812441520list_a )
& ( B2 = bot_bo3186585308812441520list_a ) ) ) ).
% Un_empty
thf(fact_776_add_Ofinprod__mono__neutral__cong__left,axiom,
! [B2: set_list_list_a,A: set_list_list_a,H: list_list_a > a,G: list_list_a > a] :
( ( finite1660835950917165235list_a @ B2 )
=> ( ( ord_le8488217952732425610list_a @ A @ B2 )
=> ( ! [I2: list_list_a] :
( ( member_list_list_a @ I2 @ ( minus_5335179877275218001list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_list_a_a @ H
@ ( pi_list_list_a_a @ B2
@ ^ [Uu: list_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum1795837918752241516list_a @ r @ G @ A )
= ( finsum1795837918752241516list_a @ r @ H @ B2 ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_left
thf(fact_777_add_Ofinprod__mono__neutral__cong__left,axiom,
! [B2: set_a,A: set_a,H: a > a,G: a > a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_a @ H
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a @ r @ G @ A )
= ( finsum_a_b_a @ r @ H @ B2 ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_left
thf(fact_778_add_Ofinprod__mono__neutral__cong__left,axiom,
! [B2: set_list_a,A: set_list_a,H: list_a > a,G: list_a > a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ ( minus_646659088055828811list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( zero_a_b @ r ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_a_a @ H
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_list_a @ r @ G @ A )
= ( finsum_a_b_list_a @ r @ H @ B2 ) ) ) ) ) ) ) ).
% add.finprod_mono_neutral_cong_left
thf(fact_779_finsum__Un__disjoint,axiom,
! [A: set_a,B2: set_a,G: a > a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( ( inf_inf_set_a @ A @ B2 )
= bot_bot_set_a )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_a @ r @ G @ ( sup_sup_set_a @ A @ B2 ) )
= ( add_a_b @ r @ ( finsum_a_b_a @ r @ G @ A ) @ ( finsum_a_b_a @ r @ G @ B2 ) ) ) ) ) ) ) ) ).
% finsum_Un_disjoint
thf(fact_780_finsum__Un__disjoint,axiom,
! [A: set_list_a,B2: set_list_a,G: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_list_a @ r @ G @ ( sup_sup_set_list_a @ A @ B2 ) )
= ( add_a_b @ r @ ( finsum_a_b_list_a @ r @ G @ A ) @ ( finsum_a_b_list_a @ r @ G @ B2 ) ) ) ) ) ) ) ) ).
% finsum_Un_disjoint
thf(fact_781_finsum__Un__disjoint,axiom,
! [A: set_set_a,B2: set_set_a,G: set_a > a] :
( ( finite_finite_set_a @ A )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( ( inf_inf_set_set_a @ A @ B2 )
= bot_bot_set_set_a )
=> ( ( member_set_a_a @ G
@ ( pi_set_a_a @ A
@ ^ [Uu: set_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_set_a_a @ G
@ ( pi_set_a_a @ B2
@ ^ [Uu: set_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum_a_b_set_a @ r @ G @ ( sup_sup_set_set_a @ A @ B2 ) )
= ( add_a_b @ r @ ( finsum_a_b_set_a @ r @ G @ A ) @ ( finsum_a_b_set_a @ r @ G @ B2 ) ) ) ) ) ) ) ) ).
% finsum_Un_disjoint
thf(fact_782_finsum__Un__disjoint,axiom,
! [A: set_set_list_a,B2: set_set_list_a,G: set_list_a > a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( finite5282473924520328461list_a @ B2 )
=> ( ( ( inf_in4657809108759609906list_a @ A @ B2 )
= bot_bo3186585308812441520list_a )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ B2
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finsum7367453022336983110list_a @ r @ G @ ( sup_su4537662296134749976list_a @ A @ B2 ) )
= ( add_a_b @ r @ ( finsum7367453022336983110list_a @ r @ G @ A ) @ ( finsum7367453022336983110list_a @ r @ G @ B2 ) ) ) ) ) ) ) ) ).
% finsum_Un_disjoint
thf(fact_783_Diff__eq__empty__iff,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A @ B2 )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_784_Diff__eq__empty__iff,axiom,
! [A: set_set_list_a,B2: set_set_list_a] :
( ( ( minus_4782336368215558443list_a @ A @ B2 )
= bot_bo3186585308812441520list_a )
= ( ord_le8877086941679407844list_a @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_785_Diff__eq__empty__iff,axiom,
! [A: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_786_Diff__eq__empty__iff,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ( minus_646659088055828811list_a @ A @ B2 )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_787_Diff__disjoint,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( minus_5736297505244876581_set_a @ B2 @ A ) )
= bot_bot_set_set_a ) ).
% Diff_disjoint
thf(fact_788_Diff__disjoint,axiom,
! [A: set_set_list_a,B2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ A @ ( minus_4782336368215558443list_a @ B2 @ A ) )
= bot_bo3186585308812441520list_a ) ).
% Diff_disjoint
thf(fact_789_Diff__disjoint,axiom,
! [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B2 @ A ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_790_Diff__disjoint,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( minus_646659088055828811list_a @ B2 @ A ) )
= bot_bot_set_list_a ) ).
% Diff_disjoint
thf(fact_791_x_Ozero__closed,axiom,
member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.zero_closed
thf(fact_792_finsum__empty,axiom,
! [F: a > a] :
( ( finsum_a_b_a @ r @ F @ bot_bot_set_a )
= ( zero_a_b @ r ) ) ).
% finsum_empty
thf(fact_793_finsum__empty,axiom,
! [F: list_a > a] :
( ( finsum_a_b_list_a @ r @ F @ bot_bot_set_list_a )
= ( zero_a_b @ r ) ) ).
% finsum_empty
thf(fact_794_finsum__empty,axiom,
! [F: set_a > a] :
( ( finsum_a_b_set_a @ r @ F @ bot_bot_set_set_a )
= ( zero_a_b @ r ) ) ).
% finsum_empty
thf(fact_795_finsum__empty,axiom,
! [F: set_list_a > a] :
( ( finsum7367453022336983110list_a @ r @ F @ bot_bo3186585308812441520list_a )
= ( zero_a_b @ r ) ) ).
% finsum_empty
thf(fact_796_finsum__infinite,axiom,
! [A: set_a,F: a > a] :
( ~ ( finite_finite_a @ A )
=> ( ( finsum_a_b_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% finsum_infinite
thf(fact_797_finsum__infinite,axiom,
! [A: set_list_a,F: list_a > a] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finsum_a_b_list_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% finsum_infinite
thf(fact_798_finprod__empty,axiom,
! [F: a > a] :
( ( finpro205304725090349623_a_b_a @ r @ F @ bot_bot_set_a )
= ( one_a_ring_ext_a_b @ r ) ) ).
% finprod_empty
thf(fact_799_finprod__empty,axiom,
! [F: list_a > a] :
( ( finpro6052973074229812797list_a @ r @ F @ bot_bot_set_list_a )
= ( one_a_ring_ext_a_b @ r ) ) ).
% finprod_empty
thf(fact_800_finprod__empty,axiom,
! [F: set_a > a] :
( ( finpro934595834566309783_set_a @ r @ F @ bot_bot_set_set_a )
= ( one_a_ring_ext_a_b @ r ) ) ).
% finprod_empty
thf(fact_801_finprod__empty,axiom,
! [F: set_list_a > a] :
( ( finpro3826550488720007709list_a @ r @ F @ bot_bo3186585308812441520list_a )
= ( one_a_ring_ext_a_b @ r ) ) ).
% finprod_empty
thf(fact_802_x_Oring_Ohom__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_a @ ( eval_a_b @ r @ X @ x ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.ring.hom_closed
thf(fact_803_s_Oring_Ohom__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_a @ ( eval_a_b @ r @ X @ s2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% s.ring.hom_closed
thf(fact_804_x_Oring_Ohom__zero,axiom,
( ( eval_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x )
= ( zero_a_b @ r ) ) ).
% x.ring.hom_zero
thf(fact_805_s_Oring_Ohom__zero,axiom,
( ( eval_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ s2 )
= ( zero_a_b @ r ) ) ).
% s.ring.hom_zero
thf(fact_806_ex__in__conv,axiom,
! [A: set_list_list_a] :
( ( ? [X3: list_list_a] : ( member_list_list_a @ X3 @ A ) )
= ( A != bot_bo1875519244922727510list_a ) ) ).
% ex_in_conv
thf(fact_807_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_808_ex__in__conv,axiom,
! [A: set_list_a] :
( ( ? [X3: list_a] : ( member_list_a @ X3 @ A ) )
= ( A != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_809_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_810_ex__in__conv,axiom,
! [A: set_set_list_a] :
( ( ? [X3: set_list_a] : ( member_set_list_a @ X3 @ A ) )
= ( A != bot_bo3186585308812441520list_a ) ) ).
% ex_in_conv
thf(fact_811_equals0I,axiom,
! [A: set_list_list_a] :
( ! [Y3: list_list_a] :
~ ( member_list_list_a @ Y3 @ A )
=> ( A = bot_bo1875519244922727510list_a ) ) ).
% equals0I
thf(fact_812_equals0I,axiom,
! [A: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_813_equals0I,axiom,
! [A: set_list_a] :
( ! [Y3: list_a] :
~ ( member_list_a @ Y3 @ A )
=> ( A = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_814_equals0I,axiom,
! [A: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_815_equals0I,axiom,
! [A: set_set_list_a] :
( ! [Y3: set_list_a] :
~ ( member_set_list_a @ Y3 @ A )
=> ( A = bot_bo3186585308812441520list_a ) ) ).
% equals0I
thf(fact_816_equals0D,axiom,
! [A: set_list_list_a,A2: list_list_a] :
( ( A = bot_bo1875519244922727510list_a )
=> ~ ( member_list_list_a @ A2 @ A ) ) ).
% equals0D
thf(fact_817_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_818_equals0D,axiom,
! [A: set_list_a,A2: list_a] :
( ( A = bot_bot_set_list_a )
=> ~ ( member_list_a @ A2 @ A ) ) ).
% equals0D
thf(fact_819_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_820_equals0D,axiom,
! [A: set_set_list_a,A2: set_list_a] :
( ( A = bot_bo3186585308812441520list_a )
=> ~ ( member_set_list_a @ A2 @ A ) ) ).
% equals0D
thf(fact_821_emptyE,axiom,
! [A2: list_list_a] :
~ ( member_list_list_a @ A2 @ bot_bo1875519244922727510list_a ) ).
% emptyE
thf(fact_822_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_823_emptyE,axiom,
! [A2: list_a] :
~ ( member_list_a @ A2 @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_824_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_825_emptyE,axiom,
! [A2: set_list_a] :
~ ( member_set_list_a @ A2 @ bot_bo3186585308812441520list_a ) ).
% emptyE
thf(fact_826_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X3: a] : $false ) ) ).
% empty_def
thf(fact_827_empty__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a
@ ^ [X3: list_a] : $false ) ) ).
% empty_def
thf(fact_828_empty__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a
@ ^ [X3: set_a] : $false ) ) ).
% empty_def
thf(fact_829_empty__def,axiom,
( bot_bo3186585308812441520list_a
= ( collect_set_list_a
@ ^ [X3: set_list_a] : $false ) ) ).
% empty_def
thf(fact_830_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_831_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_832_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_833_finite_OemptyI,axiom,
finite5282473924520328461list_a @ bot_bo3186585308812441520list_a ).
% finite.emptyI
thf(fact_834_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_835_infinite__imp__nonempty,axiom,
! [S: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ( S != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_836_infinite__imp__nonempty,axiom,
! [S: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ( S != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_837_infinite__imp__nonempty,axiom,
! [S: set_set_list_a] :
( ~ ( finite5282473924520328461list_a @ S )
=> ( S != bot_bo3186585308812441520list_a ) ) ).
% infinite_imp_nonempty
thf(fact_838_Int__emptyI,axiom,
! [A: set_list_list_a,B2: set_list_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ~ ( member_list_list_a @ X2 @ B2 ) )
=> ( ( inf_in7423150557312423384list_a @ A @ B2 )
= bot_bo1875519244922727510list_a ) ) ).
% Int_emptyI
thf(fact_839_Int__emptyI,axiom,
! [A: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_a @ A @ B2 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_840_Int__emptyI,axiom,
! [A: set_list_a,B2: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ~ ( member_list_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a ) ) ).
% Int_emptyI
thf(fact_841_Int__emptyI,axiom,
! [A: set_set_a,B2: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ~ ( member_set_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_set_a @ A @ B2 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_842_Int__emptyI,axiom,
! [A: set_set_list_a,B2: set_set_list_a] :
( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
=> ~ ( member_set_list_a @ X2 @ B2 ) )
=> ( ( inf_in4657809108759609906list_a @ A @ B2 )
= bot_bo3186585308812441520list_a ) ) ).
% Int_emptyI
thf(fact_843_disjoint__iff,axiom,
! [A: set_list_list_a,B2: set_list_list_a] :
( ( ( inf_in7423150557312423384list_a @ A @ B2 )
= bot_bo1875519244922727510list_a )
= ( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A )
=> ~ ( member_list_list_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_844_disjoint__iff,axiom,
! [A: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_845_disjoint__iff,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ~ ( member_list_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_846_disjoint__iff,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B2 )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ( member_set_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_847_disjoint__iff,axiom,
! [A: set_set_list_a,B2: set_set_list_a] :
( ( ( inf_in4657809108759609906list_a @ A @ B2 )
= bot_bo3186585308812441520list_a )
= ( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A )
=> ~ ( member_set_list_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_848_Int__empty__left,axiom,
! [B2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B2 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_849_Int__empty__left,axiom,
! [B2: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ B2 )
= bot_bot_set_list_a ) ).
% Int_empty_left
thf(fact_850_Int__empty__left,axiom,
! [B2: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B2 )
= bot_bot_set_set_a ) ).
% Int_empty_left
thf(fact_851_Int__empty__left,axiom,
! [B2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ bot_bo3186585308812441520list_a @ B2 )
= bot_bo3186585308812441520list_a ) ).
% Int_empty_left
thf(fact_852_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_853_Int__empty__right,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% Int_empty_right
thf(fact_854_Int__empty__right,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% Int_empty_right
thf(fact_855_Int__empty__right,axiom,
! [A: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ A @ bot_bo3186585308812441520list_a )
= bot_bo3186585308812441520list_a ) ).
% Int_empty_right
thf(fact_856_disjoint__iff__not__equal,axiom,
! [A: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ! [Y4: a] :
( ( member_a @ Y4 @ B2 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_857_disjoint__iff__not__equal,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ! [Y4: list_a] :
( ( member_list_a @ Y4 @ B2 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_858_disjoint__iff__not__equal,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B2 )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ B2 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_859_disjoint__iff__not__equal,axiom,
! [A: set_set_list_a,B2: set_set_list_a] :
( ( ( inf_in4657809108759609906list_a @ A @ B2 )
= bot_bo3186585308812441520list_a )
= ( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A )
=> ! [Y4: set_list_a] :
( ( member_set_list_a @ Y4 @ B2 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_860_Un__empty__left,axiom,
! [B2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_861_Un__empty__left,axiom,
! [B2: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_862_Un__empty__left,axiom,
! [B2: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_863_Un__empty__left,axiom,
! [B2: set_set_list_a] :
( ( sup_su4537662296134749976list_a @ bot_bo3186585308812441520list_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_864_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_865_Un__empty__right,axiom,
! [A: set_list_a] :
( ( sup_sup_set_list_a @ A @ bot_bot_set_list_a )
= A ) ).
% Un_empty_right
thf(fact_866_Un__empty__right,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% Un_empty_right
thf(fact_867_Un__empty__right,axiom,
! [A: set_set_list_a] :
( ( sup_su4537662296134749976list_a @ A @ bot_bo3186585308812441520list_a )
= A ) ).
% Un_empty_right
thf(fact_868_sup__set__def,axiom,
( sup_su1566401739067690942list_a
= ( ^ [A7: set_list_list_a,B: set_list_list_a] :
( collect_list_list_a
@ ( sup_su1850317741939728543st_a_o
@ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ A7 )
@ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ B ) ) ) ) ) ).
% sup_set_def
thf(fact_869_sup__set__def,axiom,
( sup_sup_set_list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
( collect_list_a
@ ( sup_sup_list_a_o
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A7 )
@ ^ [X3: list_a] : ( member_list_a @ X3 @ B ) ) ) ) ) ).
% sup_set_def
thf(fact_870_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A7: set_a,B: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A7 )
@ ^ [X3: a] : ( member_a @ X3 @ B ) ) ) ) ) ).
% sup_set_def
thf(fact_871_sup__set__def,axiom,
( sup_sup_set_set_a
= ( ^ [A7: set_set_a,B: set_set_a] :
( collect_set_a
@ ( sup_sup_set_a_o
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A7 )
@ ^ [X3: set_a] : ( member_set_a @ X3 @ B ) ) ) ) ) ).
% sup_set_def
thf(fact_872_sup__set__def,axiom,
( sup_su4537662296134749976list_a
= ( ^ [A7: set_set_list_a,B: set_set_list_a] :
( collect_set_list_a
@ ( sup_sup_set_list_a_o
@ ^ [X3: set_list_a] : ( member_set_list_a @ X3 @ A7 )
@ ^ [X3: set_list_a] : ( member_set_list_a @ X3 @ B ) ) ) ) ) ).
% sup_set_def
thf(fact_873_minus__set__def,axiom,
( minus_5335179877275218001list_a
= ( ^ [A7: set_list_list_a,B: set_list_list_a] :
( collect_list_list_a
@ ( minus_5004031662284927436st_a_o
@ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ A7 )
@ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ B ) ) ) ) ) ).
% minus_set_def
thf(fact_874_minus__set__def,axiom,
( minus_5736297505244876581_set_a
= ( ^ [A7: set_set_a,B: set_set_a] :
( collect_set_a
@ ( minus_minus_set_a_o
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A7 )
@ ^ [X3: set_a] : ( member_set_a @ X3 @ B ) ) ) ) ) ).
% minus_set_def
thf(fact_875_minus__set__def,axiom,
( minus_4782336368215558443list_a
= ( ^ [A7: set_set_list_a,B: set_set_list_a] :
( collect_set_list_a
@ ( minus_1481011451066941298st_a_o
@ ^ [X3: set_list_a] : ( member_set_list_a @ X3 @ A7 )
@ ^ [X3: set_list_a] : ( member_set_list_a @ X3 @ B ) ) ) ) ) ).
% minus_set_def
thf(fact_876_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A7: set_a,B: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A7 )
@ ^ [X3: a] : ( member_a @ X3 @ B ) ) ) ) ) ).
% minus_set_def
thf(fact_877_minus__set__def,axiom,
( minus_646659088055828811list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
( collect_list_a
@ ( minus_minus_list_a_o
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A7 )
@ ^ [X3: list_a] : ( member_list_a @ X3 @ B ) ) ) ) ) ).
% minus_set_def
thf(fact_878_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_879_finite__has__maximal,axiom,
! [A: set_set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( A != bot_bo3186585308812441520list_a )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_880_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_881_finite__has__minimal,axiom,
! [A: set_set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( A != bot_bo3186585308812441520list_a )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_882_Diff__triv,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B2 )
= bot_bot_set_set_a )
=> ( ( minus_5736297505244876581_set_a @ A @ B2 )
= A ) ) ).
% Diff_triv
thf(fact_883_Diff__triv,axiom,
! [A: set_set_list_a,B2: set_set_list_a] :
( ( ( inf_in4657809108759609906list_a @ A @ B2 )
= bot_bo3186585308812441520list_a )
=> ( ( minus_4782336368215558443list_a @ A @ B2 )
= A ) ) ).
% Diff_triv
thf(fact_884_Diff__triv,axiom,
! [A: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A @ B2 )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A @ B2 )
= A ) ) ).
% Diff_triv
thf(fact_885_Diff__triv,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
=> ( ( minus_646659088055828811list_a @ A @ B2 )
= A ) ) ).
% Diff_triv
thf(fact_886_Int__Diff__disjoint,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ ( inf_inf_set_set_a @ A @ B2 ) @ ( minus_5736297505244876581_set_a @ A @ B2 ) )
= bot_bot_set_set_a ) ).
% Int_Diff_disjoint
thf(fact_887_Int__Diff__disjoint,axiom,
! [A: set_set_list_a,B2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ ( inf_in4657809108759609906list_a @ A @ B2 ) @ ( minus_4782336368215558443list_a @ A @ B2 ) )
= bot_bo3186585308812441520list_a ) ).
% Int_Diff_disjoint
thf(fact_888_Int__Diff__disjoint,axiom,
! [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( minus_minus_set_a @ A @ B2 ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_889_Int__Diff__disjoint,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ ( minus_646659088055828811list_a @ A @ B2 ) )
= bot_bot_set_list_a ) ).
% Int_Diff_disjoint
thf(fact_890_PiE,axiom,
! [F: a > a,A: set_a,B2: a > set_a,X: a] :
( ( member_a_a @ F @ ( pi_a_a @ A @ B2 ) )
=> ( ~ ( member_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% PiE
thf(fact_891_PiE,axiom,
! [F: list_a > a,A: set_list_a,B2: list_a > set_a,X: list_a] :
( ( member_list_a_a @ F @ ( pi_list_a_a @ A @ B2 ) )
=> ( ~ ( member_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_list_a @ X @ A ) ) ) ).
% PiE
thf(fact_892_PiE,axiom,
! [F: list_list_a > a,A: set_list_list_a,B2: list_list_a > set_a,X: list_list_a] :
( ( member_list_list_a_a @ F @ ( pi_list_list_a_a @ A @ B2 ) )
=> ( ~ ( member_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_list_list_a @ X @ A ) ) ) ).
% PiE
thf(fact_893_PiE,axiom,
! [F: a > list_a,A: set_a,B2: a > set_list_a,X: a] :
( ( member_a_list_a @ F @ ( pi_a_list_a @ A @ B2 ) )
=> ( ~ ( member_list_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% PiE
thf(fact_894_PiE,axiom,
! [F: list_a > list_a,A: set_list_a,B2: list_a > set_list_a,X: list_a] :
( ( member_list_a_list_a @ F @ ( pi_list_a_list_a @ A @ B2 ) )
=> ( ~ ( member_list_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_list_a @ X @ A ) ) ) ).
% PiE
thf(fact_895_PiE,axiom,
! [F: list_list_a > list_a,A: set_list_list_a,B2: list_list_a > set_list_a,X: list_list_a] :
( ( member7168557129179038582list_a @ F @ ( pi_lis8207908228422549957list_a @ A @ B2 ) )
=> ( ~ ( member_list_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_list_list_a @ X @ A ) ) ) ).
% PiE
thf(fact_896_PiE,axiom,
! [F: a > list_list_a,A: set_a,B2: a > set_list_list_a,X: a] :
( ( member_a_list_list_a @ F @ ( pi_a_list_list_a @ A @ B2 ) )
=> ( ~ ( member_list_list_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% PiE
thf(fact_897_PiE,axiom,
! [F: list_a > list_list_a,A: set_list_a,B2: list_a > set_list_list_a,X: list_a] :
( ( member6714375691612171394list_a @ F @ ( pi_lis3067418140807155665list_a @ A @ B2 ) )
=> ( ~ ( member_list_list_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_list_a @ X @ A ) ) ) ).
% PiE
thf(fact_898_PiE,axiom,
! [F: list_list_a > list_list_a,A: set_list_list_a,B2: list_list_a > set_list_list_a,X: list_list_a] :
( ( member8231385768148312316list_a @ F @ ( pi_lis7180132755996294475list_a @ A @ B2 ) )
=> ( ~ ( member_list_list_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_list_list_a @ X @ A ) ) ) ).
% PiE
thf(fact_899_Pi__I_H,axiom,
! [A: set_a,F: a > a,B2: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_a @ F @ ( pi_a_a @ A @ B2 ) ) ) ).
% Pi_I'
thf(fact_900_Pi__I_H,axiom,
! [A: set_a,F: a > list_a,B2: a > set_list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_list_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_list_a @ F @ ( pi_a_list_a @ A @ B2 ) ) ) ).
% Pi_I'
thf(fact_901_Pi__I_H,axiom,
! [A: set_a,F: a > list_list_a,B2: a > set_list_list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_list_list_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_list_list_a @ F @ ( pi_a_list_list_a @ A @ B2 ) ) ) ).
% Pi_I'
thf(fact_902_Pi__I_H,axiom,
! [A: set_list_a,F: list_a > a,B2: list_a > set_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_list_a_a @ F @ ( pi_list_a_a @ A @ B2 ) ) ) ).
% Pi_I'
thf(fact_903_Pi__I_H,axiom,
! [A: set_list_a,F: list_a > list_a,B2: list_a > set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_list_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_list_a_list_a @ F @ ( pi_list_a_list_a @ A @ B2 ) ) ) ).
% Pi_I'
thf(fact_904_Pi__I_H,axiom,
! [A: set_list_a,F: list_a > list_list_a,B2: list_a > set_list_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_list_list_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member6714375691612171394list_a @ F @ ( pi_lis3067418140807155665list_a @ A @ B2 ) ) ) ).
% Pi_I'
thf(fact_905_Pi__I_H,axiom,
! [A: set_list_list_a,F: list_list_a > a,B2: list_list_a > set_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_list_list_a_a @ F @ ( pi_list_list_a_a @ A @ B2 ) ) ) ).
% Pi_I'
thf(fact_906_Pi__I_H,axiom,
! [A: set_list_list_a,F: list_list_a > list_a,B2: list_list_a > set_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( member_list_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member7168557129179038582list_a @ F @ ( pi_lis8207908228422549957list_a @ A @ B2 ) ) ) ).
% Pi_I'
thf(fact_907_Pi__I_H,axiom,
! [A: set_list_list_a,F: list_list_a > list_list_a,B2: list_list_a > set_list_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( member_list_list_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member8231385768148312316list_a @ F @ ( pi_lis7180132755996294475list_a @ A @ B2 ) ) ) ).
% Pi_I'
thf(fact_908_Pi__mem,axiom,
! [F: a > a,A: set_a,B2: a > set_a,X: a] :
( ( member_a_a @ F @ ( pi_a_a @ A @ B2 ) )
=> ( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_909_Pi__mem,axiom,
! [F: a > list_a,A: set_a,B2: a > set_list_a,X: a] :
( ( member_a_list_a @ F @ ( pi_a_list_a @ A @ B2 ) )
=> ( ( member_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_910_Pi__mem,axiom,
! [F: a > list_list_a,A: set_a,B2: a > set_list_list_a,X: a] :
( ( member_a_list_list_a @ F @ ( pi_a_list_list_a @ A @ B2 ) )
=> ( ( member_a @ X @ A )
=> ( member_list_list_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_911_Pi__mem,axiom,
! [F: list_a > a,A: set_list_a,B2: list_a > set_a,X: list_a] :
( ( member_list_a_a @ F @ ( pi_list_a_a @ A @ B2 ) )
=> ( ( member_list_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_912_Pi__mem,axiom,
! [F: list_a > list_a,A: set_list_a,B2: list_a > set_list_a,X: list_a] :
( ( member_list_a_list_a @ F @ ( pi_list_a_list_a @ A @ B2 ) )
=> ( ( member_list_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_913_Pi__mem,axiom,
! [F: list_a > list_list_a,A: set_list_a,B2: list_a > set_list_list_a,X: list_a] :
( ( member6714375691612171394list_a @ F @ ( pi_lis3067418140807155665list_a @ A @ B2 ) )
=> ( ( member_list_a @ X @ A )
=> ( member_list_list_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_914_Pi__mem,axiom,
! [F: list_list_a > a,A: set_list_list_a,B2: list_list_a > set_a,X: list_list_a] :
( ( member_list_list_a_a @ F @ ( pi_list_list_a_a @ A @ B2 ) )
=> ( ( member_list_list_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_915_Pi__mem,axiom,
! [F: list_list_a > list_a,A: set_list_list_a,B2: list_list_a > set_list_a,X: list_list_a] :
( ( member7168557129179038582list_a @ F @ ( pi_lis8207908228422549957list_a @ A @ B2 ) )
=> ( ( member_list_list_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_916_Pi__mem,axiom,
! [F: list_list_a > list_list_a,A: set_list_list_a,B2: list_list_a > set_list_list_a,X: list_list_a] :
( ( member8231385768148312316list_a @ F @ ( pi_lis7180132755996294475list_a @ A @ B2 ) )
=> ( ( member_list_list_a @ X @ A )
=> ( member_list_list_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_917_funcset__mem,axiom,
! [F: a > a,A: set_a,B2: set_a,X: a] :
( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : B2 ) )
=> ( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_918_funcset__mem,axiom,
! [F: a > list_a,A: set_a,B2: set_list_a,X: a] :
( ( member_a_list_a @ F
@ ( pi_a_list_a @ A
@ ^ [Uu: a] : B2 ) )
=> ( ( member_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_919_funcset__mem,axiom,
! [F: a > list_list_a,A: set_a,B2: set_list_list_a,X: a] :
( ( member_a_list_list_a @ F
@ ( pi_a_list_list_a @ A
@ ^ [Uu: a] : B2 ) )
=> ( ( member_a @ X @ A )
=> ( member_list_list_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_920_funcset__mem,axiom,
! [F: list_a > a,A: set_list_a,B2: set_a,X: list_a] :
( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : B2 ) )
=> ( ( member_list_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_921_funcset__mem,axiom,
! [F: list_a > list_a,A: set_list_a,B2: set_list_a,X: list_a] :
( ( member_list_a_list_a @ F
@ ( pi_list_a_list_a @ A
@ ^ [Uu: list_a] : B2 ) )
=> ( ( member_list_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_922_funcset__mem,axiom,
! [F: list_a > list_list_a,A: set_list_a,B2: set_list_list_a,X: list_a] :
( ( member6714375691612171394list_a @ F
@ ( pi_lis3067418140807155665list_a @ A
@ ^ [Uu: list_a] : B2 ) )
=> ( ( member_list_a @ X @ A )
=> ( member_list_list_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_923_funcset__mem,axiom,
! [F: list_list_a > a,A: set_list_list_a,B2: set_a,X: list_list_a] :
( ( member_list_list_a_a @ F
@ ( pi_list_list_a_a @ A
@ ^ [Uu: list_list_a] : B2 ) )
=> ( ( member_list_list_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_924_funcset__mem,axiom,
! [F: list_list_a > list_a,A: set_list_list_a,B2: set_list_a,X: list_list_a] :
( ( member7168557129179038582list_a @ F
@ ( pi_lis8207908228422549957list_a @ A
@ ^ [Uu: list_list_a] : B2 ) )
=> ( ( member_list_list_a @ X @ A )
=> ( member_list_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_925_funcset__mem,axiom,
! [F: list_list_a > list_list_a,A: set_list_list_a,B2: set_list_list_a,X: list_list_a] :
( ( member8231385768148312316list_a @ F
@ ( pi_lis7180132755996294475list_a @ A
@ ^ [Uu: list_list_a] : B2 ) )
=> ( ( member_list_list_a @ X @ A )
=> ( member_list_list_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_926_funcsetI,axiom,
! [A: set_a,F: a > a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : B2 ) ) ) ).
% funcsetI
thf(fact_927_funcsetI,axiom,
! [A: set_a,F: a > list_a,B2: set_list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_list_a @ ( F @ X2 ) @ B2 ) )
=> ( member_a_list_a @ F
@ ( pi_a_list_a @ A
@ ^ [Uu: a] : B2 ) ) ) ).
% funcsetI
thf(fact_928_funcsetI,axiom,
! [A: set_a,F: a > list_list_a,B2: set_list_list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_list_list_a @ ( F @ X2 ) @ B2 ) )
=> ( member_a_list_list_a @ F
@ ( pi_a_list_list_a @ A
@ ^ [Uu: a] : B2 ) ) ) ).
% funcsetI
thf(fact_929_funcsetI,axiom,
! [A: set_list_a,F: list_a > a,B2: set_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : B2 ) ) ) ).
% funcsetI
thf(fact_930_funcsetI,axiom,
! [A: set_list_a,F: list_a > list_a,B2: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_list_a @ ( F @ X2 ) @ B2 ) )
=> ( member_list_a_list_a @ F
@ ( pi_list_a_list_a @ A
@ ^ [Uu: list_a] : B2 ) ) ) ).
% funcsetI
thf(fact_931_funcsetI,axiom,
! [A: set_list_a,F: list_a > list_list_a,B2: set_list_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_list_list_a @ ( F @ X2 ) @ B2 ) )
=> ( member6714375691612171394list_a @ F
@ ( pi_lis3067418140807155665list_a @ A
@ ^ [Uu: list_a] : B2 ) ) ) ).
% funcsetI
thf(fact_932_funcsetI,axiom,
! [A: set_list_list_a,F: list_list_a > a,B2: set_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( member_list_list_a_a @ F
@ ( pi_list_list_a_a @ A
@ ^ [Uu: list_list_a] : B2 ) ) ) ).
% funcsetI
thf(fact_933_funcsetI,axiom,
! [A: set_list_list_a,F: list_list_a > list_a,B2: set_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( member_list_a @ ( F @ X2 ) @ B2 ) )
=> ( member7168557129179038582list_a @ F
@ ( pi_lis8207908228422549957list_a @ A
@ ^ [Uu: list_list_a] : B2 ) ) ) ).
% funcsetI
thf(fact_934_funcsetI,axiom,
! [A: set_list_list_a,F: list_list_a > list_list_a,B2: set_list_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( member_list_list_a @ ( F @ X2 ) @ B2 ) )
=> ( member8231385768148312316list_a @ F
@ ( pi_lis7180132755996294475list_a @ A
@ ^ [Uu: list_list_a] : B2 ) ) ) ).
% funcsetI
thf(fact_935_Pi__mono,axiom,
! [A: set_a,B2: a > set_a,C3: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C3 @ X2 ) ) )
=> ( ord_less_eq_set_a_a @ ( pi_a_a @ A @ B2 ) @ ( pi_a_a @ A @ C3 ) ) ) ).
% Pi_mono
thf(fact_936_Pi__mono,axiom,
! [A: set_list_a,B2: list_a > set_a,C3: list_a > set_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C3 @ X2 ) ) )
=> ( ord_le6942402695062981877st_a_a @ ( pi_list_a_a @ A @ B2 ) @ ( pi_list_a_a @ A @ C3 ) ) ) ).
% Pi_mono
thf(fact_937_Pi__mono,axiom,
! [A: set_list_list_a,B2: list_list_a > set_a,C3: list_list_a > set_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C3 @ X2 ) ) )
=> ( ord_le8231825031761998959st_a_a @ ( pi_list_list_a_a @ A @ B2 ) @ ( pi_list_list_a_a @ A @ C3 ) ) ) ).
% Pi_mono
thf(fact_938_Pi__mono,axiom,
! [A: set_a,B2: a > set_list_a,C3: a > set_list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_le8861187494160871172list_a @ ( B2 @ X2 ) @ ( C3 @ X2 ) ) )
=> ( ord_le50412136050534657list_a @ ( pi_a_list_a @ A @ B2 ) @ ( pi_a_list_a @ A @ C3 ) ) ) ).
% Pi_mono
thf(fact_939_Pi__mono,axiom,
! [A: set_list_a,B2: list_a > set_list_a,C3: list_a > set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ord_le8861187494160871172list_a @ ( B2 @ X2 ) @ ( C3 @ X2 ) ) )
=> ( ord_le890037400083324667list_a @ ( pi_list_a_list_a @ A @ B2 ) @ ( pi_list_a_list_a @ A @ C3 ) ) ) ).
% Pi_mono
thf(fact_940_Pi__mono,axiom,
! [A: set_list_list_a,B2: list_list_a > set_list_a,C3: list_list_a > set_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( ord_le8861187494160871172list_a @ ( B2 @ X2 ) @ ( C3 @ X2 ) ) )
=> ( ord_le6360101854653088117list_a @ ( pi_lis8207908228422549957list_a @ A @ B2 ) @ ( pi_lis8207908228422549957list_a @ A @ C3 ) ) ) ).
% Pi_mono
thf(fact_941_ring__iso__memE_I1_J,axiom,
! [H: a > a,R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R2 @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_942_ring__iso__memE_I1_J,axiom,
! [H: a > list_a,R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R2 @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_943_ring__iso__memE_I1_J,axiom,
! [H: a > list_list_a,R2: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,X: a] :
( ( member_a_list_list_a @ H @ ( ring_i4464730343205239444t_unit @ R2 @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( member_list_list_a @ ( H @ X ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_944_ring__iso__memE_I1_J,axiom,
! [H: list_a > a,R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R2 @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_945_ring__iso__memE_I1_J,axiom,
! [H: list_a > list_a,R2: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a] :
( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R2 @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_946_ring__iso__memE_I1_J,axiom,
! [H: list_a > list_list_a,R2: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,X: list_a] :
( ( member6714375691612171394list_a @ H @ ( ring_i7582117978422105628t_unit @ R2 @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_list_list_a @ ( H @ X ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_947_ring__iso__memE_I1_J,axiom,
! [H: list_list_a > a,R2: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X: list_list_a] :
( ( member_list_list_a_a @ H @ ( ring_i5684343068699926420it_a_b @ R2 @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_948_ring__iso__memE_I1_J,axiom,
! [H: list_list_a > list_a,R2: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X: list_list_a] :
( ( member7168557129179038582list_a @ H @ ( ring_i4611353245267337884t_unit @ R2 @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_949_ring__iso__memE_I1_J,axiom,
! [H: list_list_a > list_list_a,R2: partia2956882679547061052t_unit,S: partia2956882679547061052t_unit,X: list_list_a] :
( ( member8231385768148312316list_a @ H @ ( ring_i6186174840089424918t_unit @ R2 @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( member_list_list_a @ ( H @ X ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_950_ring__iso__memE_I4_J,axiom,
! [H: a > a,R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R2 @ S ) )
=> ( ( H @ ( one_a_ring_ext_a_b @ R2 ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_951_ring__iso__memE_I4_J,axiom,
! [H: a > list_a,R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R2 @ S ) )
=> ( ( H @ ( one_a_ring_ext_a_b @ R2 ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_952_ring__iso__memE_I4_J,axiom,
! [H: list_a > a,R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R2 @ S ) )
=> ( ( H @ ( one_li8328186300101108157t_unit @ R2 ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_953_ring__iso__memE_I4_J,axiom,
! [H: list_a > list_a,R2: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit] :
( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R2 @ S ) )
=> ( ( H @ ( one_li8328186300101108157t_unit @ R2 ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_954_ring__iso__memE_I3_J,axiom,
! [H: a > a,R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R2 @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( H @ ( add_a_b @ R2 @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_955_ring__iso__memE_I3_J,axiom,
! [H: a > list_a,R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R2 @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( H @ ( add_a_b @ R2 @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_956_ring__iso__memE_I3_J,axiom,
! [H: list_a > a,R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R2 @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R2 @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_957_ring__iso__memE_I3_J,axiom,
! [H: list_a > list_a,R2: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R2 @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R2 @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_958_ring__iso__memE_I3_J,axiom,
! [H: list_list_a > a,R2: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H @ ( ring_i5684343068699926420it_a_b @ R2 @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( H @ ( add_li174743652000525320t_unit @ R2 @ X @ Y ) )
= ( add_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_959_ring__iso__memE_I3_J,axiom,
! [H: list_list_a > list_a,R2: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H @ ( ring_i4611353245267337884t_unit @ R2 @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( H @ ( add_li174743652000525320t_unit @ R2 @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_960_ring__iso__memE_I2_J,axiom,
! [H: a > a,R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R2 @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R2 @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_961_ring__iso__memE_I2_J,axiom,
! [H: a > list_a,R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R2 @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R2 @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_962_ring__iso__memE_I2_J,axiom,
! [H: a > list_list_a,R2: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,X: a,Y: a] :
( ( member_a_list_list_a @ H @ ( ring_i4464730343205239444t_unit @ R2 @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R2 ) )
=> ( ( H @ ( mult_a_ring_ext_a_b @ R2 @ X @ Y ) )
= ( mult_l4853965630390486993t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_963_ring__iso__memE_I2_J,axiom,
! [H: list_a > a,R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R2 @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( H @ ( mult_l7073676228092353617t_unit @ R2 @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_964_ring__iso__memE_I2_J,axiom,
! [H: list_a > list_a,R2: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R2 @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( H @ ( mult_l7073676228092353617t_unit @ R2 @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_965_ring__iso__memE_I2_J,axiom,
! [H: list_a > list_list_a,R2: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,X: list_a,Y: list_a] :
( ( member6714375691612171394list_a @ H @ ( ring_i7582117978422105628t_unit @ R2 @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( H @ ( mult_l7073676228092353617t_unit @ R2 @ X @ Y ) )
= ( mult_l4853965630390486993t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_966_ring__iso__memE_I2_J,axiom,
! [H: list_list_a > a,R2: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H @ ( ring_i5684343068699926420it_a_b @ R2 @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( H @ ( mult_l4853965630390486993t_unit @ R2 @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_967_ring__iso__memE_I2_J,axiom,
! [H: list_list_a > list_a,R2: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H @ ( ring_i4611353245267337884t_unit @ R2 @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( H @ ( mult_l4853965630390486993t_unit @ R2 @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_968_ring__iso__memE_I2_J,axiom,
! [H: list_list_a > list_list_a,R2: partia2956882679547061052t_unit,S: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( member8231385768148312316list_a @ H @ ( ring_i6186174840089424918t_unit @ R2 @ S ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( H @ ( mult_l4853965630390486993t_unit @ R2 @ X @ Y ) )
= ( mult_l4853965630390486993t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_969_x_Ocgenideal__is__principalideal,axiom,
! [I4: list_a] :
( ( member_list_a @ I4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.cgenideal_is_principalideal
thf(fact_970_x_Oonepideal,axiom,
princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.onepideal
thf(fact_971_abelian__monoid_Ofinsum__Un__disjoint,axiom,
! [G2: partia2175431115845679010xt_a_b,A: set_a,B2: set_a,G: a > a] :
( ( abelian_monoid_a_b @ G2 )
=> ( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( ( inf_inf_set_a @ A @ B2 )
= bot_bot_set_a )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
=> ( ( finsum_a_b_a @ G2 @ G @ ( sup_sup_set_a @ A @ B2 ) )
= ( add_a_b @ G2 @ ( finsum_a_b_a @ G2 @ G @ A ) @ ( finsum_a_b_a @ G2 @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% abelian_monoid.finsum_Un_disjoint
thf(fact_972_abelian__monoid_Ofinsum__Un__disjoint,axiom,
! [G2: partia2175431115845679010xt_a_b,A: set_list_a,B2: set_list_a,G: list_a > a] :
( ( abelian_monoid_a_b @ G2 )
=> ( ( finite_finite_list_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
=> ( ( finsum_a_b_list_a @ G2 @ G @ ( sup_sup_set_list_a @ A @ B2 ) )
= ( add_a_b @ G2 @ ( finsum_a_b_list_a @ G2 @ G @ A ) @ ( finsum_a_b_list_a @ G2 @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% abelian_monoid.finsum_Un_disjoint
thf(fact_973_abelian__monoid_Ofinsum__Un__disjoint,axiom,
! [G2: partia2175431115845679010xt_a_b,A: set_set_a,B2: set_set_a,G: set_a > a] :
( ( abelian_monoid_a_b @ G2 )
=> ( ( finite_finite_set_a @ A )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( ( inf_inf_set_set_a @ A @ B2 )
= bot_bot_set_set_a )
=> ( ( member_set_a_a @ G
@ ( pi_set_a_a @ A
@ ^ [Uu: set_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
=> ( ( member_set_a_a @ G
@ ( pi_set_a_a @ B2
@ ^ [Uu: set_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
=> ( ( finsum_a_b_set_a @ G2 @ G @ ( sup_sup_set_set_a @ A @ B2 ) )
= ( add_a_b @ G2 @ ( finsum_a_b_set_a @ G2 @ G @ A ) @ ( finsum_a_b_set_a @ G2 @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% abelian_monoid.finsum_Un_disjoint
thf(fact_974_abelian__monoid_Ofinsum__Un__disjoint,axiom,
! [G2: partia2670972154091845814t_unit,A: set_a,B2: set_a,G: a > list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( ( inf_inf_set_a @ A @ B2 )
= bot_bot_set_a )
=> ( ( member_a_list_a @ G
@ ( pi_a_list_a @ A
@ ^ [Uu: a] : ( partia5361259788508890537t_unit @ G2 ) ) )
=> ( ( member_a_list_a @ G
@ ( pi_a_list_a @ B2
@ ^ [Uu: a] : ( partia5361259788508890537t_unit @ G2 ) ) )
=> ( ( finsum7322697649718157656unit_a @ G2 @ G @ ( sup_sup_set_a @ A @ B2 ) )
= ( add_li7652885771158616974t_unit @ G2 @ ( finsum7322697649718157656unit_a @ G2 @ G @ A ) @ ( finsum7322697649718157656unit_a @ G2 @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% abelian_monoid.finsum_Un_disjoint
thf(fact_975_abelian__monoid_Ofinsum__Un__disjoint,axiom,
! [G2: partia2175431115845679010xt_a_b,A: set_set_list_a,B2: set_set_list_a,G: set_list_a > a] :
( ( abelian_monoid_a_b @ G2 )
=> ( ( finite5282473924520328461list_a @ A )
=> ( ( finite5282473924520328461list_a @ B2 )
=> ( ( ( inf_in4657809108759609906list_a @ A @ B2 )
= bot_bo3186585308812441520list_a )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ B2
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
=> ( ( finsum7367453022336983110list_a @ G2 @ G @ ( sup_su4537662296134749976list_a @ A @ B2 ) )
= ( add_a_b @ G2 @ ( finsum7367453022336983110list_a @ G2 @ G @ A ) @ ( finsum7367453022336983110list_a @ G2 @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% abelian_monoid.finsum_Un_disjoint
thf(fact_976_abelian__monoid_Ofinsum__Un__disjoint,axiom,
! [G2: partia2670972154091845814t_unit,A: set_list_a,B2: set_list_a,G: list_a > list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( finite_finite_list_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
=> ( ( member_list_a_list_a @ G
@ ( pi_list_a_list_a @ A
@ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ G2 ) ) )
=> ( ( member_list_a_list_a @ G
@ ( pi_list_a_list_a @ B2
@ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ G2 ) ) )
=> ( ( finsum8721804980556663006list_a @ G2 @ G @ ( sup_sup_set_list_a @ A @ B2 ) )
= ( add_li7652885771158616974t_unit @ G2 @ ( finsum8721804980556663006list_a @ G2 @ G @ A ) @ ( finsum8721804980556663006list_a @ G2 @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% abelian_monoid.finsum_Un_disjoint
thf(fact_977_abelian__monoid_Ofinsum__Un__disjoint,axiom,
! [G2: partia2670972154091845814t_unit,A: set_set_a,B2: set_set_a,G: set_a > list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( finite_finite_set_a @ A )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( ( inf_inf_set_set_a @ A @ B2 )
= bot_bot_set_set_a )
=> ( ( member_set_a_list_a @ G
@ ( pi_set_a_list_a @ A
@ ^ [Uu: set_a] : ( partia5361259788508890537t_unit @ G2 ) ) )
=> ( ( member_set_a_list_a @ G
@ ( pi_set_a_list_a @ B2
@ ^ [Uu: set_a] : ( partia5361259788508890537t_unit @ G2 ) ) )
=> ( ( finsum1810151912383652536_set_a @ G2 @ G @ ( sup_sup_set_set_a @ A @ B2 ) )
= ( add_li7652885771158616974t_unit @ G2 @ ( finsum1810151912383652536_set_a @ G2 @ G @ A ) @ ( finsum1810151912383652536_set_a @ G2 @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% abelian_monoid.finsum_Un_disjoint
thf(fact_978_abelian__monoid_Ofinsum__Un__disjoint,axiom,
! [G2: partia2956882679547061052t_unit,A: set_a,B2: set_a,G: a > list_list_a] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( ( inf_inf_set_a @ A @ B2 )
= bot_bot_set_a )
=> ( ( member_a_list_list_a @ G
@ ( pi_a_list_list_a @ A
@ ^ [Uu: a] : ( partia2464479390973590831t_unit @ G2 ) ) )
=> ( ( member_a_list_list_a @ G
@ ( pi_a_list_list_a @ B2
@ ^ [Uu: a] : ( partia2464479390973590831t_unit @ G2 ) ) )
=> ( ( finsum463596448938265310unit_a @ G2 @ G @ ( sup_sup_set_a @ A @ B2 ) )
= ( add_li174743652000525320t_unit @ G2 @ ( finsum463596448938265310unit_a @ G2 @ G @ A ) @ ( finsum463596448938265310unit_a @ G2 @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% abelian_monoid.finsum_Un_disjoint
thf(fact_979_abelian__monoid_Ofinsum__Un__disjoint,axiom,
! [G2: partia2670972154091845814t_unit,A: set_set_list_a,B2: set_set_list_a,G: set_list_a > list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( finite5282473924520328461list_a @ A )
=> ( ( finite5282473924520328461list_a @ B2 )
=> ( ( ( inf_in4657809108759609906list_a @ A @ B2 )
= bot_bo3186585308812441520list_a )
=> ( ( member5910328476188217884list_a @ G
@ ( pi_set_list_a_list_a @ A
@ ^ [Uu: set_list_a] : ( partia5361259788508890537t_unit @ G2 ) ) )
=> ( ( member5910328476188217884list_a @ G
@ ( pi_set_list_a_list_a @ B2
@ ^ [Uu: set_list_a] : ( partia5361259788508890537t_unit @ G2 ) ) )
=> ( ( finsum1236349457110730942list_a @ G2 @ G @ ( sup_su4537662296134749976list_a @ A @ B2 ) )
= ( add_li7652885771158616974t_unit @ G2 @ ( finsum1236349457110730942list_a @ G2 @ G @ A ) @ ( finsum1236349457110730942list_a @ G2 @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% abelian_monoid.finsum_Un_disjoint
thf(fact_980_abelian__monoid_Ofinsum__Un__disjoint,axiom,
! [G2: partia2956882679547061052t_unit,A: set_list_a,B2: set_list_a,G: list_a > list_list_a] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( finite_finite_list_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
=> ( ( member6714375691612171394list_a @ G
@ ( pi_lis3067418140807155665list_a @ A
@ ^ [Uu: list_a] : ( partia2464479390973590831t_unit @ G2 ) ) )
=> ( ( member6714375691612171394list_a @ G
@ ( pi_lis3067418140807155665list_a @ B2
@ ^ [Uu: list_a] : ( partia2464479390973590831t_unit @ G2 ) ) )
=> ( ( finsum159916373282764644list_a @ G2 @ G @ ( sup_sup_set_list_a @ A @ B2 ) )
= ( add_li174743652000525320t_unit @ G2 @ ( finsum159916373282764644list_a @ G2 @ G @ A ) @ ( finsum159916373282764644list_a @ G2 @ G @ B2 ) ) ) ) ) ) ) ) ) ).
% abelian_monoid.finsum_Un_disjoint
thf(fact_981_s_Oring__hom__cring__axioms,axiom,
( ring_h1547129875642963619it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 ) ) ).
% s.ring_hom_cring_axioms
thf(fact_982_x_Oring__hom__cring__axioms,axiom,
( ring_h1547129875642963619it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ).
% x.ring_hom_cring_axioms
thf(fact_983_x_Oabelian__monoid__axioms,axiom,
abelia226231641709521465t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.abelian_monoid_axioms
thf(fact_984_x_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X2 )
= X2 ) )
=> ( U
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.one_unique
thf(fact_985_x_Oinv__unique,axiom,
! [Y: list_a,X: list_a,Y2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% x.inv_unique
thf(fact_986_x_Or__distr,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ Y ) ) ) ) ) ) ).
% x.r_distr
thf(fact_987_x_Ol__distr,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.l_distr
thf(fact_988_x_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.m_assoc
thf(fact_989_x_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).
% x.m_comm
thf(fact_990_x_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) ) ) ) ) ) ).
% x.m_lcomm
thf(fact_991_x_Oadd_Ol__cancel,axiom,
! [C: list_a,A2: list_a,B3: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B3 ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A2 = B3 ) ) ) ) ) ).
% x.add.l_cancel
thf(fact_992_x_Oadd_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.add.m_assoc
thf(fact_993_x_Oadd_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).
% x.add.m_comm
thf(fact_994_x_Oadd_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) ) ) ) ) ) ).
% x.add.m_lcomm
thf(fact_995_x_Oadd_Or__cancel,axiom,
! [A2: list_a,C: list_a,B3: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ C )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 @ C ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A2 = B3 ) ) ) ) ) ).
% x.add.r_cancel
thf(fact_996_x_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_a,F: a > list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( F @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.finprod_one_eqI
thf(fact_997_x_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_list_a,F: list_a > list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( F @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.finprod_one_eqI
thf(fact_998_x_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_list_list_a,F: list_list_a > list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( ( F @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum666616272051086308list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.finprod_one_eqI
thf(fact_999_x_Ofinsum__cong_H,axiom,
! [A: set_a,B2: set_a,G: a > list_a,F: a > list_a] :
( ( A = B2 )
=> ( ( member_a_list_a @ G
@ ( pi_a_list_a @ B2
@ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [I2: a] :
( ( member_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B2 ) ) ) ) ) ).
% x.finsum_cong'
thf(fact_1000_x_Ofinsum__cong_H,axiom,
! [A: set_list_a,B2: set_list_a,G: list_a > list_a,F: list_a > list_a] :
( ( A = B2 )
=> ( ( member_list_a_list_a @ G
@ ( pi_list_a_list_a @ B2
@ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B2 ) ) ) ) ) ).
% x.finsum_cong'
thf(fact_1001_x_Ofinsum__cong_H,axiom,
! [A: set_list_list_a,B2: set_list_list_a,G: list_list_a > list_a,F: list_list_a > list_a] :
( ( A = B2 )
=> ( ( member7168557129179038582list_a @ G
@ ( pi_lis8207908228422549957list_a @ B2
@ ^ [Uu: list_list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [I2: list_list_a] :
( ( member_list_list_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finsum666616272051086308list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( finsum666616272051086308list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B2 ) ) ) ) ) ).
% x.finsum_cong'
thf(fact_1002_x_Ocring__fieldI2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A6: list_a] :
( ( member_list_a @ A6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A6
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A6 @ X4 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.cring_fieldI2
thf(fact_1003_x_Ominus__unique,axiom,
! [Y: list_a,X: list_a,Y2: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% x.minus_unique
thf(fact_1004_x_Oadd_Or__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.r_inv_ex
thf(fact_1005_x_Oadd_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.one_unique
thf(fact_1006_x_Oadd_Ol__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.l_inv_ex
thf(fact_1007_x_Oadd_Oinv__comm,axiom,
! [X: list_a,Y: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.inv_comm
thf(fact_1008_x_Ofinsum__rdistr,axiom,
! [A: set_a,A2: list_a,F: a > list_a] :
( ( finite_finite_a @ A )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a_list_a @ F
@ ( pi_a_list_a @ A
@ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) )
= ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [I: a] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ ( F @ I ) )
@ A ) ) ) ) ) ).
% x.finsum_rdistr
thf(fact_1009_x_Ofinsum__rdistr,axiom,
! [A: set_list_a,A2: list_a,F: list_a > list_a] :
( ( finite_finite_list_a @ A )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a_list_a @ F
@ ( pi_list_a_list_a @ A
@ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) )
= ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [I: list_a] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ ( F @ I ) )
@ A ) ) ) ) ) ).
% x.finsum_rdistr
thf(fact_1010_x_Ofinsum__ldistr,axiom,
! [A: set_a,A2: list_a,F: a > list_a] :
( ( finite_finite_a @ A )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a_list_a @ F
@ ( pi_a_list_a @ A
@ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) @ A2 )
= ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [I: a] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ I ) @ A2 )
@ A ) ) ) ) ) ).
% x.finsum_ldistr
thf(fact_1011_x_Ofinsum__ldistr,axiom,
! [A: set_list_a,A2: list_a,F: list_a > list_a] :
( ( finite_finite_list_a @ A )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a_list_a @ F
@ ( pi_list_a_list_a @ A
@ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) @ A2 )
= ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [I: list_a] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ I ) @ A2 )
@ A ) ) ) ) ) ).
% x.finsum_ldistr
thf(fact_1012_x_Ofinsum__singleton,axiom,
! [I4: list_list_a,A: set_list_list_a,F: list_list_a > list_a] :
( ( member_list_list_a @ I4 @ A )
=> ( ( finite1660835950917165235list_a @ A )
=> ( ( member7168557129179038582list_a @ F
@ ( pi_lis8207908228422549957list_a @ A
@ ^ [Uu: list_list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum666616272051086308list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [J: list_list_a] : ( if_list_a @ ( I4 = J ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% x.finsum_singleton
thf(fact_1013_x_Ofinsum__singleton,axiom,
! [I4: a,A: set_a,F: a > list_a] :
( ( member_a @ I4 @ A )
=> ( ( finite_finite_a @ A )
=> ( ( member_a_list_a @ F
@ ( pi_a_list_a @ A
@ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [J: a] : ( if_list_a @ ( I4 = J ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% x.finsum_singleton
thf(fact_1014_x_Ofinsum__singleton,axiom,
! [I4: list_a,A: set_list_a,F: list_a > list_a] :
( ( member_list_a @ I4 @ A )
=> ( ( finite_finite_list_a @ A )
=> ( ( member_list_a_list_a @ F
@ ( pi_list_a_list_a @ A
@ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [J: list_a] : ( if_list_a @ ( I4 = J ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% x.finsum_singleton
thf(fact_1015_x_Oadd_Ofinprod__singleton__swap,axiom,
! [I4: list_list_a,A: set_list_list_a,F: list_list_a > list_a] :
( ( member_list_list_a @ I4 @ A )
=> ( ( finite1660835950917165235list_a @ A )
=> ( ( member7168557129179038582list_a @ F
@ ( pi_lis8207908228422549957list_a @ A
@ ^ [Uu: list_list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum666616272051086308list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [J: list_list_a] : ( if_list_a @ ( J = I4 ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% x.add.finprod_singleton_swap
thf(fact_1016_x_Oadd_Ofinprod__singleton__swap,axiom,
! [I4: a,A: set_a,F: a > list_a] :
( ( member_a @ I4 @ A )
=> ( ( finite_finite_a @ A )
=> ( ( member_a_list_a @ F
@ ( pi_a_list_a @ A
@ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [J: a] : ( if_list_a @ ( J = I4 ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% x.add.finprod_singleton_swap
thf(fact_1017_x_Oadd_Ofinprod__singleton__swap,axiom,
! [I4: list_a,A: set_list_a,F: list_a > list_a] :
( ( member_list_a @ I4 @ A )
=> ( ( finite_finite_list_a @ A )
=> ( ( member_list_a_list_a @ F
@ ( pi_list_a_list_a @ A
@ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [J: list_a] : ( if_list_a @ ( J = I4 ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% x.add.finprod_singleton_swap
thf(fact_1018_x_Ofinsum__Un__Int,axiom,
! [A: set_a,B2: set_a,G: a > list_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( member_a_list_a @ G
@ ( pi_a_list_a @ A
@ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_a_list_a @ G
@ ( pi_a_list_a @ B2
@ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_a @ A @ B2 ) ) @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( inf_inf_set_a @ A @ B2 ) ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A ) @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B2 ) ) ) ) ) ) ) ).
% x.finsum_Un_Int
thf(fact_1019_x_Ofinsum__Un__Int,axiom,
! [A: set_list_a,B2: set_list_a,G: list_a > list_a] :
( ( finite_finite_list_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ( member_list_a_list_a @ G
@ ( pi_list_a_list_a @ A
@ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a_list_a @ G
@ ( pi_list_a_list_a @ B2
@ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_list_a @ A @ B2 ) ) @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( inf_inf_set_list_a @ A @ B2 ) ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A ) @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B2 ) ) ) ) ) ) ) ).
% x.finsum_Un_Int
thf(fact_1020_x_Oadd_Ofinprod__mono__neutral__cong__right,axiom,
! [B2: set_list_list_a,A: set_list_list_a,G: list_list_a > list_a,H: list_list_a > list_a] :
( ( finite1660835950917165235list_a @ B2 )
=> ( ( ord_le8488217952732425610list_a @ A @ B2 )
=> ( ! [I2: list_list_a] :
( ( member_list_list_a @ I2 @ ( minus_5335179877275218001list_a @ B2 @ A ) )
=> ( ( G @ I2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member7168557129179038582list_a @ G
@ ( pi_lis8207908228422549957list_a @ B2
@ ^ [Uu: list_list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum666616272051086308list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B2 )
= ( finsum666616272051086308list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A ) ) ) ) ) ) ) ).
% x.add.finprod_mono_neutral_cong_right
thf(fact_1021_x_Oadd_Ofinprod__mono__neutral__cong__right,axiom,
! [B2: set_a,A: set_a,G: a > list_a,H: a > list_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ B2 @ A ) )
=> ( ( G @ I2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_list_a @ G
@ ( pi_a_list_a @ B2
@ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B2 )
= ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A ) ) ) ) ) ) ) ).
% x.add.finprod_mono_neutral_cong_right
thf(fact_1022_x_Oadd_Ofinprod__mono__neutral__cong__right,axiom,
! [B2: set_list_a,A: set_list_a,G: list_a > list_a,H: list_a > list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ ( minus_646659088055828811list_a @ B2 @ A ) )
=> ( ( G @ I2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_a_list_a @ G
@ ( pi_list_a_list_a @ B2
@ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B2 )
= ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A ) ) ) ) ) ) ) ).
% x.add.finprod_mono_neutral_cong_right
thf(fact_1023_x_Oadd_Ofinprod__mono__neutral__cong__left,axiom,
! [B2: set_list_list_a,A: set_list_list_a,H: list_list_a > list_a,G: list_list_a > list_a] :
( ( finite1660835950917165235list_a @ B2 )
=> ( ( ord_le8488217952732425610list_a @ A @ B2 )
=> ( ! [I2: list_list_a] :
( ( member_list_list_a @ I2 @ ( minus_5335179877275218001list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member7168557129179038582list_a @ H
@ ( pi_lis8207908228422549957list_a @ B2
@ ^ [Uu: list_list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum666616272051086308list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A )
= ( finsum666616272051086308list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B2 ) ) ) ) ) ) ) ).
% x.add.finprod_mono_neutral_cong_left
thf(fact_1024_x_Oadd_Ofinprod__mono__neutral__cong__left,axiom,
! [B2: set_a,A: set_a,H: a > list_a,G: a > list_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_list_a @ H
@ ( pi_a_list_a @ B2
@ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A )
= ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B2 ) ) ) ) ) ) ) ).
% x.add.finprod_mono_neutral_cong_left
thf(fact_1025_x_Oadd_Ofinprod__mono__neutral__cong__left,axiom,
! [B2: set_list_a,A: set_list_a,H: list_a > list_a,G: list_a > list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ ( minus_646659088055828811list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_a_list_a @ H
@ ( pi_list_a_list_a @ B2
@ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A )
= ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B2 ) ) ) ) ) ) ) ).
% x.add.finprod_mono_neutral_cong_left
thf(fact_1026_x_Ofinsum__Un__disjoint,axiom,
! [A: set_a,B2: set_a,G: a > list_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( ( inf_inf_set_a @ A @ B2 )
= bot_bot_set_a )
=> ( ( member_a_list_a @ G
@ ( pi_a_list_a @ A
@ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_a_list_a @ G
@ ( pi_a_list_a @ B2
@ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_a @ A @ B2 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A ) @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B2 ) ) ) ) ) ) ) ) ).
% x.finsum_Un_disjoint
thf(fact_1027_x_Ofinsum__Un__disjoint,axiom,
! [A: set_list_a,B2: set_list_a,G: list_a > list_a] :
( ( finite_finite_list_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
=> ( ( member_list_a_list_a @ G
@ ( pi_list_a_list_a @ A
@ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a_list_a @ G
@ ( pi_list_a_list_a @ B2
@ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_list_a @ A @ B2 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A ) @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B2 ) ) ) ) ) ) ) ) ).
% x.finsum_Un_disjoint
thf(fact_1028_x_Ofinsum__Un__disjoint,axiom,
! [A: set_set_a,B2: set_set_a,G: set_a > list_a] :
( ( finite_finite_set_a @ A )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( ( inf_inf_set_set_a @ A @ B2 )
= bot_bot_set_set_a )
=> ( ( member_set_a_list_a @ G
@ ( pi_set_a_list_a @ A
@ ^ [Uu: set_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_set_a_list_a @ G
@ ( pi_set_a_list_a @ B2
@ ^ [Uu: set_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum1810151912383652536_set_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_set_a @ A @ B2 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum1810151912383652536_set_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A ) @ ( finsum1810151912383652536_set_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B2 ) ) ) ) ) ) ) ) ).
% x.finsum_Un_disjoint
thf(fact_1029_x_Ofinsum__Un__disjoint,axiom,
! [A: set_set_list_a,B2: set_set_list_a,G: set_list_a > list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( finite5282473924520328461list_a @ B2 )
=> ( ( ( inf_in4657809108759609906list_a @ A @ B2 )
= bot_bo3186585308812441520list_a )
=> ( ( member5910328476188217884list_a @ G
@ ( pi_set_list_a_list_a @ A
@ ^ [Uu: set_list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member5910328476188217884list_a @ G
@ ( pi_set_list_a_list_a @ B2
@ ^ [Uu: set_list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum1236349457110730942list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_su4537662296134749976list_a @ A @ B2 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum1236349457110730942list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A ) @ ( finsum1236349457110730942list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B2 ) ) ) ) ) ) ) ) ).
% x.finsum_Un_disjoint
thf(fact_1030_x_Oadd_Ofinprod__mono__neutral__cong,axiom,
! [B2: set_list_list_a,A: set_list_list_a,H: list_list_a > list_a,G: list_list_a > list_a] :
( ( finite1660835950917165235list_a @ B2 )
=> ( ( finite1660835950917165235list_a @ A )
=> ( ! [I2: list_list_a] :
( ( member_list_list_a @ I2 @ ( minus_5335179877275218001list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [I2: list_list_a] :
( ( member_list_list_a @ I2 @ ( minus_5335179877275218001list_a @ A @ B2 ) )
=> ( ( G @ I2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( inf_in7423150557312423384list_a @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member7168557129179038582list_a @ G
@ ( pi_lis8207908228422549957list_a @ A
@ ^ [Uu: list_list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member7168557129179038582list_a @ H
@ ( pi_lis8207908228422549957list_a @ B2
@ ^ [Uu: list_list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum666616272051086308list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A )
= ( finsum666616272051086308list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% x.add.finprod_mono_neutral_cong
thf(fact_1031_x_Oadd_Ofinprod__mono__neutral__cong,axiom,
! [B2: set_a,A: set_a,H: a > list_a,G: a > list_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ A )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ A @ B2 ) )
=> ( ( G @ I2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( inf_inf_set_a @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_list_a @ G
@ ( pi_a_list_a @ A
@ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_a_list_a @ H
@ ( pi_a_list_a @ B2
@ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A )
= ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% x.add.finprod_mono_neutral_cong
thf(fact_1032_x_Oadd_Ofinprod__mono__neutral__cong,axiom,
! [B2: set_list_a,A: set_list_a,H: list_a > list_a,G: list_a > list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( finite_finite_list_a @ A )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ ( minus_646659088055828811list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ ( minus_646659088055828811list_a @ A @ B2 ) )
=> ( ( G @ I2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( inf_inf_set_list_a @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_a_list_a @ G
@ ( pi_list_a_list_a @ A
@ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a_list_a @ H
@ ( pi_list_a_list_a @ B2
@ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A )
= ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% x.add.finprod_mono_neutral_cong
thf(fact_1033_x_Or__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X ) ) ).
% x.r_one
thf(fact_1034_x_Ol__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% x.l_one
thf(fact_1035_x_Oone__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.one_closed
thf(fact_1036_x_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.m_closed
thf(fact_1037_x_Oadd_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.m_closed
thf(fact_1038_x_Oadd_Oright__cancel,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% x.add.right_cancel
thf(fact_1039_x_Ofinsum__empty,axiom,
! [F: a > list_a] :
( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ bot_bot_set_a )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.finsum_empty
thf(fact_1040_x_Ofinsum__empty,axiom,
! [F: list_a > list_a] :
( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ bot_bot_set_list_a )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.finsum_empty
thf(fact_1041_x_Ofinsum__empty,axiom,
! [F: set_a > list_a] :
( ( finsum1810151912383652536_set_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ bot_bot_set_set_a )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.finsum_empty
thf(fact_1042_x_Ofinsum__empty,axiom,
! [F: set_list_a > list_a] :
( ( finsum1236349457110730942list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ bot_bo3186585308812441520list_a )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.finsum_empty
thf(fact_1043_x_Or__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.r_null
thf(fact_1044_x_Ol__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.l_null
thf(fact_1045_x_Or__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X ) ) ).
% x.r_zero
thf(fact_1046_x_Ol__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% x.l_zero
thf(fact_1047_x_Oadd_Or__cancel__one_H,axiom,
! [X: list_a,A2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ X ) )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one'
thf(fact_1048_x_Oadd_Or__cancel__one,axiom,
! [X: list_a,A2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ X )
= X )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one
thf(fact_1049_x_Oadd_Ol__cancel__one_H,axiom,
! [X: list_a,A2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A2 ) )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one'
thf(fact_1050_x_Oadd_Ol__cancel__one,axiom,
! [X: list_a,A2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A2 )
= X )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one
thf(fact_1051_x_Ofinsum__infinite,axiom,
! [A: set_a,F: a > list_a] :
( ~ ( finite_finite_a @ A )
=> ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.finsum_infinite
thf(fact_1052_x_Ofinsum__infinite,axiom,
! [A: set_list_a,F: list_a > list_a] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.finsum_infinite
thf(fact_1053_x_Oring_Ohom__one,axiom,
( ( eval_a_b @ r @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x )
= ( one_a_ring_ext_a_b @ r ) ) ).
% x.ring.hom_one
thf(fact_1054_s_Oring_Ohom__one,axiom,
( ( eval_a_b @ r @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ s2 )
= ( one_a_ring_ext_a_b @ r ) ) ).
% s.ring.hom_one
thf(fact_1055_x_Oring_Ohom__add,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ x )
= ( add_a_b @ r @ ( eval_a_b @ r @ X @ x ) @ ( eval_a_b @ r @ Y @ x ) ) ) ) ) ).
% x.ring.hom_add
thf(fact_1056_s_Oring_Ohom__add,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ s2 )
= ( add_a_b @ r @ ( eval_a_b @ r @ X @ s2 ) @ ( eval_a_b @ r @ Y @ s2 ) ) ) ) ) ).
% s.ring.hom_add
thf(fact_1057_x_Oring_Ohom__mult,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ x )
= ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ X @ x ) @ ( eval_a_b @ r @ Y @ x ) ) ) ) ) ).
% x.ring.hom_mult
thf(fact_1058_s_Oring_Ohom__mult,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ s2 )
= ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ X @ s2 ) @ ( eval_a_b @ r @ Y @ s2 ) ) ) ) ) ).
% s.ring.hom_mult
thf(fact_1059_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_1060_bot__set__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a @ bot_bot_list_a_o ) ) ).
% bot_set_def
thf(fact_1061_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_1062_bot__set__def,axiom,
( bot_bo3186585308812441520list_a
= ( collect_set_list_a @ bot_bot_set_list_a_o ) ) ).
% bot_set_def
thf(fact_1063_ring__hom__cring_Ohom__zero,axiom,
! [R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H: a > a] :
( ( ring_h661254511236296859_b_a_b @ R2 @ S @ H )
=> ( ( H @ ( zero_a_b @ R2 ) )
= ( zero_a_b @ S ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_1064_ring__hom__cring_Ohom__zero,axiom,
! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H: a > list_a] :
( ( ring_h8279546866833948963t_unit @ R2 @ S @ H )
=> ( ( H @ ( zero_a_b @ R2 ) )
= ( zero_l4142658623432671053t_unit @ S ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_1065_ring__hom__cring_Ohom__zero,axiom,
! [R2: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H: list_a > list_a] :
( ( ring_h8282015026914974507t_unit @ R2 @ S @ H )
=> ( ( H @ ( zero_l4142658623432671053t_unit @ R2 ) )
= ( zero_l4142658623432671053t_unit @ S ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_1066_ring__hom__cring_Ohom__zero,axiom,
! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H: list_a > a] :
( ( ring_h1547129875642963619it_a_b @ R2 @ S @ H )
=> ( ( H @ ( zero_l4142658623432671053t_unit @ R2 ) )
= ( zero_a_b @ S ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_1067_ring_Ounfold__congs_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,R5: partia2175431115845679010xt_a_b,V3: a,F: a > a,F3: a > a] :
( ( R = R5 )
=> ( ( ( zero_a_b @ R5 )
= V3 )
=> ( ! [V4: a] :
( ( V4 = V3 )
=> ( ( F @ V4 )
= ( F3 @ V4 ) ) )
=> ( ( zero_update_a_b @ F @ R )
= ( zero_update_a_b @ F3 @ R5 ) ) ) ) ) ).
% ring.unfold_congs(4)
thf(fact_1068_ring_Ounfold__congs_I4_J,axiom,
! [R: partia2670972154091845814t_unit,R5: partia2670972154091845814t_unit,V3: list_a,F: list_a > list_a,F3: list_a > list_a] :
( ( R = R5 )
=> ( ( ( zero_l4142658623432671053t_unit @ R5 )
= V3 )
=> ( ! [V4: list_a] :
( ( V4 = V3 )
=> ( ( F @ V4 )
= ( F3 @ V4 ) ) )
=> ( ( zero_u1196785550890449590t_unit @ F @ R )
= ( zero_u1196785550890449590t_unit @ F3 @ R5 ) ) ) ) ) ).
% ring.unfold_congs(4)
thf(fact_1069_ring_Ofold__congs_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,R5: partia2175431115845679010xt_a_b,V3: a,F: a > a,F3: a > a] :
( ( R = R5 )
=> ( ( ( zero_a_b @ R5 )
= V3 )
=> ( ! [V4: a] :
( ( V3 = V4 )
=> ( ( F @ V4 )
= ( F3 @ V4 ) ) )
=> ( ( zero_update_a_b @ F @ R )
= ( zero_update_a_b @ F3 @ R5 ) ) ) ) ) ).
% ring.fold_congs(4)
thf(fact_1070_ring_Ofold__congs_I4_J,axiom,
! [R: partia2670972154091845814t_unit,R5: partia2670972154091845814t_unit,V3: list_a,F: list_a > list_a,F3: list_a > list_a] :
( ( R = R5 )
=> ( ( ( zero_l4142658623432671053t_unit @ R5 )
= V3 )
=> ( ! [V4: list_a] :
( ( V3 = V4 )
=> ( ( F @ V4 )
= ( F3 @ V4 ) ) )
=> ( ( zero_u1196785550890449590t_unit @ F @ R )
= ( zero_u1196785550890449590t_unit @ F3 @ R5 ) ) ) ) ) ).
% ring.fold_congs(4)
thf(fact_1071_semiring_Oaxioms_I1_J,axiom,
! [R2: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R2 )
=> ( abelia226231641709521465t_unit @ R2 ) ) ).
% semiring.axioms(1)
thf(fact_1072_finite__number__of__roots,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( finite_finite_a @ ( collect_a @ ( polyno4133073214067823460ot_a_b @ r @ P2 ) ) ) ) ).
% finite_number_of_roots
thf(fact_1073_s_Oring_Ois__abelian__group__hom,axiom,
( abelia8217020544048703197it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 ) ) ).
% s.ring.is_abelian_group_hom
thf(fact_1074_x_Oring_Ois__abelian__group__hom,axiom,
( abelia8217020544048703197it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ).
% x.ring.is_abelian_group_hom
thf(fact_1075_x_Oa__lcos__mult__one,axiom,
! [M: set_list_a] :
( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ M )
= M ) ) ).
% x.a_lcos_mult_one
thf(fact_1076_x_Osemiring__axioms,axiom,
semiri2871908745932252451t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.semiring_axioms
thf(fact_1077_x_Oa__l__coset__subset__G,axiom,
! [H2: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ H2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.a_l_coset_subset_G
thf(fact_1078_is__root__poly__mult__imp__is__root,axiom,
! [P2: list_a,Q2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q2 ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q2 @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_1079_x_Oa__lcos__m__assoc,axiom,
! [M: set_list_a,G: list_a,H: list_a] :
( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ M ) )
= ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ H ) @ M ) ) ) ) ) ).
% x.a_lcos_m_assoc
thf(fact_1080_x_Omonoid__cancelI,axiom,
( ! [A6: list_a,B4: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ A6 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ B4 ) )
=> ( ( member_list_a @ A6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A6 = B4 ) ) ) ) )
=> ( ! [A6: list_a,B4: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A6 @ C2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B4 @ C2 ) )
=> ( ( member_list_a @ A6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A6 = B4 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.monoid_cancelI
thf(fact_1081_x_Oideal__is__subalgebra,axiom,
! [K: set_list_a,I3: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ideal_8896367198367571637t_unit @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( embedd1768981623711841426t_unit @ K @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.ideal_is_subalgebra
thf(fact_1082_field__intro2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) ) )
=> ( field_a_b @ r ) ) ) ).
% field_intro2
thf(fact_1083_ring__irreducibleI,axiom,
! [R: a] :
( ( member_a @ R @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R @ ( units_a_ring_ext_a_b @ r ) )
=> ( ! [A6: a,B4: a] :
( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R
= ( mult_a_ring_ext_a_b @ r @ A6 @ B4 ) )
=> ( ( member_a @ A6 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B4 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ r @ R ) ) ) ) ).
% ring_irreducibleI
thf(fact_1084_zeroideal,axiom,
ideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroideal
thf(fact_1085_genideal__self_H,axiom,
! [I4: a] :
( ( member_a @ I4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I4 @ ( genideal_a_b @ r @ ( insert_a @ I4 @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_1086_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_1087_x_Osubalgebra__inter,axiom,
! [K: set_list_a,V: set_list_a,V2: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1768981623711841426t_unit @ K @ V2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( embedd1768981623711841426t_unit @ K @ ( inf_inf_set_list_a @ V @ V2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subalgebra_inter
thf(fact_1088_zeromaximalideal,axiom,
maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeromaximalideal
thf(fact_1089_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_1090_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_1091_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_1092_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_1093_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_1094_x_Osubalgebra__in__carrier,axiom,
! [K: set_list_a,V: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ V @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subalgebra_in_carrier
thf(fact_1095_x_Ocarrier__is__subalgebra,axiom,
! [K: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( embedd1768981623711841426t_unit @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.carrier_is_subalgebra
thf(fact_1096_Idl__subset__ideal_H,axiom,
! [A2: a,B3: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B3 @ bot_bot_set_a ) ) )
= ( member_a @ A2 @ ( genideal_a_b @ r @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_1097_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_1098_cgenideal__eq__genideal,axiom,
! [I4: a] :
( ( member_a @ I4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( cgenid547466209912283029xt_a_b @ r @ I4 )
= ( genideal_a_b @ r @ ( insert_a @ I4 @ bot_bot_set_a ) ) ) ) ).
% cgenideal_eq_genideal
thf(fact_1099_zeromaximalideal__fieldI,axiom,
( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
=> ( field_a_b @ r ) ) ).
% zeromaximalideal_fieldI
thf(fact_1100_zeromaximalideal__eq__field,axiom,
( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
= ( field_a_b @ r ) ) ).
% zeromaximalideal_eq_field
thf(fact_1101_local_Ofield__Units,axiom,
( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% local.field_Units
thf(fact_1102_cring__fieldI,axiom,
( ( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( field_a_b @ r ) ) ).
% cring_fieldI
thf(fact_1103_primeideal__iff__prime,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P2 ) @ r )
= ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% primeideal_iff_prime
thf(fact_1104_field__iff__prime,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( field_6045675692312731021t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A2 ) ) )
= ( ring_ring_prime_a_b @ r @ A2 ) ) ) ).
% field_iff_prime
thf(fact_1105_s_Oring_Oimg__is__subalgebra,axiom,
! [K: set_list_a,V: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( embedd9027525575939734154ra_a_b
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 )
@ K )
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 )
@ V )
@ r ) ) ) ).
% s.ring.img_is_subalgebra
thf(fact_1106_x_OUnits__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_closed
thf(fact_1107_maximalideal__prime,axiom,
! [I3: set_a] :
( ( maximalideal_a_b @ I3 @ r )
=> ( primeideal_a_b @ I3 @ r ) ) ).
% maximalideal_prime
thf(fact_1108_x_Oprod__unit__l,axiom,
! [A2: list_a,B3: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B3 ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ B3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.prod_unit_l
thf(fact_1109_x_Oprod__unit__r,axiom,
! [A2: list_a,B3: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B3 ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.prod_unit_r
thf(fact_1110_x_Ounit__factor,axiom,
! [A2: list_a,B3: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B3 ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.unit_factor
thf(fact_1111_x_OUnits__inv__comm,axiom,
! [X: list_a,Y: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.Units_inv_comm
thf(fact_1112_x_Oideal__eq__carrier__iff,axiom,
! [A2: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 ) )
= ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.ideal_eq_carrier_iff
thf(fact_1113_zeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroprimeideal
thf(fact_1114_x_OUnits__r__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Units_r_inv_ex
thf(fact_1115_x_OUnits__l__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Units_l_inv_ex
thf(fact_1116_x_Ozeroideal,axiom,
ideal_8896367198367571637t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.zeroideal
thf(fact_1117_x_Ogenideal__self_H,axiom,
! [I4: list_a] :
( ( member_list_a @ I4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ I4 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I4 @ bot_bot_set_list_a ) ) ) ) ).
% x.genideal_self'
thf(fact_1118_x_Ogenideal__zero,axiom,
( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ).
% x.genideal_zero
thf(fact_1119_x_Ozeromaximalideal__fieldI,axiom,
( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.zeromaximalideal_fieldI
thf(fact_1120_x_Ozeromaximalideal__eq__field,axiom,
( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.zeromaximalideal_eq_field
thf(fact_1121_all__ideals,axiom,
( ( collect_set_a
@ ^ [I6: set_a] : ( ideal_a_b @ I6 @ r ) )
= ( insert_set_a @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ ( insert_set_a @ ( partia707051561876973205xt_a_b @ r ) @ bot_bot_set_set_a ) ) ) ).
% all_ideals
thf(fact_1122_x_Ozeropideal,axiom,
princi8786919440553033881t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.zeropideal
thf(fact_1123_x_Oone__zeroI,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
=> ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.one_zeroI
thf(fact_1124_x_Oone__zeroD,axiom,
( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ).
% x.one_zeroD
thf(fact_1125_x_Ocarrier__one__zero,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.carrier_one_zero
thf(fact_1126_x_Ocarrier__one__not__zero,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.carrier_one_not_zero
thf(fact_1127_x_OIdl__subset__ideal_H,axiom,
! [A2: list_a,B3: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B3 @ bot_bot_set_list_a ) ) )
= ( member_list_a @ A2 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B3 @ bot_bot_set_list_a ) ) ) ) ) ) ).
% x.Idl_subset_ideal'
thf(fact_1128_x_Ocring__fieldI,axiom,
( ( ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.cring_fieldI
thf(fact_1129_x_Ogenideal__one,axiom,
( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.genideal_one
thf(fact_1130_x_Ocgenideal__eq__genideal,axiom,
! [I4: list_a] :
( ( member_list_a @ I4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 )
= ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I4 @ bot_bot_set_list_a ) ) ) ) ).
% x.cgenideal_eq_genideal
thf(fact_1131_trivialideals__eq__field,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( ( ( collect_set_a
@ ^ [I6: set_a] : ( ideal_a_b @ I6 @ r ) )
= ( insert_set_a @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ ( insert_set_a @ ( partia707051561876973205xt_a_b @ r ) @ bot_bot_set_set_a ) ) )
= ( field_a_b @ r ) ) ) ).
% trivialideals_eq_field
thf(fact_1132_trivialideals__fieldI,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( ( ( collect_set_a
@ ^ [I6: set_a] : ( ideal_a_b @ I6 @ r ) )
= ( insert_set_a @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ ( insert_set_a @ ( partia707051561876973205xt_a_b @ r ) @ bot_bot_set_set_a ) ) )
=> ( field_a_b @ r ) ) ) ).
% trivialideals_fieldI
thf(fact_1133_x_Ofield__intro2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.field_intro2
thf(fact_1134_x_Otrivialideals__fieldI,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
=> ( ( ( collect_set_list_a
@ ^ [I6: set_list_a] : ( ideal_8896367198367571637t_unit @ I6 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( insert_set_list_a @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( insert_set_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bo3186585308812441520list_a ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.trivialideals_fieldI
thf(fact_1135_x_Otrivialideals__eq__field,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
=> ( ( ( collect_set_list_a
@ ^ [I6: set_list_a] : ( ideal_8896367198367571637t_unit @ I6 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( insert_set_list_a @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( insert_set_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bo3186585308812441520list_a ) ) )
= ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.trivialideals_eq_field
thf(fact_1136_x_Oring_Oimg__is__subalgebra,axiom,
! [K: set_list_a,V: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( embedd9027525575939734154ra_a_b
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ K )
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ V )
@ r ) ) ) ).
% x.ring.img_is_subalgebra
thf(fact_1137_x_OUnits__m__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Units_m_closed
thf(fact_1138_x_OUnits__one__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.Units_one_closed
thf(fact_1139_x_OUnits__l__cancel,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% x.Units_l_cancel
thf(fact_1140_x_Ofinite__ring__finite__units,axiom,
( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( finite_finite_list_a @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.finite_ring_finite_units
thf(fact_1141_FactRing__zeroideal_I2_J,axiom,
is_rin9099215527551818550t_unit @ r @ ( factRing_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% FactRing_zeroideal(2)
thf(fact_1142_FactRing__zeroideal_I1_J,axiom,
is_rin6001486760346555702it_a_b @ ( factRing_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) @ r ).
% FactRing_zeroideal(1)
thf(fact_1143_s_Oring_Onon__trivial__field__hom__imp__inj,axiom,
( ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% s.ring.non_trivial_field_hom_imp_inj
thf(fact_1144_x_Oring_Onon__trivial__field__hom__imp__inj,axiom,
( ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.ring.non_trivial_field_hom_imp_inj
thf(fact_1145_domain__iff__prime,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A2 ) ) )
= ( ring_ring_prime_a_b @ r @ A2 ) ) ) ).
% domain_iff_prime
thf(fact_1146_add_Osurj__const__mult,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( image_a_a @ ( add_a_b @ r @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) )
= ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.surj_const_mult
thf(fact_1147_x_Oadd_Osurj__const__mult,axiom,
! [A2: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( image_list_a_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.surj_const_mult
thf(fact_1148_quot__domain__iff__primeideal,axiom,
! [P: set_a] :
( ( ideal_a_b @ P @ r )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ P ) )
= ( primeideal_a_b @ P @ r ) ) ) ).
% quot_domain_iff_primeideal
thf(fact_1149_quot__domain__imp__primeideal,axiom,
! [P: set_a] :
( ( ideal_a_b @ P @ r )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ P ) )
=> ( primeideal_a_b @ P @ r ) ) ) ).
% quot_domain_imp_primeideal
thf(fact_1150_x_OFactRing__zeroideal_I2_J,axiom,
is_rin2993610189962786360t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ).
% x.FactRing_zeroideal(2)
thf(fact_1151_x_OFactRing__zeroideal_I1_J,axiom,
is_rin4843644836746533432t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.FactRing_zeroideal(1)
thf(fact_1152_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_1153_add_Oinj__on__cmult,axiom,
! [C: a] :
( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( inj_on_a_a @ ( add_a_b @ r @ C ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.inj_on_cmult
thf(fact_1154_add_Oinj__on__multc,axiom,
! [C: a] :
( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( inj_on_a_a
@ ^ [X3: a] : ( add_a_b @ r @ X3 @ C )
@ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.inj_on_multc
thf(fact_1155_x_Oeval__in__carrier__2,axiom,
! [X: list_list_a,Y: list_a] :
( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.eval_in_carrier_2
thf(fact_1156_add_Oinj__on__g,axiom,
! [H2: set_a,A2: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( inj_on_a_a
@ ^ [Y4: a] : ( add_a_b @ r @ Y4 @ A2 )
@ H2 ) ) ) ).
% add.inj_on_g
thf(fact_1157_x_Omaximalideal__prime,axiom,
! [I3: set_list_a] :
( ( maxima6585700282301356660t_unit @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( primei6309817859076077608t_unit @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.maximalideal_prime
thf(fact_1158_x_Oadd_Oinj__on__cmult,axiom,
! [C: list_a] :
( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( inj_on_list_a_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.inj_on_cmult
thf(fact_1159_x_Oadd_Oinj__on__multc,axiom,
! [C: list_a] :
( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( inj_on_list_a_list_a
@ ^ [X3: list_a] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ C )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.inj_on_multc
thf(fact_1160_x_Oquot__domain__imp__primeideal,axiom,
! [P: set_list_a] :
( ( ideal_8896367198367571637t_unit @ P @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( domain1617769409708967785t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) )
=> ( primei6309817859076077608t_unit @ P @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.quot_domain_imp_primeideal
thf(fact_1161_x_Oquot__domain__iff__primeideal,axiom,
! [P: set_list_a] :
( ( ideal_8896367198367571637t_unit @ P @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( domain1617769409708967785t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) )
= ( primei6309817859076077608t_unit @ P @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.quot_domain_iff_primeideal
thf(fact_1162_domain__eq__zeroprimeideal,axiom,
( ( domain_a_b @ r )
= ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ) ) ).
% domain_eq_zeroprimeideal
thf(fact_1163_zeroprimeideal__domainI,axiom,
( ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
=> ( domain_a_b @ r ) ) ).
% zeroprimeideal_domainI
thf(fact_1164_x_Ozeroprimeideal__domainI,axiom,
( ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.zeroprimeideal_domainI
thf(fact_1165_x_Odomain__eq__zeroprimeideal,axiom,
( ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.domain_eq_zeroprimeideal
thf(fact_1166_x_Oadd_Oinj__on__g,axiom,
! [H2: set_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( inj_on_list_a_list_a
@ ^ [Y4: list_a] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y4 @ A2 )
@ H2 ) ) ) ).
% x.add.inj_on_g
thf(fact_1167_x_Oring_Oinj__on__domain,axiom,
( ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( domain_a_b @ r )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.ring.inj_on_domain
thf(fact_1168_s_Oring_Oinj__on__domain,axiom,
( ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( domain_a_b @ r )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% s.ring.inj_on_domain
thf(fact_1169_x_Ominus__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.minus_closed
thf(fact_1170_x_Or__right__minus__eq,axiom,
! [A2: list_a,B3: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( A2 = B3 ) ) ) ) ).
% x.r_right_minus_eq
thf(fact_1171_x_Ohom__sub,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ x )
= ( a_minus_a_b @ r @ ( eval_a_b @ r @ X @ x ) @ ( eval_a_b @ r @ Y @ x ) ) ) ) ) ).
% x.hom_sub
thf(fact_1172_s_Ohom__sub,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ s2 )
= ( a_minus_a_b @ r @ ( eval_a_b @ r @ X @ s2 ) @ ( eval_a_b @ r @ Y @ s2 ) ) ) ) ) ).
% s.hom_sub
thf(fact_1173_x_Oeval__var,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% x.eval_var
thf(fact_1174_x_Osubfield__m__inv__simprule,axiom,
! [K: set_list_a,K2: list_a,A2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A2 ) @ K )
=> ( member_list_a @ A2 @ K ) ) ) ) ) ).
% x.subfield_m_inv_simprule
thf(fact_1175_x_Osubring__props_I2_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) ) ).
% x.subring_props(2)
thf(fact_1176_x_Osubring__props_I7_J,axiom,
! [K: set_list_a,H1: list_a,H22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H1 @ K )
=> ( ( member_list_a @ H22 @ K )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 ) @ K ) ) ) ) ).
% x.subring_props(7)
thf(fact_1177_x_Osubring__props_I6_J,axiom,
! [K: set_list_a,H1: list_a,H22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H1 @ K )
=> ( ( member_list_a @ H22 @ K )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 ) @ K ) ) ) ) ).
% x.subring_props(6)
thf(fact_1178_x_Osubring__props_I4_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( K != bot_bot_set_list_a ) ) ).
% x.subring_props(4)
thf(fact_1179_x_Osubring__props_I3_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) ) ).
% x.subring_props(3)
thf(fact_1180_x_Osubring__props_I1_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subring_props(1)
thf(fact_1181_x_Oring_Oimg__is__subfield_I1_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ K ) ) ) ).
% x.ring.img_is_subfield(1)
thf(fact_1182_s_Oring_Oimg__is__subfield_I1_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 )
@ K ) ) ) ).
% s.ring.img_is_subfield(1)
thf(fact_1183_x_Olagrange__basis__polynomial__aux__def,axiom,
! [S: set_list_a] :
( ( lagran3534788790333317459t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S )
= ( finpro3417560807142560175list_a @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ ^ [S2: list_a] : ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) )
@ S ) ) ).
% x.lagrange_basis_polynomial_aux_def
thf(fact_1184_x_Opoly__of__const__in__carrier,axiom,
! [S3: list_a] :
( ( member_list_a @ S3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S3 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.poly_of_const_in_carrier
thf(fact_1185_x_Oeval__poly__of__const,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Y )
= X ) ) ).
% x.eval_poly_of_const
thf(fact_1186_x_Oline__extension__smult__closed,axiom,
! [K: set_list_a,E: set_list_a,A2: list_a,K2: list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [K3: list_a,V4: list_a] :
( ( member_list_a @ K3 @ K )
=> ( ( member_list_a @ V4 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ V4 ) @ E ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K2 @ K )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A2 @ E ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ U ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A2 @ E ) ) ) ) ) ) ) ) ).
% x.line_extension_smult_closed
thf(fact_1187_s_Oring_Oimg__is__subfield_I2_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( subfield_a_b
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 )
@ K )
@ r ) ) ) ).
% s.ring.img_is_subfield(2)
thf(fact_1188_carrier__is__subfield,axiom,
subfield_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subfield
thf(fact_1189_univ__poly__is__principal,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_principal
thf(fact_1190_subring__props_I2_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K ) ) ).
% subring_props(2)
thf(fact_1191_subring__props_I7_J,axiom,
! [K: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H22 @ K )
=> ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ K ) ) ) ) ).
% subring_props(7)
thf(fact_1192_subring__props_I6_J,axiom,
! [K: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H22 @ K )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ K ) ) ) ) ).
% subring_props(6)
thf(fact_1193_subring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( K != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_1194_subring__props_I3_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K ) ) ).
% subring_props(3)
thf(fact_1195_subring__props_I1_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_1196_pprime__iff__pirreducible,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_1197_x_Oline__extension__in__carrier,axiom,
! [K: set_list_a,A2: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A2 @ E ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.line_extension_in_carrier
thf(fact_1198_x_Oline__extension__mem__iff,axiom,
! [U: list_a,K: set_list_a,A2: list_a,E: set_list_a] :
( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A2 @ E ) )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ K )
& ? [Y4: list_a] :
( ( member_list_a @ Y4 @ E )
& ( U
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ A2 ) @ Y4 ) ) ) ) ) ) ).
% x.line_extension_mem_iff
thf(fact_1199_pprimeE_I2_J,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pprimeE(2)
thf(fact_1200_subfield__m__inv__simprule,axiom,
! [K: set_a,K2: a,A2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ A2 ) @ K )
=> ( member_a @ A2 @ K ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_1201_x_Oring_Oimg__is__subfield_I2_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( subfield_a_b
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ K )
@ r ) ) ) ).
% x.ring.img_is_subfield(2)
thf(fact_1202_s_Oring_Oline__extension__hom,axiom,
! [K: set_list_a,A2: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( embedd971793762689825387on_a_b @ r
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 )
@ K )
@ ( eval_a_b @ r @ A2 @ s2 )
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 )
@ E ) )
= ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 )
@ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A2 @ E ) ) ) ) ) ) ).
% s.ring.line_extension_hom
thf(fact_1203_x_Oring_Oline__extension__hom,axiom,
! [K: set_list_a,A2: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( embedd971793762689825387on_a_b @ r
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ K )
@ ( eval_a_b @ r @ A2 @ x )
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ E ) )
= ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A2 @ E ) ) ) ) ) ) ).
% x.ring.line_extension_hom
thf(fact_1204_line__extension__in__carrier,axiom,
! [K: set_a,A2: a,E: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K @ A2 @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_1205_line__extension__mem__iff,axiom,
! [U: a,K: set_a,A2: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A2 @ E ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ K )
& ? [Y4: a] :
( ( member_a @ Y4 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ A2 ) @ Y4 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_1206_line__extension__smult__closed,axiom,
! [K: set_a,E: set_a,A2: a,K2: a,U: a] :
( ( subfield_a_b @ K @ r )
=> ( ! [K3: a,V4: a] :
( ( member_a @ K3 @ K )
=> ( ( member_a @ V4 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ V4 ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A2 @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ U ) @ ( embedd971793762689825387on_a_b @ r @ K @ A2 @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_1207_rupture__is__field__iff__pirreducible,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ).
% rupture_is_field_iff_pirreducible
thf(fact_1208_long__division__add_I1_J,axiom,
! [K: set_a,A2: list_a,B3: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A2 @ B3 ) @ Q2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ A2 @ Q2 ) @ ( polynomial_pdiv_a_b @ r @ B3 @ Q2 ) ) ) ) ) ) ) ).
% long_division_add(1)
thf(fact_1209_long__division__closed_I1_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_1210_lagrange__basis__polynomial__aux__def,axiom,
! [S: set_a] :
( ( lagran9092808442999052491ux_a_b @ r @ S )
= ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [S2: a] : ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ ( poly_of_const_a_b @ r @ S2 ) )
@ S ) ) ).
% lagrange_basis_polynomial_aux_def
thf(fact_1211_eval__poly__of__const,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_of_const_a_b @ r @ X ) @ Y )
= X ) ) ).
% eval_poly_of_const
thf(fact_1212_poly__of__const__in__carrier,axiom,
! [S3: a] :
( ( member_a @ S3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( poly_of_const_a_b @ r @ S3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% poly_of_const_in_carrier
thf(fact_1213_univ__poly__carrier__subfield__of__consts,axiom,
subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% univ_poly_carrier_subfield_of_consts
thf(fact_1214_univ__poly__subfield__of__consts,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_subfield_of_consts
thf(fact_1215_x_Ouniv__poly__subfield__of__consts,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subfie4546268998243038636t_unit @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ).
% x.univ_poly_subfield_of_consts
thf(fact_1216__092_060open_062local_Oeval_A_Ilagrange__basis__polynomial_AS_Ax_J_Ax_A_061_Afinprod_AR_A_Ia__minus_AR_Ax_J_AS_A_092_060otimes_062_Alocal_Oeval_A_Ipoly__of__const_A_Iinv_Afinprod_AR_A_Ia__minus_AR_Ax_J_AS_J_J_Ax_092_060close_062,axiom,
( ( eval_a_b @ r @ ( lagran2649660974587678107al_a_b @ r @ s @ x ) @ x )
= ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) @ ( eval_a_b @ r @ ( poly_of_const_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) ) ) @ x ) ) ) ).
% \<open>local.eval (lagrange_basis_polynomial S x) x = finprod R (a_minus R x) S \<otimes> local.eval (poly_of_const (inv finprod R (a_minus R x) S)) x\<close>
thf(fact_1217__092_060open_062finprod_AR_A_Ia__minus_AR_Ax_J_AS_A_092_060otimes_062_Alocal_Oeval_A_Ipoly__of__const_A_Iinv_Afinprod_AR_A_Ia__minus_AR_Ax_J_AS_J_J_Ax_A_061_A_092_060one_062_092_060close_062,axiom,
( ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) @ ( eval_a_b @ r @ ( poly_of_const_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) ) ) @ x ) )
= ( one_a_ring_ext_a_b @ r ) ) ).
% \<open>finprod R (a_minus R x) S \<otimes> local.eval (poly_of_const (inv finprod R (a_minus R x) S)) x = \<one>\<close>
thf(fact_1218_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_1219_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_1220_inv__inj__on__Units,axiom,
inj_on_a_a @ ( m_inv_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).
% inv_inj_on_Units
thf(fact_1221_comm__inv__char,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ X )
= Y ) ) ) ) ).
% comm_inv_char
thf(fact_1222_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_1223_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_1224_subfield__m__inv_I1_J,axiom,
! [K: set_a,K2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ K2 ) @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ).
% subfield_m_inv(1)
thf(fact_1225_subfield__m__inv_I2_J,axiom,
! [K: set_a,K2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ K2 @ ( m_inv_a_ring_ext_a_b @ r @ K2 ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% subfield_m_inv(2)
thf(fact_1226_subfield__m__inv_I3_J,axiom,
! [K: set_a,K2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ K2 ) @ K2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% subfield_m_inv(3)
thf(fact_1227_lagrange__basis__polynomial__def,axiom,
! [S: set_a,X: a] :
( ( lagran2649660974587678107al_a_b @ r @ S @ X )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( lagran9092808442999052491ux_a_b @ r @ S ) @ ( poly_of_const_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ X ) @ S ) ) ) ) ) ).
% lagrange_basis_polynomial_def
thf(fact_1228_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_1229_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_1230_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_1231_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_1232_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_1233_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_1234_x_Osubfield__m__inv_I2_J,axiom,
! [K: set_list_a,K2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.subfield_m_inv(2)
thf(fact_1235_x_Osubfield__m__inv_I3_J,axiom,
! [K: set_list_a,K2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ K2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.subfield_m_inv(3)
thf(fact_1236_x_Oinv__eq__imp__eq,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) )
=> ( X = Y ) ) ) ) ).
% x.inv_eq_imp_eq
thf(fact_1237_x_Oinv__eq__one__eq,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( X
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.inv_eq_one_eq
thf(fact_1238_x_Oinv__inj__on__Units,axiom,
inj_on_list_a_list_a @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.inv_inj_on_Units
thf(fact_1239_x_Ocomm__inv__char,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= Y ) ) ) ) ).
% x.comm_inv_char
thf(fact_1240_x_Oinv__char,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= Y ) ) ) ) ) ).
% x.inv_char
thf(fact_1241_x_Oinv__unique_H,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y
= ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) ) ) ) ) ) ).
% x.inv_unique'
thf(fact_1242_x_Osubfield__m__inv_I1_J,axiom,
! [K: set_list_a,K2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ) ).
% x.subfield_m_inv(1)
thf(fact_1243_x_Olagrange__basis__polynomial__def,axiom,
! [S: set_list_a,X: list_a] :
( ( lagran6985349428869127715t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S @ X )
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( lagran3534788790333317459t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ S ) ) ) ) ) ).
% x.lagrange_basis_polynomial_def
thf(fact_1244_x_OUnits__inv__Units,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_inv_Units
thf(fact_1245_x_OUnits__inv__inv,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
= X ) ) ).
% x.Units_inv_inv
thf(fact_1246_x_Oinv__one,axiom,
( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.inv_one
thf(fact_1247_x_OUnits__inv__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_inv_closed
thf(fact_1248_x_OUnits__r__inv,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_r_inv
thf(fact_1249_x_OUnits__l__inv,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_l_inv
thf(fact_1250_pdiv__pmod,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ).
% pdiv_pmod
thf(fact_1251_univ__poly__infinite__dimension,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ~ ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ K ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).
% univ_poly_infinite_dimension
thf(fact_1252_long__division__closed_I2_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(2)
thf(fact_1253_x_Otelescopic__base__dim_I1_J,axiom,
! [K: set_list_a,F2: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subfie1779122896746047282t_unit @ F2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ F2 )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F2 @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E ) ) ) ) ) ).
% x.telescopic_base_dim(1)
thf(fact_1254_long__division__add__iff,axiom,
! [K: set_a,A2: list_a,B3: list_a,C: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A2 @ Q2 )
= ( polynomial_pmod_a_b @ r @ B3 @ Q2 ) )
= ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A2 @ C ) @ Q2 )
= ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ B3 @ C ) @ Q2 ) ) ) ) ) ) ) ) ).
% long_division_add_iff
thf(fact_1255_long__division__add_I2_J,axiom,
! [K: set_a,A2: list_a,B3: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A2 @ B3 ) @ Q2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ A2 @ Q2 ) @ ( polynomial_pmod_a_b @ r @ B3 @ Q2 ) ) ) ) ) ) ) ).
% long_division_add(2)
thf(fact_1256_x_Ofinite__dimension__imp__subalgebra,axiom,
! [K: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
=> ( embedd1768981623711841426t_unit @ K @ E @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.finite_dimension_imp_subalgebra
thf(fact_1257_x_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K: set_list_a,E: set_list_a,V: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
=> ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ V @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ V ) ) ) ) ) ).
% x.subalbegra_incl_imp_finite_dimension
thf(fact_1258_s_Oring_Oinfinite__dimension__hom,axiom,
! [K: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 )
@ E )
=> ( ( embedd1768981623711841426t_unit @ K @ E @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ~ ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
=> ~ ( embedd8708762675212832759on_a_b @ r
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 )
@ K )
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 )
@ E ) ) ) ) ) ) ) ).
% s.ring.infinite_dimension_hom
thf(fact_1259_x_Oring_Oinfinite__dimension__hom,axiom,
! [K: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ E )
=> ( ( embedd1768981623711841426t_unit @ K @ E @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ~ ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
=> ~ ( embedd8708762675212832759on_a_b @ r
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ K )
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ E ) ) ) ) ) ) ) ).
% x.ring.infinite_dimension_hom
thf(fact_1260_telescopic__base__dim_I1_J,axiom,
! [K: set_a,F2: set_a,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( subfield_a_b @ F2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ F2 )
=> ( ( embedd8708762675212832759on_a_b @ r @ F2 @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K @ E ) ) ) ) ) ).
% telescopic_base_dim(1)
thf(fact_1261_finite__dimension__imp__subalgebra,axiom,
! [K: set_a,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ( embedd9027525575939734154ra_a_b @ K @ E @ r ) ) ) ).
% finite_dimension_imp_subalgebra
thf(fact_1262_subalbegra__incl__imp__finite__dimension,axiom,
! [K: set_a,E: set_a,V: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ( ord_less_eq_set_a @ V @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K @ V ) ) ) ) ) ).
% subalbegra_incl_imp_finite_dimension
thf(fact_1263_x_Ospace__subgroup__props_I6_J,axiom,
! [K: set_list_a,N: nat,E: set_list_a,K2: list_a,A2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K @ E )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A2 ) @ E )
=> ( member_list_a @ A2 @ E ) ) ) ) ) ) ).
% x.space_subgroup_props(6)
thf(fact_1264_x_Odimension__is__inj,axiom,
! [K: set_list_a,N: nat,E: set_list_a,M2: nat] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K @ E )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M2 @ K @ E )
=> ( N = M2 ) ) ) ) ).
% x.dimension_is_inj
thf(fact_1265_x_Ofinite__dimension__def,axiom,
! [K: set_list_a,E: set_list_a] :
( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
= ( ? [N2: nat] : ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N2 @ K @ E ) ) ) ).
% x.finite_dimension_def
thf(fact_1266_x_Ofinite__dimensionI,axiom,
! [N: nat,K: set_list_a,E: set_list_a] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E ) ) ).
% x.finite_dimensionI
thf(fact_1267_x_Ofinite__dimensionE_H,axiom,
! [K: set_list_a,E: set_list_a] :
( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
=> ~ ! [N3: nat] :
~ ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N3 @ K @ E ) ) ).
% x.finite_dimensionE'
thf(fact_1268_x_Ospace__subgroup__props_I2_J,axiom,
! [K: set_list_a,N: nat,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K @ E )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ E ) ) ) ).
% x.space_subgroup_props(2)
thf(fact_1269_x_Ospace__subgroup__props_I3_J,axiom,
! [K: set_list_a,N: nat,E: set_list_a,V1: list_a,V22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K @ E )
=> ( ( member_list_a @ V1 @ E )
=> ( ( member_list_a @ V22 @ E )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ V1 @ V22 ) @ E ) ) ) ) ) ).
% x.space_subgroup_props(3)
thf(fact_1270_x_Ospace__subgroup__props_I5_J,axiom,
! [K: set_list_a,N: nat,E: set_list_a,K2: list_a,V5: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K @ E )
=> ( ( member_list_a @ K2 @ K )
=> ( ( member_list_a @ V5 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ V5 ) @ E ) ) ) ) ) ).
% x.space_subgroup_props(5)
thf(fact_1271_x_Ounique__dimension,axiom,
! [K: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
=> ? [X2: nat] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ K @ E )
& ! [Y6: nat] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y6 @ K @ E )
=> ( Y6 = X2 ) ) ) ) ) ).
% x.unique_dimension
thf(fact_1272_x_Ospace__subgroup__props_I1_J,axiom,
! [K: set_list_a,N: nat,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K @ E )
=> ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.space_subgroup_props(1)
thf(fact_1273_x_Odimension__zero,axiom,
! [K: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ zero_zero_nat @ K @ E )
=> ( E
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ).
% x.dimension_zero
thf(fact_1274_s_Oring_Oinj__hom__dimension,axiom,
! [K: set_list_a,E: set_list_a,N: nat] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 )
@ E )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K @ E )
=> ( embedd2795209813406577254on_a_b @ r @ N
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 )
@ K )
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ s2 )
@ E ) ) ) ) ) ) ).
% s.ring.inj_hom_dimension
thf(fact_1275_dimension__is__inj,axiom,
! [K: set_a,N: nat,E: set_a,M2: nat] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M2 @ K @ E )
=> ( N = M2 ) ) ) ) ).
% dimension_is_inj
thf(fact_1276_finite__dimensionE_H,axiom,
! [K: set_a,E: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ~ ! [N3: nat] :
~ ( embedd2795209813406577254on_a_b @ r @ N3 @ K @ E ) ) ).
% finite_dimensionE'
thf(fact_1277_finite__dimensionI,axiom,
! [N: nat,K: set_a,E: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K @ E ) ) ).
% finite_dimensionI
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( eval_a_b @ r @ p @ s2 )
= ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ s2 ) @ s ) ) ).
%------------------------------------------------------------------------------