TPTP Problem File: SLH0116^1.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 : Commuting_Hermitian/0001_Spectral_Theory_Complements/prob_00674_024322__19295058_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1340 ( 504 unt; 172 typ; 0 def)
% Number of atoms : 3269 (1195 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10119 ( 279 ~; 38 |; 267 &;7939 @)
% ( 0 <=>;1596 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 783 ( 783 >; 0 *; 0 +; 0 <<)
% Number of symbols : 162 ( 159 usr; 19 con; 0-4 aty)
% Number of variables : 3430 ( 413 ^;2927 !; 90 ?;3430 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:35:35.863
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Matrix__Omat_Itf__a_J_J_J,type,
set_set_mat_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_It__Nat__Onat_J_J,type,
set_mat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Ovec_Itf__a_J_J,type,
set_vec_a: $tType ).
thf(ty_n_t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
set_mat_a: $tType ).
thf(ty_n_t__Matrix__Omat_It__Nat__Onat_J,type,
mat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Matrix__Ovec_Itf__a_J,type,
vec_a: $tType ).
thf(ty_n_t__Matrix__Omat_Itf__a_J,type,
mat_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (159)
thf(sy_c_Complex__Matrix_Ohermitian_001tf__a,type,
complex_hermitian_a: mat_a > $o ).
thf(sy_c_Complex__Matrix_Otrace_001tf__a,type,
complex_trace_a: mat_a > a ).
thf(sy_c_Complex__Matrix_Ounitary_001tf__a,type,
complex_unitary_a: mat_a > $o ).
thf(sy_c_Finite__Set_Ocard_001t__Matrix__Omat_Itf__a_J,type,
finite_card_mat_a: set_mat_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite2115694454571419734at_nat: set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Matrix__Omat_Itf__a_J,type,
finite_finite_mat_a: set_mat_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
finite5775620362878804648_mat_a: set_set_mat_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Gauss__Jordan__Elimination_Ogauss__jordan__single_001tf__a,type,
gauss_4684855476144371464ngle_a: mat_a > mat_a ).
thf(sy_c_Gauss__Jordan__Elimination_Orow__echelon__form_001tf__a,type,
gauss_5855338539171749649form_a: mat_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_It__Nat__Onat_J,type,
minus_minus_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Matrix__Omat_Itf__a_J,type,
minus_minus_mat_a: mat_a > mat_a > mat_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
minus_4757590266979429866_mat_a: set_mat_a > set_mat_a > set_mat_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001tf__a,type,
minus_minus_a: a > a > a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001tf__a,type,
one_one_a: a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_It__Nat__Onat_J,type,
plus_plus_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Matrix__Omat_Itf__a_J,type,
plus_plus_mat_a: mat_a > mat_a > mat_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
plus_plus_set_mat_a: set_mat_a > set_mat_a > set_mat_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001tf__a,type,
plus_plus_a: a > a > a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_It__Nat__Onat_J,type,
times_times_mat_nat: mat_nat > mat_nat > mat_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Matrix__Omat_Itf__a_J,type,
times_times_mat_a: mat_a > mat_a > mat_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
times_1230744552615602198_mat_a: set_mat_a > set_mat_a > set_mat_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001tf__a,type,
times_times_a: a > a > a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Nat__Onat_J,type,
zero_zero_set_nat: set_nat ).
thf(sy_c_HOL_OEx1_001t__Matrix__Omat_Itf__a_J,type,
ex1_mat_a: ( mat_a > $o ) > $o ).
thf(sy_c_HOL_OEx1_001t__Nat__Onat,type,
ex1_nat: ( nat > $o ) > $o ).
thf(sy_c_HOL_OThe_001t__Matrix__Omat_Itf__a_J,type,
the_mat_a: ( mat_a > $o ) > mat_a ).
thf(sy_c_HOL_OThe_001t__Nat__Onat,type,
the_nat: ( nat > $o ) > nat ).
thf(sy_c_HOL_OThe_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
the_set_mat_a: ( set_mat_a > $o ) > set_mat_a ).
thf(sy_c_HOL_OUniq_001t__Matrix__Omat_It__Nat__Onat_J,type,
uniq_mat_nat: ( mat_nat > $o ) > $o ).
thf(sy_c_HOL_OUniq_001t__Matrix__Omat_Itf__a_J,type,
uniq_mat_a: ( mat_a > $o ) > $o ).
thf(sy_c_HOL_OUniq_001t__Nat__Onat,type,
uniq_nat: ( nat > $o ) > $o ).
thf(sy_c_HOL_OUniq_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
uniq_set_mat_a: ( set_mat_a > $o ) > $o ).
thf(sy_c_If_001t__Matrix__Omat_Itf__a_J,type,
if_mat_a: $o > mat_a > mat_a > mat_a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
if_set_mat_a: $o > set_mat_a > set_mat_a > set_mat_a ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
sup_sup_set_mat_a: set_mat_a > set_mat_a > set_mat_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Matrix__Omat_Itf__a_J_001t__Nat__Onat,type,
lattic3922145225401787590_a_nat: ( mat_a > nat ) > set_mat_a > mat_a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Linear__Algebra__Complements_Oprojector_001tf__a,type,
linear2821214051344812439ctor_a: mat_a > $o ).
thf(sy_c_Matrix_Ocarrier__mat_001t__Nat__Onat,type,
carrier_mat_nat: nat > nat > set_mat_nat ).
thf(sy_c_Matrix_Ocarrier__mat_001tf__a,type,
carrier_mat_a: nat > nat > set_mat_a ).
thf(sy_c_Matrix_Ocomm__monoid__add__class_Osum__mat_001t__Nat__Onat,type,
comm_m4056229327131402372at_nat: mat_nat > nat ).
thf(sy_c_Matrix_Ocomm__monoid__add__class_Osum__mat_001tf__a,type,
comm_m5291664705200495434_mat_a: mat_a > a ).
thf(sy_c_Matrix_Odim__col_001t__Nat__Onat,type,
dim_col_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__col_001tf__a,type,
dim_col_a: mat_a > nat ).
thf(sy_c_Matrix_Odim__row_001t__Nat__Onat,type,
dim_row_nat: mat_nat > nat ).
thf(sy_c_Matrix_Odim__row_001tf__a,type,
dim_row_a: mat_a > nat ).
thf(sy_c_Matrix_Oindex__mat_001t__Nat__Onat,type,
index_mat_nat: mat_nat > product_prod_nat_nat > nat ).
thf(sy_c_Matrix_Oindex__mat_001tf__a,type,
index_mat_a: mat_a > product_prod_nat_nat > a ).
thf(sy_c_Matrix_Oinvertible__mat_001tf__a,type,
invertible_mat_a: mat_a > $o ).
thf(sy_c_Matrix_Oinverts__mat_001t__Nat__Onat,type,
inverts_mat_nat: mat_nat > mat_nat > $o ).
thf(sy_c_Matrix_Oinverts__mat_001tf__a,type,
inverts_mat_a: mat_a > mat_a > $o ).
thf(sy_c_Matrix_Omat_001t__Nat__Onat,type,
mat_nat2: nat > nat > ( product_prod_nat_nat > nat ) > mat_nat ).
thf(sy_c_Matrix_Omat_001tf__a,type,
mat_a2: nat > nat > ( product_prod_nat_nat > a ) > mat_a ).
thf(sy_c_Matrix_Omat__diag_001t__Nat__Onat,type,
mat_diag_nat: nat > ( nat > nat ) > mat_nat ).
thf(sy_c_Matrix_Omat__diag_001tf__a,type,
mat_diag_a: nat > ( nat > a ) > mat_a ).
thf(sy_c_Matrix_Omat__of__row_001tf__a,type,
mat_of_row_a: vec_a > mat_a ).
thf(sy_c_Matrix_Oone__mat_001t__Nat__Onat,type,
one_mat_nat: nat > mat_nat ).
thf(sy_c_Matrix_Oone__mat_001tf__a,type,
one_mat_a: nat > mat_a ).
thf(sy_c_Matrix_Osimilar__mat_001t__Nat__Onat,type,
similar_mat_nat: mat_nat > mat_nat > $o ).
thf(sy_c_Matrix_Osimilar__mat_001tf__a,type,
similar_mat_a: mat_a > mat_a > $o ).
thf(sy_c_Matrix_Osimilar__mat__wit_001t__Nat__Onat,type,
similar_mat_wit_nat: mat_nat > mat_nat > mat_nat > mat_nat > $o ).
thf(sy_c_Matrix_Osimilar__mat__wit_001tf__a,type,
similar_mat_wit_a: mat_a > mat_a > mat_a > mat_a > $o ).
thf(sy_c_Matrix_Osmult__mat_001t__Nat__Onat,type,
smult_mat_nat: nat > mat_nat > mat_nat ).
thf(sy_c_Matrix_Osmult__mat_001tf__a,type,
smult_mat_a: a > mat_a > mat_a ).
thf(sy_c_Matrix_Osquare__mat_001tf__a,type,
square_mat_a: mat_a > $o ).
thf(sy_c_Matrix_Oupdate__mat_001tf__a,type,
update_mat_a: mat_a > product_prod_nat_nat > a > mat_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
bot_bot_nat_nat_o: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Matrix__Omat_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_mat_nat_o: mat_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Matrix__Omat_Itf__a_J_M_Eo_J,type,
bot_bot_mat_a_o: mat_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Matrix__Omat_Itf__a_J_J_M_Eo_J,type,
bot_bot_set_mat_a_o: set_mat_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_set_nat_o: set_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Matrix__Omat_It__Nat__Onat_J_J,type,
bot_bot_set_mat_nat: set_mat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
bot_bot_set_mat_a: set_mat_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Matrix__Omat_Itf__a_J_J_J,type,
bot_bo8661580253428394715_mat_a: set_set_mat_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
ord_Least_set_mat_a: ( set_mat_a > $o ) > set_mat_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Matrix__Omat_It__Nat__Onat_J_M_Eo_J,type,
ord_le1720399365423063892_nat_o: ( mat_nat > $o ) > ( mat_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Matrix__Omat_Itf__a_J_M_Eo_J,type,
ord_less_eq_mat_a_o: ( mat_a > $o ) > ( mat_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Matrix__Omat_Itf__a_J_J_M_Eo_J,type,
ord_le2661774091922174110at_a_o: ( set_mat_a > $o ) > ( set_mat_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Omat_It__Nat__Onat_J_J,type,
ord_le7789122042438455497at_nat: set_mat_nat > set_mat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
ord_le3318621148231462513_mat_a: set_mat_a > set_mat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Matrix__Omat_Itf__a_J_J_J,type,
ord_le2341747070211005607_mat_a: set_set_mat_a > set_set_mat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_It__Matrix__Omat_Itf__a_J_M_Eo_J,type,
order_1137619110770037133at_a_o: ( ( mat_a > $o ) > $o ) > mat_a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_It__Nat__Onat_M_Eo_J,type,
order_Greatest_nat_o: ( ( nat > $o ) > $o ) > nat > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
order_3466378972280292088_mat_a: ( set_mat_a > $o ) > set_mat_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Nat__Onat_J,type,
order_5724808138429204845et_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
top_top_set_mat_a: set_mat_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Schur__Decomposition_Ocorthogonal__mat_001tf__a,type,
schur_4042290226164342457_mat_a: mat_a > $o ).
thf(sy_c_Schur__Decomposition_Omat__adjoint_001tf__a,type,
schur_mat_adjoint_a: mat_a > mat_a ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_It__Nat__Onat_J,type,
collect_mat_nat: ( mat_nat > $o ) > set_mat_nat ).
thf(sy_c_Set_OCollect_001t__Matrix__Omat_Itf__a_J,type,
collect_mat_a: ( mat_a > $o ) > set_mat_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
collect_set_mat_a: ( set_mat_a > $o ) > set_set_mat_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert_nat_nat: ( nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oinsert_001t__Matrix__Omat_It__Nat__Onat_J,type,
insert_mat_nat: mat_nat > set_mat_nat > set_mat_nat ).
thf(sy_c_Set_Oinsert_001t__Matrix__Omat_Itf__a_J,type,
insert_mat_a: mat_a > set_mat_a > set_mat_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
insert_set_mat_a: set_mat_a > set_set_mat_a > set_set_mat_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Ois__empty_001t__Matrix__Omat_Itf__a_J,type,
is_empty_mat_a: set_mat_a > $o ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
is_empty_set_mat_a: set_set_mat_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Matrix__Omat_It__Nat__Onat_J,type,
is_singleton_mat_nat: set_mat_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Matrix__Omat_Itf__a_J,type,
is_singleton_mat_a: set_mat_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
is_sin4571450623289582109_mat_a: set_set_mat_a > $o ).
thf(sy_c_Set_Oremove_001t__Matrix__Omat_Itf__a_J,type,
remove_mat_a: mat_a > set_mat_a > set_mat_a ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Othe__elem_001t__Matrix__Omat_Itf__a_J,type,
the_elem_mat_a: set_mat_a > mat_a ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
the_elem_set_mat_a: set_set_mat_a > set_mat_a ).
thf(sy_c_Spectral__Theory__Complements_Omat__conj_001tf__a,type,
spectr5828033140197310157conj_a: mat_a > mat_a > mat_a ).
thf(sy_c_Spectral__Theory__Complements_Ounitarily__equiv_001tf__a,type,
spectr4825054497075562704quiv_a: mat_a > mat_a > mat_a > $o ).
thf(sy_c_VS__Connect_Ovec__space_Orow__space_001tf__a,type,
vS_vec_row_space_a: nat > mat_a > set_vec_a ).
thf(sy_c_fChoice_001t__Matrix__Omat_Itf__a_J,type,
fChoice_mat_a: ( mat_a > $o ) > mat_a ).
thf(sy_c_fChoice_001t__Nat__Onat,type,
fChoice_nat: ( nat > $o ) > nat ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001t__Matrix__Omat_It__Nat__Onat_J,type,
member_mat_nat: mat_nat > set_mat_nat > $o ).
thf(sy_c_member_001t__Matrix__Omat_Itf__a_J,type,
member_mat_a: mat_a > set_mat_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J,type,
member_set_mat_a: set_mat_a > set_set_mat_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_A,type,
a2: mat_a ).
thf(sy_v_B,type,
b: mat_a ).
thf(sy_v_U,type,
u: mat_a ).
% Relevant facts (1156)
thf(fact_0__092_060open_062Complex__Matrix_Oadjoint_AU_A_K_AU_A_061_A1_092_060_094sub_062m_A_Idim__row_AB_J_092_060close_062,axiom,
( ( times_times_mat_a @ ( schur_mat_adjoint_a @ u ) @ u )
= ( one_mat_a @ ( dim_row_a @ b ) ) ) ).
% \<open>Complex_Matrix.adjoint U * U = 1\<^sub>m (dim_row B)\<close>
thf(fact_1_assms,axiom,
spectr4825054497075562704quiv_a @ a2 @ b @ u ).
% assms
thf(fact_2__092_060open_062Complex__Matrix_Ounitary_A_IComplex__Matrix_Oadjoint_AU_J_092_060close_062,axiom,
complex_unitary_a @ ( schur_mat_adjoint_a @ u ) ).
% \<open>Complex_Matrix.unitary (Complex_Matrix.adjoint U)\<close>
thf(fact_3_left__mult__one__mat_H,axiom,
! [A: mat_nat,N: nat] :
( ( ( dim_row_nat @ A )
= N )
=> ( ( times_times_mat_nat @ ( one_mat_nat @ N ) @ A )
= A ) ) ).
% left_mult_one_mat'
thf(fact_4_left__mult__one__mat_H,axiom,
! [A: mat_a,N: nat] :
( ( ( dim_row_a @ A )
= N )
=> ( ( times_times_mat_a @ ( one_mat_a @ N ) @ A )
= A ) ) ).
% left_mult_one_mat'
thf(fact_5_index__one__mat_I2_J,axiom,
! [N: nat] :
( ( dim_row_nat @ ( one_mat_nat @ N ) )
= N ) ).
% index_one_mat(2)
thf(fact_6_index__one__mat_I2_J,axiom,
! [N: nat] :
( ( dim_row_a @ ( one_mat_a @ N ) )
= N ) ).
% index_one_mat(2)
thf(fact_7_index__mult__mat_I2_J,axiom,
! [A: mat_nat,B: mat_nat] :
( ( dim_row_nat @ ( times_times_mat_nat @ A @ B ) )
= ( dim_row_nat @ A ) ) ).
% index_mult_mat(2)
thf(fact_8_index__mult__mat_I2_J,axiom,
! [A: mat_a,B: mat_a] :
( ( dim_row_a @ ( times_times_mat_a @ A @ B ) )
= ( dim_row_a @ A ) ) ).
% index_mult_mat(2)
thf(fact_9_mat__conj__def,axiom,
( spectr5828033140197310157conj_a
= ( ^ [U: mat_a,V: mat_a] : ( times_times_mat_a @ ( times_times_mat_a @ U @ V ) @ ( schur_mat_adjoint_a @ U ) ) ) ) ).
% mat_conj_def
thf(fact_10_mat__conj__adjoint,axiom,
! [U2: mat_a,V2: mat_a] :
( ( spectr5828033140197310157conj_a @ ( schur_mat_adjoint_a @ U2 ) @ V2 )
= ( times_times_mat_a @ ( times_times_mat_a @ ( schur_mat_adjoint_a @ U2 ) @ V2 ) @ U2 ) ) ).
% mat_conj_adjoint
thf(fact_11_Complex__Matrix_Oadjoint__adjoint,axiom,
! [A: mat_a] :
( ( schur_mat_adjoint_a @ ( schur_mat_adjoint_a @ A ) )
= A ) ).
% Complex_Matrix.adjoint_adjoint
thf(fact_12_unitarily__equiv__eq,axiom,
! [A: mat_a,B: mat_a,U2: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U2 )
=> ( A
= ( times_times_mat_a @ ( times_times_mat_a @ U2 @ B ) @ ( schur_mat_adjoint_a @ U2 ) ) ) ) ).
% unitarily_equiv_eq
thf(fact_13_inverts__mat__def,axiom,
( inverts_mat_nat
= ( ^ [A2: mat_nat,B2: mat_nat] :
( ( times_times_mat_nat @ A2 @ B2 )
= ( one_mat_nat @ ( dim_row_nat @ A2 ) ) ) ) ) ).
% inverts_mat_def
thf(fact_14_inverts__mat__def,axiom,
( inverts_mat_a
= ( ^ [A2: mat_a,B2: mat_a] :
( ( times_times_mat_a @ A2 @ B2 )
= ( one_mat_a @ ( dim_row_a @ A2 ) ) ) ) ) ).
% inverts_mat_def
thf(fact_15_Groups_Omult__ac_I3_J,axiom,
! [B3: a,A3: a,C: a] :
( ( times_times_a @ B3 @ ( times_times_a @ A3 @ C ) )
= ( times_times_a @ A3 @ ( times_times_a @ B3 @ C ) ) ) ).
% Groups.mult_ac(3)
thf(fact_16_Groups_Omult__ac_I3_J,axiom,
! [B3: set_nat,A3: set_nat,C: set_nat] :
( ( times_times_set_nat @ B3 @ ( times_times_set_nat @ A3 @ C ) )
= ( times_times_set_nat @ A3 @ ( times_times_set_nat @ B3 @ C ) ) ) ).
% Groups.mult_ac(3)
thf(fact_17_Groups_Omult__ac_I3_J,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( times_times_nat @ B3 @ ( times_times_nat @ A3 @ C ) )
= ( times_times_nat @ A3 @ ( times_times_nat @ B3 @ C ) ) ) ).
% Groups.mult_ac(3)
thf(fact_18_Groups_Omult__ac_I2_J,axiom,
( times_times_a
= ( ^ [A4: a,B4: a] : ( times_times_a @ B4 @ A4 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_19_Groups_Omult__ac_I2_J,axiom,
( times_times_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( times_times_set_nat @ B4 @ A4 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_20_Groups_Omult__ac_I2_J,axiom,
( times_times_nat
= ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% Groups.mult_ac(2)
thf(fact_21_Groups_Omult__ac_I1_J,axiom,
! [A3: a,B3: a,C: a] :
( ( times_times_a @ ( times_times_a @ A3 @ B3 ) @ C )
= ( times_times_a @ A3 @ ( times_times_a @ B3 @ C ) ) ) ).
% Groups.mult_ac(1)
thf(fact_22_Groups_Omult__ac_I1_J,axiom,
! [A3: set_nat,B3: set_nat,C: set_nat] :
( ( times_times_set_nat @ ( times_times_set_nat @ A3 @ B3 ) @ C )
= ( times_times_set_nat @ A3 @ ( times_times_set_nat @ B3 @ C ) ) ) ).
% Groups.mult_ac(1)
thf(fact_23_Groups_Omult__ac_I1_J,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B3 ) @ C )
= ( times_times_nat @ A3 @ ( times_times_nat @ B3 @ C ) ) ) ).
% Groups.mult_ac(1)
thf(fact_24_unitarily__equivD_I1_J,axiom,
! [A: mat_a,B: mat_a,U2: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U2 )
=> ( complex_unitary_a @ U2 ) ) ).
% unitarily_equivD(1)
thf(fact_25_unitary__one,axiom,
! [N: nat] : ( complex_unitary_a @ ( one_mat_a @ N ) ) ).
% unitary_one
thf(fact_26_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: a,B3: a,C: a] :
( ( times_times_a @ ( times_times_a @ A3 @ B3 ) @ C )
= ( times_times_a @ A3 @ ( times_times_a @ B3 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_27_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: set_nat,B3: set_nat,C: set_nat] :
( ( times_times_set_nat @ ( times_times_set_nat @ A3 @ B3 ) @ C )
= ( times_times_set_nat @ A3 @ ( times_times_set_nat @ B3 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_28_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A3 @ B3 ) @ C )
= ( times_times_nat @ A3 @ ( times_times_nat @ B3 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_29_Complex__Matrix_Ounitary__def,axiom,
( complex_unitary_a
= ( ^ [A2: mat_a] :
( ( member_mat_a @ A2 @ ( carrier_mat_a @ ( dim_row_a @ A2 ) @ ( dim_row_a @ A2 ) ) )
& ( inverts_mat_a @ A2 @ ( schur_mat_adjoint_a @ A2 ) ) ) ) ) ).
% Complex_Matrix.unitary_def
thf(fact_30_unitary__simps_I1_J,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ A )
=> ( ( times_times_mat_a @ ( schur_mat_adjoint_a @ A ) @ A )
= ( one_mat_a @ N ) ) ) ) ).
% unitary_simps(1)
thf(fact_31_unitary__simps_I2_J,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ A )
=> ( ( times_times_mat_a @ A @ ( schur_mat_adjoint_a @ A ) )
= ( one_mat_a @ N ) ) ) ) ).
% unitary_simps(2)
thf(fact_32_unitarily__equiv__def,axiom,
( spectr4825054497075562704quiv_a
= ( ^ [A2: mat_a,B2: mat_a,U: mat_a] :
( ( complex_unitary_a @ U )
& ( similar_mat_wit_a @ A2 @ B2 @ U @ ( schur_mat_adjoint_a @ U ) ) ) ) ) ).
% unitarily_equiv_def
thf(fact_33_unitarily__equivI,axiom,
! [A: mat_a,B: mat_a,U2: mat_a] :
( ( similar_mat_wit_a @ A @ B @ U2 @ ( schur_mat_adjoint_a @ U2 ) )
=> ( ( complex_unitary_a @ U2 )
=> ( spectr4825054497075562704quiv_a @ A @ B @ U2 ) ) ) ).
% unitarily_equivI
thf(fact_34_unitarily__equivI_H,axiom,
! [A: mat_a,U2: mat_a,B: mat_a,N: nat] :
( ( A
= ( spectr5828033140197310157conj_a @ U2 @ B ) )
=> ( ( complex_unitary_a @ U2 )
=> ( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( spectr4825054497075562704quiv_a @ A @ B @ U2 ) ) ) ) ) ).
% unitarily_equivI'
thf(fact_35_similar__mat__witD_I1_J,axiom,
! [N: nat,A: mat_nat,B: mat_nat,P: mat_nat,Q: mat_nat] :
( ( N
= ( dim_row_nat @ A ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P @ Q )
=> ( ( times_times_mat_nat @ P @ Q )
= ( one_mat_nat @ N ) ) ) ) ).
% similar_mat_witD(1)
thf(fact_36_similar__mat__witD_I1_J,axiom,
! [N: nat,A: mat_a,B: mat_a,P: mat_a,Q: mat_a] :
( ( N
= ( dim_row_a @ A ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( ( times_times_mat_a @ P @ Q )
= ( one_mat_a @ N ) ) ) ) ).
% similar_mat_witD(1)
thf(fact_37_similar__mat__witD_I2_J,axiom,
! [N: nat,A: mat_nat,B: mat_nat,P: mat_nat,Q: mat_nat] :
( ( N
= ( dim_row_nat @ A ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P @ Q )
=> ( ( times_times_mat_nat @ Q @ P )
= ( one_mat_nat @ N ) ) ) ) ).
% similar_mat_witD(2)
thf(fact_38_similar__mat__witD_I2_J,axiom,
! [N: nat,A: mat_a,B: mat_a,P: mat_a,Q: mat_a] :
( ( N
= ( dim_row_a @ A ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( ( times_times_mat_a @ Q @ P )
= ( one_mat_a @ N ) ) ) ) ).
% similar_mat_witD(2)
thf(fact_39_mat__conj__unit__commute,axiom,
! [U2: mat_a,A: mat_a,N: nat] :
( ( complex_unitary_a @ U2 )
=> ( ( ( times_times_mat_a @ U2 @ A )
= ( times_times_mat_a @ A @ U2 ) )
=> ( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ U2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( spectr5828033140197310157conj_a @ U2 @ A )
= A ) ) ) ) ) ).
% mat_conj_unit_commute
thf(fact_40_unitaryD2,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ A )
=> ( inverts_mat_a @ ( schur_mat_adjoint_a @ A ) @ A ) ) ) ).
% unitaryD2
thf(fact_41_car,axiom,
ord_le3318621148231462513_mat_a @ ( insert_mat_a @ b @ ( insert_mat_a @ a2 @ ( insert_mat_a @ ( schur_mat_adjoint_a @ u ) @ ( insert_mat_a @ u @ bot_bot_set_mat_a ) ) ) ) @ ( carrier_mat_a @ ( dim_row_a @ b ) @ ( dim_row_a @ b ) ) ).
% car
thf(fact_42_unitary__elim,axiom,
! [A: mat_a,N: nat,B: mat_a,P: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ P @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ P )
=> ( ( ( times_times_mat_a @ ( times_times_mat_a @ P @ A ) @ ( schur_mat_adjoint_a @ P ) )
= ( times_times_mat_a @ ( times_times_mat_a @ P @ B ) @ ( schur_mat_adjoint_a @ P ) ) )
=> ( A = B ) ) ) ) ) ) ).
% unitary_elim
thf(fact_43_similar__mat__witD2_I7_J,axiom,
! [A: mat_nat,N: nat,M: nat,B: mat_nat,P: mat_nat,Q: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ M ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P @ Q )
=> ( member_mat_nat @ Q @ ( carrier_mat_nat @ N @ N ) ) ) ) ).
% similar_mat_witD2(7)
thf(fact_44_similar__mat__witD2_I7_J,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,P: mat_a,Q: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( member_mat_a @ Q @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD2(7)
thf(fact_45_similar__mat__witD2_I6_J,axiom,
! [A: mat_nat,N: nat,M: nat,B: mat_nat,P: mat_nat,Q: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ M ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P @ Q )
=> ( member_mat_nat @ P @ ( carrier_mat_nat @ N @ N ) ) ) ) ).
% similar_mat_witD2(6)
thf(fact_46_similar__mat__witD2_I6_J,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,P: mat_a,Q: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( member_mat_a @ P @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD2(6)
thf(fact_47_similar__mat__witD2_I5_J,axiom,
! [A: mat_nat,N: nat,M: nat,B: mat_nat,P: mat_nat,Q: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ M ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P @ Q )
=> ( member_mat_nat @ B @ ( carrier_mat_nat @ N @ N ) ) ) ) ).
% similar_mat_witD2(5)
thf(fact_48_similar__mat__witD2_I5_J,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,P: mat_a,Q: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD2(5)
thf(fact_49_similar__mat__witD2_I4_J,axiom,
! [A: mat_nat,N: nat,M: nat,B: mat_nat,P: mat_nat,Q: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ M ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P @ Q )
=> ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ N ) ) ) ) ).
% similar_mat_witD2(4)
thf(fact_50_similar__mat__witD2_I4_J,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,P: mat_a,Q: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD2(4)
thf(fact_51_similar__mat__wit__sym,axiom,
! [A: mat_a,B: mat_a,P: mat_a,Q: mat_a] :
( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( similar_mat_wit_a @ B @ A @ Q @ P ) ) ).
% similar_mat_wit_sym
thf(fact_52_similar__mat__witD_I4_J,axiom,
! [N: nat,A: mat_nat,B: mat_nat,P: mat_nat,Q: mat_nat] :
( ( N
= ( dim_row_nat @ A ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P @ Q )
=> ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ N ) ) ) ) ).
% similar_mat_witD(4)
thf(fact_53_similar__mat__witD_I4_J,axiom,
! [N: nat,A: mat_a,B: mat_a,P: mat_a,Q: mat_a] :
( ( N
= ( dim_row_a @ A ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD(4)
thf(fact_54_similar__mat__witD_I5_J,axiom,
! [N: nat,A: mat_nat,B: mat_nat,P: mat_nat,Q: mat_nat] :
( ( N
= ( dim_row_nat @ A ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P @ Q )
=> ( member_mat_nat @ B @ ( carrier_mat_nat @ N @ N ) ) ) ) ).
% similar_mat_witD(5)
thf(fact_55_similar__mat__witD_I5_J,axiom,
! [N: nat,A: mat_a,B: mat_a,P: mat_a,Q: mat_a] :
( ( N
= ( dim_row_a @ A ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD(5)
thf(fact_56_similar__mat__witD_I6_J,axiom,
! [N: nat,A: mat_nat,B: mat_nat,P: mat_nat,Q: mat_nat] :
( ( N
= ( dim_row_nat @ A ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P @ Q )
=> ( member_mat_nat @ P @ ( carrier_mat_nat @ N @ N ) ) ) ) ).
% similar_mat_witD(6)
thf(fact_57_similar__mat__witD_I6_J,axiom,
! [N: nat,A: mat_a,B: mat_a,P: mat_a,Q: mat_a] :
( ( N
= ( dim_row_a @ A ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( member_mat_a @ P @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD(6)
thf(fact_58_similar__mat__witD_I7_J,axiom,
! [N: nat,A: mat_nat,B: mat_nat,P: mat_nat,Q: mat_nat] :
( ( N
= ( dim_row_nat @ A ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P @ Q )
=> ( member_mat_nat @ Q @ ( carrier_mat_nat @ N @ N ) ) ) ) ).
% similar_mat_witD(7)
thf(fact_59_similar__mat__witD_I7_J,axiom,
! [N: nat,A: mat_a,B: mat_a,P: mat_a,Q: mat_a] :
( ( N
= ( dim_row_a @ A ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( member_mat_a @ Q @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% similar_mat_witD(7)
thf(fact_60_similar__mat__witD2_I3_J,axiom,
! [A: mat_nat,N: nat,M: nat,B: mat_nat,P: mat_nat,Q: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ M ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P @ Q )
=> ( A
= ( times_times_mat_nat @ ( times_times_mat_nat @ P @ B ) @ Q ) ) ) ) ).
% similar_mat_witD2(3)
thf(fact_61_similar__mat__witD2_I3_J,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,P: mat_a,Q: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( A
= ( times_times_mat_a @ ( times_times_mat_a @ P @ B ) @ Q ) ) ) ) ).
% similar_mat_witD2(3)
thf(fact_62_similar__mat__wit__refl,axiom,
! [A: mat_nat,N: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ N ) )
=> ( similar_mat_wit_nat @ A @ A @ ( one_mat_nat @ N ) @ ( one_mat_nat @ N ) ) ) ).
% similar_mat_wit_refl
thf(fact_63_similar__mat__wit__refl,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( similar_mat_wit_a @ A @ A @ ( one_mat_a @ N ) @ ( one_mat_a @ N ) ) ) ).
% similar_mat_wit_refl
thf(fact_64_mem__Collect__eq,axiom,
! [A3: mat_nat,P: mat_nat > $o] :
( ( member_mat_nat @ A3 @ ( collect_mat_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
! [A3: mat_a,P: mat_a > $o] :
( ( member_mat_a @ A3 @ ( collect_mat_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
! [A3: set_mat_a,P: set_mat_a > $o] :
( ( member_set_mat_a @ A3 @ ( collect_set_mat_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A3 @ ( collect_set_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
! [A3: nat > nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ A3 @ ( collect_nat_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
! [A3: nat,P: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A: set_mat_nat] :
( ( collect_mat_nat
@ ^ [X: mat_nat] : ( member_mat_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A: set_mat_a] :
( ( collect_mat_a
@ ^ [X: mat_a] : ( member_mat_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A: set_set_mat_a] :
( ( collect_set_mat_a
@ ^ [X: set_mat_a] : ( member_set_mat_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( member_set_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_75_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_76_Collect__cong,axiom,
! [P: mat_a > $o,Q: mat_a > $o] :
( ! [X2: mat_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_mat_a @ P )
= ( collect_mat_a @ Q ) ) ) ).
% Collect_cong
thf(fact_77_Collect__cong,axiom,
! [P: set_mat_a > $o,Q: set_mat_a > $o] :
( ! [X2: set_mat_a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_mat_a @ P )
= ( collect_set_mat_a @ Q ) ) ) ).
% Collect_cong
thf(fact_78_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X2: set_nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_79_Collect__cong,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X2: nat > nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat_nat @ P )
= ( collect_nat_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_80_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_81_similar__mat__witD2_I2_J,axiom,
! [A: mat_nat,N: nat,M: nat,B: mat_nat,P: mat_nat,Q: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ M ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P @ Q )
=> ( ( times_times_mat_nat @ Q @ P )
= ( one_mat_nat @ N ) ) ) ) ).
% similar_mat_witD2(2)
thf(fact_82_similar__mat__witD2_I2_J,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,P: mat_a,Q: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( ( times_times_mat_a @ Q @ P )
= ( one_mat_a @ N ) ) ) ) ).
% similar_mat_witD2(2)
thf(fact_83_similar__mat__witD2_I1_J,axiom,
! [A: mat_nat,N: nat,M: nat,B: mat_nat,P: mat_nat,Q: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ M ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P @ Q )
=> ( ( times_times_mat_nat @ P @ Q )
= ( one_mat_nat @ N ) ) ) ) ).
% similar_mat_witD2(1)
thf(fact_84_similar__mat__witD2_I1_J,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,P: mat_a,Q: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( ( times_times_mat_a @ P @ Q )
= ( one_mat_a @ N ) ) ) ) ).
% similar_mat_witD2(1)
thf(fact_85_similar__mat__witI,axiom,
! [P: mat_nat,Q: mat_nat,N: nat,A: mat_nat,B: mat_nat] :
( ( ( times_times_mat_nat @ P @ Q )
= ( one_mat_nat @ N ) )
=> ( ( ( times_times_mat_nat @ Q @ P )
= ( one_mat_nat @ N ) )
=> ( ( A
= ( times_times_mat_nat @ ( times_times_mat_nat @ P @ B ) @ Q ) )
=> ( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ N ) )
=> ( ( member_mat_nat @ B @ ( carrier_mat_nat @ N @ N ) )
=> ( ( member_mat_nat @ P @ ( carrier_mat_nat @ N @ N ) )
=> ( ( member_mat_nat @ Q @ ( carrier_mat_nat @ N @ N ) )
=> ( similar_mat_wit_nat @ A @ B @ P @ Q ) ) ) ) ) ) ) ) ).
% similar_mat_witI
thf(fact_86_similar__mat__witI,axiom,
! [P: mat_a,Q: mat_a,N: nat,A: mat_a,B: mat_a] :
( ( ( times_times_mat_a @ P @ Q )
= ( one_mat_a @ N ) )
=> ( ( ( times_times_mat_a @ Q @ P )
= ( one_mat_a @ N ) )
=> ( ( A
= ( times_times_mat_a @ ( times_times_mat_a @ P @ B ) @ Q ) )
=> ( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ P @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ Q @ ( carrier_mat_a @ N @ N ) )
=> ( similar_mat_wit_a @ A @ B @ P @ Q ) ) ) ) ) ) ) ) ).
% similar_mat_witI
thf(fact_87_carrier__matD_I1_J,axiom,
! [A: mat_nat,Nr: nat,Nc: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( dim_row_nat @ A )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_88_carrier__matD_I1_J,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( dim_row_a @ A )
= Nr ) ) ).
% carrier_matD(1)
thf(fact_89_assoc__mult__mat,axiom,
! [A: mat_nat,N_1: nat,N_2: nat,B: mat_nat,N_3: nat,C2: mat_nat,N_4: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N_1 @ N_2 ) )
=> ( ( member_mat_nat @ B @ ( carrier_mat_nat @ N_2 @ N_3 ) )
=> ( ( member_mat_nat @ C2 @ ( carrier_mat_nat @ N_3 @ N_4 ) )
=> ( ( times_times_mat_nat @ ( times_times_mat_nat @ A @ B ) @ C2 )
= ( times_times_mat_nat @ A @ ( times_times_mat_nat @ B @ C2 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_90_assoc__mult__mat,axiom,
! [A: mat_a,N_1: nat,N_2: nat,B: mat_a,N_3: nat,C2: mat_a,N_4: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N_1 @ N_2 ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N_2 @ N_3 ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N_3 @ N_4 ) )
=> ( ( times_times_mat_a @ ( times_times_mat_a @ A @ B ) @ C2 )
= ( times_times_mat_a @ A @ ( times_times_mat_a @ B @ C2 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_91_mult__carrier__mat,axiom,
! [A: mat_nat,Nr: nat,N: nat,B: mat_nat,Nc: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ N ) )
=> ( ( member_mat_nat @ B @ ( carrier_mat_nat @ N @ Nc ) )
=> ( member_mat_nat @ ( times_times_mat_nat @ A @ B ) @ ( carrier_mat_nat @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_92_mult__carrier__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,B: mat_a,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ Nc ) )
=> ( member_mat_a @ ( times_times_mat_a @ A @ B ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_93_one__carrier__mat,axiom,
! [N: nat] : ( member_mat_nat @ ( one_mat_nat @ N ) @ ( carrier_mat_nat @ N @ N ) ) ).
% one_carrier_mat
thf(fact_94_one__carrier__mat,axiom,
! [N: nat] : ( member_mat_a @ ( one_mat_a @ N ) @ ( carrier_mat_a @ N @ N ) ) ).
% one_carrier_mat
thf(fact_95_adjoint__dim_H,axiom,
! [A: mat_a,N: nat,M: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( member_mat_a @ ( schur_mat_adjoint_a @ A ) @ ( carrier_mat_a @ M @ N ) ) ) ).
% adjoint_dim'
thf(fact_96_adjoint__dim,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( member_mat_a @ ( schur_mat_adjoint_a @ A ) @ ( carrier_mat_a @ N @ N ) ) ) ).
% adjoint_dim
thf(fact_97_similar__mat__wit__trans,axiom,
! [A: mat_nat,B: mat_nat,P: mat_nat,Q: mat_nat,C2: mat_nat,P2: mat_nat,Q2: mat_nat] :
( ( similar_mat_wit_nat @ A @ B @ P @ Q )
=> ( ( similar_mat_wit_nat @ B @ C2 @ P2 @ Q2 )
=> ( similar_mat_wit_nat @ A @ C2 @ ( times_times_mat_nat @ P @ P2 ) @ ( times_times_mat_nat @ Q2 @ Q ) ) ) ) ).
% similar_mat_wit_trans
thf(fact_98_similar__mat__wit__trans,axiom,
! [A: mat_a,B: mat_a,P: mat_a,Q: mat_a,C2: mat_a,P2: mat_a,Q2: mat_a] :
( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( ( similar_mat_wit_a @ B @ C2 @ P2 @ Q2 )
=> ( similar_mat_wit_a @ A @ C2 @ ( times_times_mat_a @ P @ P2 ) @ ( times_times_mat_a @ Q2 @ Q ) ) ) ) ).
% similar_mat_wit_trans
thf(fact_99_unitarily__equiv__carrier_I1_J,axiom,
! [A: mat_a,N: nat,B: mat_a,U2: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( spectr4825054497075562704quiv_a @ A @ B @ U2 )
=> ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% unitarily_equiv_carrier(1)
thf(fact_100_unitarily__equiv__carrier_I2_J,axiom,
! [A: mat_a,N: nat,B: mat_a,U2: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( spectr4825054497075562704quiv_a @ A @ B @ U2 )
=> ( member_mat_a @ U2 @ ( carrier_mat_a @ N @ N ) ) ) ) ).
% unitarily_equiv_carrier(2)
thf(fact_101_inverts__mat__unique,axiom,
! [A: mat_a,N: nat,B: mat_a,C2: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( inverts_mat_a @ A @ B )
=> ( ( inverts_mat_a @ A @ C2 )
=> ( B = C2 ) ) ) ) ) ) ).
% inverts_mat_unique
thf(fact_102_inverts__mat__symm,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( inverts_mat_a @ A @ B )
=> ( inverts_mat_a @ B @ A ) ) ) ) ).
% inverts_mat_symm
thf(fact_103_left__mult__one__mat,axiom,
! [A: mat_nat,Nr: nat,Nc: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( times_times_mat_nat @ ( one_mat_nat @ Nr ) @ A )
= A ) ) ).
% left_mult_one_mat
thf(fact_104_left__mult__one__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( times_times_mat_a @ ( one_mat_a @ Nr ) @ A )
= A ) ) ).
% left_mult_one_mat
thf(fact_105_right__mult__one__mat,axiom,
! [A: mat_nat,Nr: nat,Nc: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( times_times_mat_nat @ A @ ( one_mat_nat @ Nc ) )
= A ) ) ).
% right_mult_one_mat
thf(fact_106_right__mult__one__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( times_times_mat_a @ A @ ( one_mat_a @ Nc ) )
= A ) ) ).
% right_mult_one_mat
thf(fact_107_adjoint__mult,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,L: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ M @ L ) )
=> ( ( schur_mat_adjoint_a @ ( times_times_mat_a @ A @ B ) )
= ( times_times_mat_a @ ( schur_mat_adjoint_a @ B ) @ ( schur_mat_adjoint_a @ A ) ) ) ) ) ).
% adjoint_mult
thf(fact_108_unitary__times__unitary,axiom,
! [P: mat_a,N: nat,Q: mat_a] :
( ( member_mat_a @ P @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ Q @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ P )
=> ( ( complex_unitary_a @ Q )
=> ( complex_unitary_a @ ( times_times_mat_a @ P @ Q ) ) ) ) ) ) ).
% unitary_times_unitary
thf(fact_109_unitary__adjoint,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ A )
=> ( complex_unitary_a @ ( schur_mat_adjoint_a @ A ) ) ) ) ).
% unitary_adjoint
thf(fact_110_unitarily__equiv__carrier_H_I3_J,axiom,
! [A: mat_a,B: mat_a,U2: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U2 )
=> ( member_mat_a @ U2 @ ( carrier_mat_a @ ( dim_row_a @ A ) @ ( dim_row_a @ A ) ) ) ) ).
% unitarily_equiv_carrier'(3)
thf(fact_111_unitarily__equiv__carrier_H_I2_J,axiom,
! [A: mat_a,B: mat_a,U2: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U2 )
=> ( member_mat_a @ B @ ( carrier_mat_a @ ( dim_row_a @ A ) @ ( dim_row_a @ A ) ) ) ) ).
% unitarily_equiv_carrier'(2)
thf(fact_112_unitarily__equiv__carrier_H_I1_J,axiom,
! [A: mat_a,B: mat_a,U2: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U2 )
=> ( member_mat_a @ A @ ( carrier_mat_a @ ( dim_row_a @ A ) @ ( dim_row_a @ A ) ) ) ) ).
% unitarily_equiv_carrier'(1)
thf(fact_113_similar__mat__witD_I3_J,axiom,
! [N: nat,A: mat_nat,B: mat_nat,P: mat_nat,Q: mat_nat] :
( ( N
= ( dim_row_nat @ A ) )
=> ( ( similar_mat_wit_nat @ A @ B @ P @ Q )
=> ( A
= ( times_times_mat_nat @ ( times_times_mat_nat @ P @ B ) @ Q ) ) ) ) ).
% similar_mat_witD(3)
thf(fact_114_similar__mat__witD_I3_J,axiom,
! [N: nat,A: mat_a,B: mat_a,P: mat_a,Q: mat_a] :
( ( N
= ( dim_row_a @ A ) )
=> ( ( similar_mat_wit_a @ A @ B @ P @ Q )
=> ( A
= ( times_times_mat_a @ ( times_times_mat_a @ P @ B ) @ Q ) ) ) ) ).
% similar_mat_witD(3)
thf(fact_115_unitarily__equivD_I2_J,axiom,
! [A: mat_a,B: mat_a,U2: mat_a] :
( ( spectr4825054497075562704quiv_a @ A @ B @ U2 )
=> ( similar_mat_wit_a @ A @ B @ U2 @ ( schur_mat_adjoint_a @ U2 ) ) ) ).
% unitarily_equivD(2)
thf(fact_116_mat__mult__left__right__inverse,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( ( times_times_mat_a @ A @ B )
= ( one_mat_a @ N ) )
=> ( ( times_times_mat_a @ B @ A )
= ( one_mat_a @ N ) ) ) ) ) ).
% mat_mult_left_right_inverse
thf(fact_117_last__subset,axiom,
! [A: set_set_mat_a,A3: set_mat_a,B3: set_mat_a] :
( ( ord_le2341747070211005607_mat_a @ A @ ( insert_set_mat_a @ A3 @ ( insert_set_mat_a @ B3 @ bot_bo8661580253428394715_mat_a ) ) )
=> ( ( A3 != B3 )
=> ( ( A
!= ( insert_set_mat_a @ A3 @ ( insert_set_mat_a @ B3 @ bot_bo8661580253428394715_mat_a ) ) )
=> ( ( A != bot_bo8661580253428394715_mat_a )
=> ( ( A
!= ( insert_set_mat_a @ A3 @ bot_bo8661580253428394715_mat_a ) )
=> ( A
= ( insert_set_mat_a @ B3 @ bot_bo8661580253428394715_mat_a ) ) ) ) ) ) ) ).
% last_subset
thf(fact_118_last__subset,axiom,
! [A: set_nat,A3: nat,B3: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ A3 @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) )
=> ( ( A3 != B3 )
=> ( ( A
!= ( insert_nat @ A3 @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) )
=> ( ( A != bot_bot_set_nat )
=> ( ( A
!= ( insert_nat @ A3 @ bot_bot_set_nat ) )
=> ( A
= ( insert_nat @ B3 @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% last_subset
thf(fact_119_last__subset,axiom,
! [A: set_mat_a,A3: mat_a,B3: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ ( insert_mat_a @ A3 @ ( insert_mat_a @ B3 @ bot_bot_set_mat_a ) ) )
=> ( ( A3 != B3 )
=> ( ( A
!= ( insert_mat_a @ A3 @ ( insert_mat_a @ B3 @ bot_bot_set_mat_a ) ) )
=> ( ( A != bot_bot_set_mat_a )
=> ( ( A
!= ( insert_mat_a @ A3 @ bot_bot_set_mat_a ) )
=> ( A
= ( insert_mat_a @ B3 @ bot_bot_set_mat_a ) ) ) ) ) ) ) ).
% last_subset
thf(fact_120_singleton__insert__inj__eq_H,axiom,
! [A3: set_mat_a,A: set_set_mat_a,B3: set_mat_a] :
( ( ( insert_set_mat_a @ A3 @ A )
= ( insert_set_mat_a @ B3 @ bot_bo8661580253428394715_mat_a ) )
= ( ( A3 = B3 )
& ( ord_le2341747070211005607_mat_a @ A @ ( insert_set_mat_a @ B3 @ bot_bo8661580253428394715_mat_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_121_singleton__insert__inj__eq_H,axiom,
! [A3: nat,A: set_nat,B3: nat] :
( ( ( insert_nat @ A3 @ A )
= ( insert_nat @ B3 @ bot_bot_set_nat ) )
= ( ( A3 = B3 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_122_singleton__insert__inj__eq_H,axiom,
! [A3: mat_a,A: set_mat_a,B3: mat_a] :
( ( ( insert_mat_a @ A3 @ A )
= ( insert_mat_a @ B3 @ bot_bot_set_mat_a ) )
= ( ( A3 = B3 )
& ( ord_le3318621148231462513_mat_a @ A @ ( insert_mat_a @ B3 @ bot_bot_set_mat_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_123_singleton__insert__inj__eq,axiom,
! [B3: set_mat_a,A3: set_mat_a,A: set_set_mat_a] :
( ( ( insert_set_mat_a @ B3 @ bot_bo8661580253428394715_mat_a )
= ( insert_set_mat_a @ A3 @ A ) )
= ( ( A3 = B3 )
& ( ord_le2341747070211005607_mat_a @ A @ ( insert_set_mat_a @ B3 @ bot_bo8661580253428394715_mat_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_124_singleton__insert__inj__eq,axiom,
! [B3: nat,A3: nat,A: set_nat] :
( ( ( insert_nat @ B3 @ bot_bot_set_nat )
= ( insert_nat @ A3 @ A ) )
= ( ( A3 = B3 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_125_singleton__insert__inj__eq,axiom,
! [B3: mat_a,A3: mat_a,A: set_mat_a] :
( ( ( insert_mat_a @ B3 @ bot_bot_set_mat_a )
= ( insert_mat_a @ A3 @ A ) )
= ( ( A3 = B3 )
& ( ord_le3318621148231462513_mat_a @ A @ ( insert_mat_a @ B3 @ bot_bot_set_mat_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_126_subset__singleton__iff,axiom,
! [X3: set_set_mat_a,A3: set_mat_a] :
( ( ord_le2341747070211005607_mat_a @ X3 @ ( insert_set_mat_a @ A3 @ bot_bo8661580253428394715_mat_a ) )
= ( ( X3 = bot_bo8661580253428394715_mat_a )
| ( X3
= ( insert_set_mat_a @ A3 @ bot_bo8661580253428394715_mat_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_127_subset__singleton__iff,axiom,
! [X3: set_nat,A3: nat] :
( ( ord_less_eq_set_nat @ X3 @ ( insert_nat @ A3 @ bot_bot_set_nat ) )
= ( ( X3 = bot_bot_set_nat )
| ( X3
= ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_128_subset__singleton__iff,axiom,
! [X3: set_mat_a,A3: mat_a] :
( ( ord_le3318621148231462513_mat_a @ X3 @ ( insert_mat_a @ A3 @ bot_bot_set_mat_a ) )
= ( ( X3 = bot_bot_set_mat_a )
| ( X3
= ( insert_mat_a @ A3 @ bot_bot_set_mat_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_129_subset__singletonD,axiom,
! [A: set_set_mat_a,X4: set_mat_a] :
( ( ord_le2341747070211005607_mat_a @ A @ ( insert_set_mat_a @ X4 @ bot_bo8661580253428394715_mat_a ) )
=> ( ( A = bot_bo8661580253428394715_mat_a )
| ( A
= ( insert_set_mat_a @ X4 @ bot_bo8661580253428394715_mat_a ) ) ) ) ).
% subset_singletonD
thf(fact_130_subset__singletonD,axiom,
! [A: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_131_subset__singletonD,axiom,
! [A: set_mat_a,X4: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) )
=> ( ( A = bot_bot_set_mat_a )
| ( A
= ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) ) ) ) ).
% subset_singletonD
thf(fact_132_similar__matD,axiom,
! [A: mat_nat,B: mat_nat] :
( ( similar_mat_nat @ A @ B )
=> ? [N2: nat,P3: mat_nat,Q3: mat_nat] :
( ( ord_le7789122042438455497at_nat @ ( insert_mat_nat @ A @ ( insert_mat_nat @ B @ ( insert_mat_nat @ P3 @ ( insert_mat_nat @ Q3 @ bot_bot_set_mat_nat ) ) ) ) @ ( carrier_mat_nat @ N2 @ N2 ) )
& ( ( times_times_mat_nat @ P3 @ Q3 )
= ( one_mat_nat @ N2 ) )
& ( ( times_times_mat_nat @ Q3 @ P3 )
= ( one_mat_nat @ N2 ) )
& ( A
= ( times_times_mat_nat @ ( times_times_mat_nat @ P3 @ B ) @ Q3 ) ) ) ) ).
% similar_matD
thf(fact_133_similar__matD,axiom,
! [A: mat_a,B: mat_a] :
( ( similar_mat_a @ A @ B )
=> ? [N2: nat,P3: mat_a,Q3: mat_a] :
( ( ord_le3318621148231462513_mat_a @ ( insert_mat_a @ A @ ( insert_mat_a @ B @ ( insert_mat_a @ P3 @ ( insert_mat_a @ Q3 @ bot_bot_set_mat_a ) ) ) ) @ ( carrier_mat_a @ N2 @ N2 ) )
& ( ( times_times_mat_a @ P3 @ Q3 )
= ( one_mat_a @ N2 ) )
& ( ( times_times_mat_a @ Q3 @ P3 )
= ( one_mat_a @ N2 ) )
& ( A
= ( times_times_mat_a @ ( times_times_mat_a @ P3 @ B ) @ Q3 ) ) ) ) ).
% similar_matD
thf(fact_134_similar__matI,axiom,
! [A: mat_nat,B: mat_nat,P: mat_nat,Q: mat_nat,N: nat] :
( ( ord_le7789122042438455497at_nat @ ( insert_mat_nat @ A @ ( insert_mat_nat @ B @ ( insert_mat_nat @ P @ ( insert_mat_nat @ Q @ bot_bot_set_mat_nat ) ) ) ) @ ( carrier_mat_nat @ N @ N ) )
=> ( ( ( times_times_mat_nat @ P @ Q )
= ( one_mat_nat @ N ) )
=> ( ( ( times_times_mat_nat @ Q @ P )
= ( one_mat_nat @ N ) )
=> ( ( A
= ( times_times_mat_nat @ ( times_times_mat_nat @ P @ B ) @ Q ) )
=> ( similar_mat_nat @ A @ B ) ) ) ) ) ).
% similar_matI
thf(fact_135_similar__matI,axiom,
! [A: mat_a,B: mat_a,P: mat_a,Q: mat_a,N: nat] :
( ( ord_le3318621148231462513_mat_a @ ( insert_mat_a @ A @ ( insert_mat_a @ B @ ( insert_mat_a @ P @ ( insert_mat_a @ Q @ bot_bot_set_mat_a ) ) ) ) @ ( carrier_mat_a @ N @ N ) )
=> ( ( ( times_times_mat_a @ P @ Q )
= ( one_mat_a @ N ) )
=> ( ( ( times_times_mat_a @ Q @ P )
= ( one_mat_a @ N ) )
=> ( ( A
= ( times_times_mat_a @ ( times_times_mat_a @ P @ B ) @ Q ) )
=> ( similar_mat_a @ A @ B ) ) ) ) ) ).
% similar_matI
thf(fact_136_similar__mat__wit__dim__row,axiom,
! [A: mat_nat,B: mat_nat,Q: mat_nat,R: mat_nat] :
( ( similar_mat_wit_nat @ A @ B @ Q @ R )
=> ( ( dim_row_nat @ B )
= ( dim_row_nat @ A ) ) ) ).
% similar_mat_wit_dim_row
thf(fact_137_similar__mat__wit__dim__row,axiom,
! [A: mat_a,B: mat_a,Q: mat_a,R: mat_a] :
( ( similar_mat_wit_a @ A @ B @ Q @ R )
=> ( ( dim_row_a @ B )
= ( dim_row_a @ A ) ) ) ).
% similar_mat_wit_dim_row
thf(fact_138_singleton__inject,axiom,
! [A3: nat,B3: nat] :
( ( ( insert_nat @ A3 @ bot_bot_set_nat )
= ( insert_nat @ B3 @ bot_bot_set_nat ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_139_singleton__inject,axiom,
! [A3: set_mat_a,B3: set_mat_a] :
( ( ( insert_set_mat_a @ A3 @ bot_bo8661580253428394715_mat_a )
= ( insert_set_mat_a @ B3 @ bot_bo8661580253428394715_mat_a ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_140_singleton__inject,axiom,
! [A3: mat_a,B3: mat_a] :
( ( ( insert_mat_a @ A3 @ bot_bot_set_mat_a )
= ( insert_mat_a @ B3 @ bot_bot_set_mat_a ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_141_insert__not__empty,axiom,
! [A3: nat,A: set_nat] :
( ( insert_nat @ A3 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_142_insert__not__empty,axiom,
! [A3: set_mat_a,A: set_set_mat_a] :
( ( insert_set_mat_a @ A3 @ A )
!= bot_bo8661580253428394715_mat_a ) ).
% insert_not_empty
thf(fact_143_insert__not__empty,axiom,
! [A3: mat_a,A: set_mat_a] :
( ( insert_mat_a @ A3 @ A )
!= bot_bot_set_mat_a ) ).
% insert_not_empty
thf(fact_144_similar__mat__trans,axiom,
! [A: mat_a,B: mat_a,C2: mat_a] :
( ( similar_mat_a @ A @ B )
=> ( ( similar_mat_a @ B @ C2 )
=> ( similar_mat_a @ A @ C2 ) ) ) ).
% similar_mat_trans
thf(fact_145_similar__mat__sym,axiom,
! [A: mat_a,B: mat_a] :
( ( similar_mat_a @ A @ B )
=> ( similar_mat_a @ B @ A ) ) ).
% similar_mat_sym
thf(fact_146_similar__mat__refl,axiom,
! [A: mat_nat,N: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ N @ N ) )
=> ( similar_mat_nat @ A @ A ) ) ).
% similar_mat_refl
thf(fact_147_similar__mat__refl,axiom,
! [A: mat_a,N: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( similar_mat_a @ A @ A ) ) ).
% similar_mat_refl
thf(fact_148_similar__mat__def,axiom,
( similar_mat_a
= ( ^ [A2: mat_a,B2: mat_a] :
? [P4: mat_a,X5: mat_a] : ( similar_mat_wit_a @ A2 @ B2 @ P4 @ X5 ) ) ) ).
% similar_mat_def
thf(fact_149_basic__trans__rules_I26_J,axiom,
! [A3: set_nat,B3: set_nat,C: set_nat] :
( ( A3 = B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% basic_trans_rules(26)
thf(fact_150_basic__trans__rules_I26_J,axiom,
! [A3: mat_a > $o,B3: mat_a > $o,C: mat_a > $o] :
( ( A3 = B3 )
=> ( ( ord_less_eq_mat_a_o @ B3 @ C )
=> ( ord_less_eq_mat_a_o @ A3 @ C ) ) ) ).
% basic_trans_rules(26)
thf(fact_151_basic__trans__rules_I26_J,axiom,
! [A3: nat > $o,B3: nat > $o,C: nat > $o] :
( ( A3 = B3 )
=> ( ( ord_less_eq_nat_o @ B3 @ C )
=> ( ord_less_eq_nat_o @ A3 @ C ) ) ) ).
% basic_trans_rules(26)
thf(fact_152_basic__trans__rules_I26_J,axiom,
! [A3: set_mat_a,B3: set_mat_a,C: set_mat_a] :
( ( A3 = B3 )
=> ( ( ord_le3318621148231462513_mat_a @ B3 @ C )
=> ( ord_le3318621148231462513_mat_a @ A3 @ C ) ) ) ).
% basic_trans_rules(26)
thf(fact_153_basic__trans__rules_I26_J,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( A3 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% basic_trans_rules(26)
thf(fact_154_basic__trans__rules_I25_J,axiom,
! [A3: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% basic_trans_rules(25)
thf(fact_155_basic__trans__rules_I25_J,axiom,
! [A3: mat_a > $o,B3: mat_a > $o,C: mat_a > $o] :
( ( ord_less_eq_mat_a_o @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_mat_a_o @ A3 @ C ) ) ) ).
% basic_trans_rules(25)
thf(fact_156_basic__trans__rules_I25_J,axiom,
! [A3: nat > $o,B3: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat_o @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat_o @ A3 @ C ) ) ) ).
% basic_trans_rules(25)
thf(fact_157_basic__trans__rules_I25_J,axiom,
! [A3: set_mat_a,B3: set_mat_a,C: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_le3318621148231462513_mat_a @ A3 @ C ) ) ) ).
% basic_trans_rules(25)
thf(fact_158_basic__trans__rules_I25_J,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% basic_trans_rules(25)
thf(fact_159_basic__trans__rules_I24_J,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% basic_trans_rules(24)
thf(fact_160_basic__trans__rules_I24_J,axiom,
! [A3: mat_a > $o,B3: mat_a > $o] :
( ( ord_less_eq_mat_a_o @ A3 @ B3 )
=> ( ( ord_less_eq_mat_a_o @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% basic_trans_rules(24)
thf(fact_161_basic__trans__rules_I24_J,axiom,
! [A3: nat > $o,B3: nat > $o] :
( ( ord_less_eq_nat_o @ A3 @ B3 )
=> ( ( ord_less_eq_nat_o @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% basic_trans_rules(24)
thf(fact_162_basic__trans__rules_I24_J,axiom,
! [A3: set_mat_a,B3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A3 @ B3 )
=> ( ( ord_le3318621148231462513_mat_a @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% basic_trans_rules(24)
thf(fact_163_basic__trans__rules_I24_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% basic_trans_rules(24)
thf(fact_164_basic__trans__rules_I23_J,axiom,
! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X4 @ Z ) ) ) ).
% basic_trans_rules(23)
thf(fact_165_basic__trans__rules_I23_J,axiom,
! [X4: mat_a > $o,Y: mat_a > $o,Z: mat_a > $o] :
( ( ord_less_eq_mat_a_o @ X4 @ Y )
=> ( ( ord_less_eq_mat_a_o @ Y @ Z )
=> ( ord_less_eq_mat_a_o @ X4 @ Z ) ) ) ).
% basic_trans_rules(23)
thf(fact_166_basic__trans__rules_I23_J,axiom,
! [X4: nat > $o,Y: nat > $o,Z: nat > $o] :
( ( ord_less_eq_nat_o @ X4 @ Y )
=> ( ( ord_less_eq_nat_o @ Y @ Z )
=> ( ord_less_eq_nat_o @ X4 @ Z ) ) ) ).
% basic_trans_rules(23)
thf(fact_167_basic__trans__rules_I23_J,axiom,
! [X4: set_mat_a,Y: set_mat_a,Z: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X4 @ Y )
=> ( ( ord_le3318621148231462513_mat_a @ Y @ Z )
=> ( ord_le3318621148231462513_mat_a @ X4 @ Z ) ) ) ).
% basic_trans_rules(23)
thf(fact_168_basic__trans__rules_I23_J,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X4 @ Z ) ) ) ).
% basic_trans_rules(23)
thf(fact_169_basic__trans__rules_I10_J,axiom,
! [A3: nat,F: nat > nat,B3: nat,C: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_170_basic__trans__rules_I10_J,axiom,
! [A3: nat,F: set_mat_a > nat,B3: set_mat_a,C: set_mat_a] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le3318621148231462513_mat_a @ B3 @ C )
=> ( ! [X2: set_mat_a,Y2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_171_basic__trans__rules_I10_J,axiom,
! [A3: set_mat_a,F: nat > set_mat_a,B3: nat,C: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le3318621148231462513_mat_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3318621148231462513_mat_a @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_172_basic__trans__rules_I10_J,axiom,
! [A3: set_mat_a,F: set_mat_a > set_mat_a,B3: set_mat_a,C: set_mat_a] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le3318621148231462513_mat_a @ B3 @ C )
=> ( ! [X2: set_mat_a,Y2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X2 @ Y2 )
=> ( ord_le3318621148231462513_mat_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3318621148231462513_mat_a @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_173_basic__trans__rules_I10_J,axiom,
! [A3: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_174_basic__trans__rules_I10_J,axiom,
! [A3: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_175_basic__trans__rules_I10_J,axiom,
! [A3: nat > $o,F: nat > nat > $o,B3: nat,C: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat_o @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_176_basic__trans__rules_I10_J,axiom,
! [A3: set_nat,F: set_nat > set_nat,B3: set_nat,C: set_nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_177_basic__trans__rules_I10_J,axiom,
! [A3: nat,F: ( nat > $o ) > nat,B3: nat > $o,C: nat > $o] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat_o @ B3 @ C )
=> ( ! [X2: nat > $o,Y2: nat > $o] :
( ( ord_less_eq_nat_o @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_178_basic__trans__rules_I10_J,axiom,
! [A3: set_nat,F: set_mat_a > set_nat,B3: set_mat_a,C: set_mat_a] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_le3318621148231462513_mat_a @ B3 @ C )
=> ( ! [X2: set_mat_a,Y2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_179_basic__trans__rules_I9_J,axiom,
! [A3: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_180_basic__trans__rules_I9_J,axiom,
! [A3: set_mat_a,B3: set_mat_a,F: set_mat_a > nat,C: nat] :
( ( ord_le3318621148231462513_mat_a @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_mat_a,Y2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_181_basic__trans__rules_I9_J,axiom,
! [A3: nat,B3: nat,F: nat > set_mat_a,C: set_mat_a] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le3318621148231462513_mat_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3318621148231462513_mat_a @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_182_basic__trans__rules_I9_J,axiom,
! [A3: set_mat_a,B3: set_mat_a,F: set_mat_a > set_mat_a,C: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_mat_a,Y2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X2 @ Y2 )
=> ( ord_le3318621148231462513_mat_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3318621148231462513_mat_a @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_183_basic__trans__rules_I9_J,axiom,
! [A3: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_184_basic__trans__rules_I9_J,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_185_basic__trans__rules_I9_J,axiom,
! [A3: nat,B3: nat,F: nat > nat > $o,C: nat > $o] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_186_basic__trans__rules_I9_J,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_187_basic__trans__rules_I9_J,axiom,
! [A3: nat > $o,B3: nat > $o,F: ( nat > $o ) > nat,C: nat] :
( ( ord_less_eq_nat_o @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat > $o,Y2: nat > $o] :
( ( ord_less_eq_nat_o @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_188_basic__trans__rules_I9_J,axiom,
! [A3: set_mat_a,B3: set_mat_a,F: set_mat_a > set_nat,C: set_nat] :
( ( ord_le3318621148231462513_mat_a @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: set_mat_a,Y2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_189_basic__trans__rules_I8_J,axiom,
! [A3: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_190_basic__trans__rules_I8_J,axiom,
! [A3: set_mat_a,F: nat > set_mat_a,B3: nat,C: nat] :
( ( ord_le3318621148231462513_mat_a @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le3318621148231462513_mat_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3318621148231462513_mat_a @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_191_basic__trans__rules_I8_J,axiom,
! [A3: nat,F: set_mat_a > nat,B3: set_mat_a,C: set_mat_a] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_le3318621148231462513_mat_a @ B3 @ C )
=> ( ! [X2: set_mat_a,Y2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_192_basic__trans__rules_I8_J,axiom,
! [A3: set_mat_a,F: set_mat_a > set_mat_a,B3: set_mat_a,C: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A3 @ ( F @ B3 ) )
=> ( ( ord_le3318621148231462513_mat_a @ B3 @ C )
=> ( ! [X2: set_mat_a,Y2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X2 @ Y2 )
=> ( ord_le3318621148231462513_mat_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3318621148231462513_mat_a @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_193_basic__trans__rules_I8_J,axiom,
! [A3: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_194_basic__trans__rules_I8_J,axiom,
! [A3: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_195_basic__trans__rules_I8_J,axiom,
! [A3: nat,F: ( nat > $o ) > nat,B3: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat_o @ B3 @ C )
=> ( ! [X2: nat > $o,Y2: nat > $o] :
( ( ord_less_eq_nat_o @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_196_basic__trans__rules_I8_J,axiom,
! [A3: set_nat,F: set_nat > set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_197_basic__trans__rules_I8_J,axiom,
! [A3: nat > $o,F: nat > nat > $o,B3: nat,C: nat] :
( ( ord_less_eq_nat_o @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat_o @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_198_basic__trans__rules_I8_J,axiom,
! [A3: set_mat_a,F: set_nat > set_mat_a,B3: set_nat,C: set_nat] :
( ( ord_le3318621148231462513_mat_a @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_le3318621148231462513_mat_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3318621148231462513_mat_a @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_199_basic__trans__rules_I7_J,axiom,
! [A3: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_200_basic__trans__rules_I7_J,axiom,
! [A3: set_mat_a,B3: set_mat_a,F: set_mat_a > nat,C: nat] :
( ( ord_le3318621148231462513_mat_a @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: set_mat_a,Y2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_201_basic__trans__rules_I7_J,axiom,
! [A3: nat,B3: nat,F: nat > set_mat_a,C: set_mat_a] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_le3318621148231462513_mat_a @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le3318621148231462513_mat_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3318621148231462513_mat_a @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_202_basic__trans__rules_I7_J,axiom,
! [A3: set_mat_a,B3: set_mat_a,F: set_mat_a > set_mat_a,C: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A3 @ B3 )
=> ( ( ord_le3318621148231462513_mat_a @ ( F @ B3 ) @ C )
=> ( ! [X2: set_mat_a,Y2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X2 @ Y2 )
=> ( ord_le3318621148231462513_mat_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le3318621148231462513_mat_a @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_203_basic__trans__rules_I7_J,axiom,
! [A3: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_204_basic__trans__rules_I7_J,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_205_basic__trans__rules_I7_J,axiom,
! [A3: nat,B3: nat,F: nat > nat > $o,C: nat > $o] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat_o @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat_o @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_206_basic__trans__rules_I7_J,axiom,
! [A3: set_nat,B3: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_207_basic__trans__rules_I7_J,axiom,
! [A3: nat > $o,B3: nat > $o,F: ( nat > $o ) > nat,C: nat] :
( ( ord_less_eq_nat_o @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat > $o,Y2: nat > $o] :
( ( ord_less_eq_nat_o @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_208_basic__trans__rules_I7_J,axiom,
! [A3: set_mat_a,B3: set_mat_a,F: set_mat_a > set_nat,C: set_nat] :
( ( ord_le3318621148231462513_mat_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: set_mat_a,Y2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_209_Set_Obasic__monos_I7_J,axiom,
! [A: set_set_mat_a,B: set_set_mat_a,X4: set_mat_a] :
( ( ord_le2341747070211005607_mat_a @ A @ B )
=> ( ( member_set_mat_a @ X4 @ A )
=> ( member_set_mat_a @ X4 @ B ) ) ) ).
% Set.basic_monos(7)
thf(fact_210_Set_Obasic__monos_I7_J,axiom,
! [A: set_mat_nat,B: set_mat_nat,X4: mat_nat] :
( ( ord_le7789122042438455497at_nat @ A @ B )
=> ( ( member_mat_nat @ X4 @ A )
=> ( member_mat_nat @ X4 @ B ) ) ) ).
% Set.basic_monos(7)
thf(fact_211_Set_Obasic__monos_I7_J,axiom,
! [A: set_nat,B: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B ) ) ) ).
% Set.basic_monos(7)
thf(fact_212_Set_Obasic__monos_I7_J,axiom,
! [A: set_mat_a,B: set_mat_a,X4: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( ( member_mat_a @ X4 @ A )
=> ( member_mat_a @ X4 @ B ) ) ) ).
% Set.basic_monos(7)
thf(fact_213_Set_Obasic__monos_I6_J,axiom,
! [P: set_mat_a > $o,Q: set_mat_a > $o] :
( ! [X2: set_mat_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le2341747070211005607_mat_a @ ( collect_set_mat_a @ P ) @ ( collect_set_mat_a @ Q ) ) ) ).
% Set.basic_monos(6)
thf(fact_214_Set_Obasic__monos_I6_J,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X2: set_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Set.basic_monos(6)
thf(fact_215_Set_Obasic__monos_I6_J,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X2: nat > nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Set.basic_monos(6)
thf(fact_216_Set_Obasic__monos_I6_J,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Set.basic_monos(6)
thf(fact_217_Set_Obasic__monos_I6_J,axiom,
! [P: mat_a > $o,Q: mat_a > $o] :
( ! [X2: mat_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le3318621148231462513_mat_a @ ( collect_mat_a @ P ) @ ( collect_mat_a @ Q ) ) ) ).
% Set.basic_monos(6)
thf(fact_218_Set_Obasic__monos_I1_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% Set.basic_monos(1)
thf(fact_219_Set_Obasic__monos_I1_J,axiom,
! [A: set_mat_a] : ( ord_le3318621148231462513_mat_a @ A @ A ) ).
% Set.basic_monos(1)
thf(fact_220_basic__trans__rules_I31_J,axiom,
! [A: set_set_mat_a,B: set_set_mat_a,C: set_mat_a] :
( ( ord_le2341747070211005607_mat_a @ A @ B )
=> ( ( member_set_mat_a @ C @ A )
=> ( member_set_mat_a @ C @ B ) ) ) ).
% basic_trans_rules(31)
thf(fact_221_basic__trans__rules_I31_J,axiom,
! [A: set_mat_nat,B: set_mat_nat,C: mat_nat] :
( ( ord_le7789122042438455497at_nat @ A @ B )
=> ( ( member_mat_nat @ C @ A )
=> ( member_mat_nat @ C @ B ) ) ) ).
% basic_trans_rules(31)
thf(fact_222_basic__trans__rules_I31_J,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% basic_trans_rules(31)
thf(fact_223_basic__trans__rules_I31_J,axiom,
! [A: set_mat_a,B: set_mat_a,C: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( ( member_mat_a @ C @ A )
=> ( member_mat_a @ C @ B ) ) ) ).
% basic_trans_rules(31)
thf(fact_224_subsetI,axiom,
! [A: set_set_mat_a,B: set_set_mat_a] :
( ! [X2: set_mat_a] :
( ( member_set_mat_a @ X2 @ A )
=> ( member_set_mat_a @ X2 @ B ) )
=> ( ord_le2341747070211005607_mat_a @ A @ B ) ) ).
% subsetI
thf(fact_225_subsetI,axiom,
! [A: set_mat_nat,B: set_mat_nat] :
( ! [X2: mat_nat] :
( ( member_mat_nat @ X2 @ A )
=> ( member_mat_nat @ X2 @ B ) )
=> ( ord_le7789122042438455497at_nat @ A @ B ) ) ).
% subsetI
thf(fact_226_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_227_subsetI,axiom,
! [A: set_mat_a,B: set_mat_a] :
( ! [X2: mat_a] :
( ( member_mat_a @ X2 @ A )
=> ( member_mat_a @ X2 @ B ) )
=> ( ord_le3318621148231462513_mat_a @ A @ B ) ) ).
% subsetI
thf(fact_228_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_229_equalityE,axiom,
! [A: set_mat_a,B: set_mat_a] :
( ( A = B )
=> ~ ( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ~ ( ord_le3318621148231462513_mat_a @ B @ A ) ) ) ).
% equalityE
thf(fact_230_equalityI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% equalityI
thf(fact_231_equalityI,axiom,
! [A: set_mat_a,B: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( ( ord_le3318621148231462513_mat_a @ B @ A )
=> ( A = B ) ) ) ).
% equalityI
thf(fact_232_subset__eq,axiom,
( ord_le2341747070211005607_mat_a
= ( ^ [A2: set_set_mat_a,B2: set_set_mat_a] :
! [X: set_mat_a] :
( ( member_set_mat_a @ X @ A2 )
=> ( member_set_mat_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_233_subset__eq,axiom,
( ord_le7789122042438455497at_nat
= ( ^ [A2: set_mat_nat,B2: set_mat_nat] :
! [X: mat_nat] :
( ( member_mat_nat @ X @ A2 )
=> ( member_mat_nat @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_234_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_235_subset__eq,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A2: set_mat_a,B2: set_mat_a] :
! [X: mat_a] :
( ( member_mat_a @ X @ A2 )
=> ( member_mat_a @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_236_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_237_equalityD1,axiom,
! [A: set_mat_a,B: set_mat_a] :
( ( A = B )
=> ( ord_le3318621148231462513_mat_a @ A @ B ) ) ).
% equalityD1
thf(fact_238_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_239_equalityD2,axiom,
! [A: set_mat_a,B: set_mat_a] :
( ( A = B )
=> ( ord_le3318621148231462513_mat_a @ B @ A ) ) ).
% equalityD2
thf(fact_240_subset__iff,axiom,
( ord_le2341747070211005607_mat_a
= ( ^ [A2: set_set_mat_a,B2: set_set_mat_a] :
! [T: set_mat_a] :
( ( member_set_mat_a @ T @ A2 )
=> ( member_set_mat_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_241_subset__iff,axiom,
( ord_le7789122042438455497at_nat
= ( ^ [A2: set_mat_nat,B2: set_mat_nat] :
! [T: mat_nat] :
( ( member_mat_nat @ T @ A2 )
=> ( member_mat_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_242_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A2 )
=> ( member_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_243_subset__iff,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A2: set_mat_a,B2: set_mat_a] :
! [T: mat_a] :
( ( member_mat_a @ T @ A2 )
=> ( member_mat_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_244_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_245_subset__trans,axiom,
! [A: set_mat_a,B: set_mat_a,C2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( ( ord_le3318621148231462513_mat_a @ B @ C2 )
=> ( ord_le3318621148231462513_mat_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_246_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_247_set__eq__subset,axiom,
( ( ^ [Y3: set_mat_a,Z2: set_mat_a] : ( Y3 = Z2 ) )
= ( ^ [A2: set_mat_a,B2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A2 @ B2 )
& ( ord_le3318621148231462513_mat_a @ B2 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_248_Collect__mono__iff,axiom,
! [P: set_mat_a > $o,Q: set_mat_a > $o] :
( ( ord_le2341747070211005607_mat_a @ ( collect_set_mat_a @ P ) @ ( collect_set_mat_a @ Q ) )
= ( ! [X: set_mat_a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_249_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X: set_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_250_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X: nat > nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_251_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_252_Collect__mono__iff,axiom,
! [P: mat_a > $o,Q: mat_a > $o] :
( ( ord_le3318621148231462513_mat_a @ ( collect_mat_a @ P ) @ ( collect_mat_a @ Q ) )
= ( ! [X: mat_a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_253_emptyE,axiom,
! [A3: mat_nat] :
~ ( member_mat_nat @ A3 @ bot_bot_set_mat_nat ) ).
% emptyE
thf(fact_254_emptyE,axiom,
! [A3: set_mat_a] :
~ ( member_set_mat_a @ A3 @ bot_bo8661580253428394715_mat_a ) ).
% emptyE
thf(fact_255_emptyE,axiom,
! [A3: nat] :
~ ( member_nat @ A3 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_256_emptyE,axiom,
! [A3: mat_a] :
~ ( member_mat_a @ A3 @ bot_bot_set_mat_a ) ).
% emptyE
thf(fact_257_equals0D,axiom,
! [A: set_mat_nat,A3: mat_nat] :
( ( A = bot_bot_set_mat_nat )
=> ~ ( member_mat_nat @ A3 @ A ) ) ).
% equals0D
thf(fact_258_equals0D,axiom,
! [A: set_set_mat_a,A3: set_mat_a] :
( ( A = bot_bo8661580253428394715_mat_a )
=> ~ ( member_set_mat_a @ A3 @ A ) ) ).
% equals0D
thf(fact_259_equals0D,axiom,
! [A: set_nat,A3: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A3 @ A ) ) ).
% equals0D
thf(fact_260_equals0D,axiom,
! [A: set_mat_a,A3: mat_a] :
( ( A = bot_bot_set_mat_a )
=> ~ ( member_mat_a @ A3 @ A ) ) ).
% equals0D
thf(fact_261_equals0I,axiom,
! [A: set_mat_nat] :
( ! [Y2: mat_nat] :
~ ( member_mat_nat @ Y2 @ A )
=> ( A = bot_bot_set_mat_nat ) ) ).
% equals0I
thf(fact_262_equals0I,axiom,
! [A: set_set_mat_a] :
( ! [Y2: set_mat_a] :
~ ( member_set_mat_a @ Y2 @ A )
=> ( A = bot_bo8661580253428394715_mat_a ) ) ).
% equals0I
thf(fact_263_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_264_equals0I,axiom,
! [A: set_mat_a] :
( ! [Y2: mat_a] :
~ ( member_mat_a @ Y2 @ A )
=> ( A = bot_bot_set_mat_a ) ) ).
% equals0I
thf(fact_265_empty__iff,axiom,
! [C: mat_nat] :
~ ( member_mat_nat @ C @ bot_bot_set_mat_nat ) ).
% empty_iff
thf(fact_266_empty__iff,axiom,
! [C: set_mat_a] :
~ ( member_set_mat_a @ C @ bot_bo8661580253428394715_mat_a ) ).
% empty_iff
thf(fact_267_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_268_empty__iff,axiom,
! [C: mat_a] :
~ ( member_mat_a @ C @ bot_bot_set_mat_a ) ).
% empty_iff
thf(fact_269_ex__in__conv,axiom,
! [A: set_mat_nat] :
( ( ? [X: mat_nat] : ( member_mat_nat @ X @ A ) )
= ( A != bot_bot_set_mat_nat ) ) ).
% ex_in_conv
thf(fact_270_ex__in__conv,axiom,
! [A: set_set_mat_a] :
( ( ? [X: set_mat_a] : ( member_set_mat_a @ X @ A ) )
= ( A != bot_bo8661580253428394715_mat_a ) ) ).
% ex_in_conv
thf(fact_271_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_272_ex__in__conv,axiom,
! [A: set_mat_a] :
( ( ? [X: mat_a] : ( member_mat_a @ X @ A ) )
= ( A != bot_bot_set_mat_a ) ) ).
% ex_in_conv
thf(fact_273_all__not__in__conv,axiom,
! [A: set_mat_nat] :
( ( ! [X: mat_nat] :
~ ( member_mat_nat @ X @ A ) )
= ( A = bot_bot_set_mat_nat ) ) ).
% all_not_in_conv
thf(fact_274_all__not__in__conv,axiom,
! [A: set_set_mat_a] :
( ( ! [X: set_mat_a] :
~ ( member_set_mat_a @ X @ A ) )
= ( A = bot_bo8661580253428394715_mat_a ) ) ).
% all_not_in_conv
thf(fact_275_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_276_all__not__in__conv,axiom,
! [A: set_mat_a] :
( ( ! [X: mat_a] :
~ ( member_mat_a @ X @ A ) )
= ( A = bot_bot_set_mat_a ) ) ).
% all_not_in_conv
thf(fact_277_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X: set_nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_278_Collect__empty__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P )
= bot_bot_set_nat_nat )
= ( ! [X: nat > nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_279_Collect__empty__eq,axiom,
! [P: set_mat_a > $o] :
( ( ( collect_set_mat_a @ P )
= bot_bo8661580253428394715_mat_a )
= ( ! [X: set_mat_a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_280_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_281_Collect__empty__eq,axiom,
! [P: mat_a > $o] :
( ( ( collect_mat_a @ P )
= bot_bot_set_mat_a )
= ( ! [X: mat_a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_282_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X: set_nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_283_empty__Collect__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( bot_bot_set_nat_nat
= ( collect_nat_nat @ P ) )
= ( ! [X: nat > nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_284_empty__Collect__eq,axiom,
! [P: set_mat_a > $o] :
( ( bot_bo8661580253428394715_mat_a
= ( collect_set_mat_a @ P ) )
= ( ! [X: set_mat_a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_285_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_286_empty__Collect__eq,axiom,
! [P: mat_a > $o] :
( ( bot_bot_set_mat_a
= ( collect_mat_a @ P ) )
= ( ! [X: mat_a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_287_insertE,axiom,
! [A3: set_mat_a,B3: set_mat_a,A: set_set_mat_a] :
( ( member_set_mat_a @ A3 @ ( insert_set_mat_a @ B3 @ A ) )
=> ( ( A3 != B3 )
=> ( member_set_mat_a @ A3 @ A ) ) ) ).
% insertE
thf(fact_288_insertE,axiom,
! [A3: mat_nat,B3: mat_nat,A: set_mat_nat] :
( ( member_mat_nat @ A3 @ ( insert_mat_nat @ B3 @ A ) )
=> ( ( A3 != B3 )
=> ( member_mat_nat @ A3 @ A ) ) ) ).
% insertE
thf(fact_289_insertE,axiom,
! [A3: mat_a,B3: mat_a,A: set_mat_a] :
( ( member_mat_a @ A3 @ ( insert_mat_a @ B3 @ A ) )
=> ( ( A3 != B3 )
=> ( member_mat_a @ A3 @ A ) ) ) ).
% insertE
thf(fact_290_insertE,axiom,
! [A3: nat,B3: nat,A: set_nat] :
( ( member_nat @ A3 @ ( insert_nat @ B3 @ A ) )
=> ( ( A3 != B3 )
=> ( member_nat @ A3 @ A ) ) ) ).
% insertE
thf(fact_291_insertCI,axiom,
! [A3: set_mat_a,B: set_set_mat_a,B3: set_mat_a] :
( ( ~ ( member_set_mat_a @ A3 @ B )
=> ( A3 = B3 ) )
=> ( member_set_mat_a @ A3 @ ( insert_set_mat_a @ B3 @ B ) ) ) ).
% insertCI
thf(fact_292_insertCI,axiom,
! [A3: mat_nat,B: set_mat_nat,B3: mat_nat] :
( ( ~ ( member_mat_nat @ A3 @ B )
=> ( A3 = B3 ) )
=> ( member_mat_nat @ A3 @ ( insert_mat_nat @ B3 @ B ) ) ) ).
% insertCI
thf(fact_293_insertCI,axiom,
! [A3: mat_a,B: set_mat_a,B3: mat_a] :
( ( ~ ( member_mat_a @ A3 @ B )
=> ( A3 = B3 ) )
=> ( member_mat_a @ A3 @ ( insert_mat_a @ B3 @ B ) ) ) ).
% insertCI
thf(fact_294_insertCI,axiom,
! [A3: nat,B: set_nat,B3: nat] :
( ( ~ ( member_nat @ A3 @ B )
=> ( A3 = B3 ) )
=> ( member_nat @ A3 @ ( insert_nat @ B3 @ B ) ) ) ).
% insertCI
thf(fact_295_insertI1,axiom,
! [A3: set_mat_a,B: set_set_mat_a] : ( member_set_mat_a @ A3 @ ( insert_set_mat_a @ A3 @ B ) ) ).
% insertI1
thf(fact_296_insertI1,axiom,
! [A3: mat_nat,B: set_mat_nat] : ( member_mat_nat @ A3 @ ( insert_mat_nat @ A3 @ B ) ) ).
% insertI1
thf(fact_297_insertI1,axiom,
! [A3: mat_a,B: set_mat_a] : ( member_mat_a @ A3 @ ( insert_mat_a @ A3 @ B ) ) ).
% insertI1
thf(fact_298_insertI1,axiom,
! [A3: nat,B: set_nat] : ( member_nat @ A3 @ ( insert_nat @ A3 @ B ) ) ).
% insertI1
thf(fact_299_insertI2,axiom,
! [A3: set_mat_a,B: set_set_mat_a,B3: set_mat_a] :
( ( member_set_mat_a @ A3 @ B )
=> ( member_set_mat_a @ A3 @ ( insert_set_mat_a @ B3 @ B ) ) ) ).
% insertI2
thf(fact_300_insertI2,axiom,
! [A3: mat_nat,B: set_mat_nat,B3: mat_nat] :
( ( member_mat_nat @ A3 @ B )
=> ( member_mat_nat @ A3 @ ( insert_mat_nat @ B3 @ B ) ) ) ).
% insertI2
thf(fact_301_insertI2,axiom,
! [A3: mat_a,B: set_mat_a,B3: mat_a] :
( ( member_mat_a @ A3 @ B )
=> ( member_mat_a @ A3 @ ( insert_mat_a @ B3 @ B ) ) ) ).
% insertI2
thf(fact_302_insertI2,axiom,
! [A3: nat,B: set_nat,B3: nat] :
( ( member_nat @ A3 @ B )
=> ( member_nat @ A3 @ ( insert_nat @ B3 @ B ) ) ) ).
% insertI2
thf(fact_303_insert__iff,axiom,
! [A3: set_mat_a,B3: set_mat_a,A: set_set_mat_a] :
( ( member_set_mat_a @ A3 @ ( insert_set_mat_a @ B3 @ A ) )
= ( ( A3 = B3 )
| ( member_set_mat_a @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_304_insert__iff,axiom,
! [A3: mat_nat,B3: mat_nat,A: set_mat_nat] :
( ( member_mat_nat @ A3 @ ( insert_mat_nat @ B3 @ A ) )
= ( ( A3 = B3 )
| ( member_mat_nat @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_305_insert__iff,axiom,
! [A3: mat_a,B3: mat_a,A: set_mat_a] :
( ( member_mat_a @ A3 @ ( insert_mat_a @ B3 @ A ) )
= ( ( A3 = B3 )
| ( member_mat_a @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_306_insert__iff,axiom,
! [A3: nat,B3: nat,A: set_nat] :
( ( member_nat @ A3 @ ( insert_nat @ B3 @ A ) )
= ( ( A3 = B3 )
| ( member_nat @ A3 @ A ) ) ) ).
% insert_iff
thf(fact_307_Set_Oset__insert,axiom,
! [X4: set_mat_a,A: set_set_mat_a] :
( ( member_set_mat_a @ X4 @ A )
=> ~ ! [B5: set_set_mat_a] :
( ( A
= ( insert_set_mat_a @ X4 @ B5 ) )
=> ( member_set_mat_a @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_308_Set_Oset__insert,axiom,
! [X4: mat_nat,A: set_mat_nat] :
( ( member_mat_nat @ X4 @ A )
=> ~ ! [B5: set_mat_nat] :
( ( A
= ( insert_mat_nat @ X4 @ B5 ) )
=> ( member_mat_nat @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_309_Set_Oset__insert,axiom,
! [X4: mat_a,A: set_mat_a] :
( ( member_mat_a @ X4 @ A )
=> ~ ! [B5: set_mat_a] :
( ( A
= ( insert_mat_a @ X4 @ B5 ) )
=> ( member_mat_a @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_310_Set_Oset__insert,axiom,
! [X4: nat,A: set_nat] :
( ( member_nat @ X4 @ A )
=> ~ ! [B5: set_nat] :
( ( A
= ( insert_nat @ X4 @ B5 ) )
=> ( member_nat @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_311_insert__ident,axiom,
! [X4: set_mat_a,A: set_set_mat_a,B: set_set_mat_a] :
( ~ ( member_set_mat_a @ X4 @ A )
=> ( ~ ( member_set_mat_a @ X4 @ B )
=> ( ( ( insert_set_mat_a @ X4 @ A )
= ( insert_set_mat_a @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_312_insert__ident,axiom,
! [X4: mat_nat,A: set_mat_nat,B: set_mat_nat] :
( ~ ( member_mat_nat @ X4 @ A )
=> ( ~ ( member_mat_nat @ X4 @ B )
=> ( ( ( insert_mat_nat @ X4 @ A )
= ( insert_mat_nat @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_313_insert__ident,axiom,
! [X4: mat_a,A: set_mat_a,B: set_mat_a] :
( ~ ( member_mat_a @ X4 @ A )
=> ( ~ ( member_mat_a @ X4 @ B )
=> ( ( ( insert_mat_a @ X4 @ A )
= ( insert_mat_a @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_314_insert__ident,axiom,
! [X4: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X4 @ A )
=> ( ~ ( member_nat @ X4 @ B )
=> ( ( ( insert_nat @ X4 @ A )
= ( insert_nat @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_315_insert__absorb,axiom,
! [A3: set_mat_a,A: set_set_mat_a] :
( ( member_set_mat_a @ A3 @ A )
=> ( ( insert_set_mat_a @ A3 @ A )
= A ) ) ).
% insert_absorb
thf(fact_316_insert__absorb,axiom,
! [A3: mat_nat,A: set_mat_nat] :
( ( member_mat_nat @ A3 @ A )
=> ( ( insert_mat_nat @ A3 @ A )
= A ) ) ).
% insert_absorb
thf(fact_317_insert__absorb,axiom,
! [A3: mat_a,A: set_mat_a] :
( ( member_mat_a @ A3 @ A )
=> ( ( insert_mat_a @ A3 @ A )
= A ) ) ).
% insert_absorb
thf(fact_318_insert__absorb,axiom,
! [A3: nat,A: set_nat] :
( ( member_nat @ A3 @ A )
=> ( ( insert_nat @ A3 @ A )
= A ) ) ).
% insert_absorb
thf(fact_319_insert__eq__iff,axiom,
! [A3: set_mat_a,A: set_set_mat_a,B3: set_mat_a,B: set_set_mat_a] :
( ~ ( member_set_mat_a @ A3 @ A )
=> ( ~ ( member_set_mat_a @ B3 @ B )
=> ( ( ( insert_set_mat_a @ A3 @ A )
= ( insert_set_mat_a @ B3 @ B ) )
= ( ( ( A3 = B3 )
=> ( A = B ) )
& ( ( A3 != B3 )
=> ? [C3: set_set_mat_a] :
( ( A
= ( insert_set_mat_a @ B3 @ C3 ) )
& ~ ( member_set_mat_a @ B3 @ C3 )
& ( B
= ( insert_set_mat_a @ A3 @ C3 ) )
& ~ ( member_set_mat_a @ A3 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_320_insert__eq__iff,axiom,
! [A3: mat_nat,A: set_mat_nat,B3: mat_nat,B: set_mat_nat] :
( ~ ( member_mat_nat @ A3 @ A )
=> ( ~ ( member_mat_nat @ B3 @ B )
=> ( ( ( insert_mat_nat @ A3 @ A )
= ( insert_mat_nat @ B3 @ B ) )
= ( ( ( A3 = B3 )
=> ( A = B ) )
& ( ( A3 != B3 )
=> ? [C3: set_mat_nat] :
( ( A
= ( insert_mat_nat @ B3 @ C3 ) )
& ~ ( member_mat_nat @ B3 @ C3 )
& ( B
= ( insert_mat_nat @ A3 @ C3 ) )
& ~ ( member_mat_nat @ A3 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_321_insert__eq__iff,axiom,
! [A3: mat_a,A: set_mat_a,B3: mat_a,B: set_mat_a] :
( ~ ( member_mat_a @ A3 @ A )
=> ( ~ ( member_mat_a @ B3 @ B )
=> ( ( ( insert_mat_a @ A3 @ A )
= ( insert_mat_a @ B3 @ B ) )
= ( ( ( A3 = B3 )
=> ( A = B ) )
& ( ( A3 != B3 )
=> ? [C3: set_mat_a] :
( ( A
= ( insert_mat_a @ B3 @ C3 ) )
& ~ ( member_mat_a @ B3 @ C3 )
& ( B
= ( insert_mat_a @ A3 @ C3 ) )
& ~ ( member_mat_a @ A3 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_322_insert__eq__iff,axiom,
! [A3: nat,A: set_nat,B3: nat,B: set_nat] :
( ~ ( member_nat @ A3 @ A )
=> ( ~ ( member_nat @ B3 @ B )
=> ( ( ( insert_nat @ A3 @ A )
= ( insert_nat @ B3 @ B ) )
= ( ( ( A3 = B3 )
=> ( A = B ) )
& ( ( A3 != B3 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B3 @ C3 ) )
& ~ ( member_nat @ B3 @ C3 )
& ( B
= ( insert_nat @ A3 @ C3 ) )
& ~ ( member_nat @ A3 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_323_insert__absorb2,axiom,
! [X4: nat,A: set_nat] :
( ( insert_nat @ X4 @ ( insert_nat @ X4 @ A ) )
= ( insert_nat @ X4 @ A ) ) ).
% insert_absorb2
thf(fact_324_insert__absorb2,axiom,
! [X4: mat_a,A: set_mat_a] :
( ( insert_mat_a @ X4 @ ( insert_mat_a @ X4 @ A ) )
= ( insert_mat_a @ X4 @ A ) ) ).
% insert_absorb2
thf(fact_325_insert__commute,axiom,
! [X4: nat,Y: nat,A: set_nat] :
( ( insert_nat @ X4 @ ( insert_nat @ Y @ A ) )
= ( insert_nat @ Y @ ( insert_nat @ X4 @ A ) ) ) ).
% insert_commute
thf(fact_326_insert__commute,axiom,
! [X4: mat_a,Y: mat_a,A: set_mat_a] :
( ( insert_mat_a @ X4 @ ( insert_mat_a @ Y @ A ) )
= ( insert_mat_a @ Y @ ( insert_mat_a @ X4 @ A ) ) ) ).
% insert_commute
thf(fact_327_mk__disjoint__insert,axiom,
! [A3: set_mat_a,A: set_set_mat_a] :
( ( member_set_mat_a @ A3 @ A )
=> ? [B5: set_set_mat_a] :
( ( A
= ( insert_set_mat_a @ A3 @ B5 ) )
& ~ ( member_set_mat_a @ A3 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_328_mk__disjoint__insert,axiom,
! [A3: mat_nat,A: set_mat_nat] :
( ( member_mat_nat @ A3 @ A )
=> ? [B5: set_mat_nat] :
( ( A
= ( insert_mat_nat @ A3 @ B5 ) )
& ~ ( member_mat_nat @ A3 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_329_mk__disjoint__insert,axiom,
! [A3: mat_a,A: set_mat_a] :
( ( member_mat_a @ A3 @ A )
=> ? [B5: set_mat_a] :
( ( A
= ( insert_mat_a @ A3 @ B5 ) )
& ~ ( member_mat_a @ A3 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_330_mk__disjoint__insert,axiom,
! [A3: nat,A: set_nat] :
( ( member_nat @ A3 @ A )
=> ? [B5: set_nat] :
( ( A
= ( insert_nat @ A3 @ B5 ) )
& ~ ( member_nat @ A3 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_331_subset__empty,axiom,
! [A: set_set_mat_a] :
( ( ord_le2341747070211005607_mat_a @ A @ bot_bo8661580253428394715_mat_a )
= ( A = bot_bo8661580253428394715_mat_a ) ) ).
% subset_empty
thf(fact_332_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_333_subset__empty,axiom,
! [A: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ bot_bot_set_mat_a )
= ( A = bot_bot_set_mat_a ) ) ).
% subset_empty
thf(fact_334_empty__subsetI,axiom,
! [A: set_set_mat_a] : ( ord_le2341747070211005607_mat_a @ bot_bo8661580253428394715_mat_a @ A ) ).
% empty_subsetI
thf(fact_335_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_336_empty__subsetI,axiom,
! [A: set_mat_a] : ( ord_le3318621148231462513_mat_a @ bot_bot_set_mat_a @ A ) ).
% empty_subsetI
thf(fact_337_Set_Oinsert__mono,axiom,
! [C2: set_nat,D: set_nat,A3: nat] :
( ( ord_less_eq_set_nat @ C2 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A3 @ C2 ) @ ( insert_nat @ A3 @ D ) ) ) ).
% Set.insert_mono
thf(fact_338_Set_Oinsert__mono,axiom,
! [C2: set_mat_a,D: set_mat_a,A3: mat_a] :
( ( ord_le3318621148231462513_mat_a @ C2 @ D )
=> ( ord_le3318621148231462513_mat_a @ ( insert_mat_a @ A3 @ C2 ) @ ( insert_mat_a @ A3 @ D ) ) ) ).
% Set.insert_mono
thf(fact_339_insert__subset,axiom,
! [X4: set_mat_a,A: set_set_mat_a,B: set_set_mat_a] :
( ( ord_le2341747070211005607_mat_a @ ( insert_set_mat_a @ X4 @ A ) @ B )
= ( ( member_set_mat_a @ X4 @ B )
& ( ord_le2341747070211005607_mat_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_340_insert__subset,axiom,
! [X4: mat_nat,A: set_mat_nat,B: set_mat_nat] :
( ( ord_le7789122042438455497at_nat @ ( insert_mat_nat @ X4 @ A ) @ B )
= ( ( member_mat_nat @ X4 @ B )
& ( ord_le7789122042438455497at_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_341_insert__subset,axiom,
! [X4: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X4 @ A ) @ B )
= ( ( member_nat @ X4 @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_342_insert__subset,axiom,
! [X4: mat_a,A: set_mat_a,B: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ ( insert_mat_a @ X4 @ A ) @ B )
= ( ( member_mat_a @ X4 @ B )
& ( ord_le3318621148231462513_mat_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_343_subset__insert,axiom,
! [X4: set_mat_a,A: set_set_mat_a,B: set_set_mat_a] :
( ~ ( member_set_mat_a @ X4 @ A )
=> ( ( ord_le2341747070211005607_mat_a @ A @ ( insert_set_mat_a @ X4 @ B ) )
= ( ord_le2341747070211005607_mat_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_344_subset__insert,axiom,
! [X4: mat_nat,A: set_mat_nat,B: set_mat_nat] :
( ~ ( member_mat_nat @ X4 @ A )
=> ( ( ord_le7789122042438455497at_nat @ A @ ( insert_mat_nat @ X4 @ B ) )
= ( ord_le7789122042438455497at_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_345_subset__insert,axiom,
! [X4: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X4 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X4 @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_346_subset__insert,axiom,
! [X4: mat_a,A: set_mat_a,B: set_mat_a] :
( ~ ( member_mat_a @ X4 @ A )
=> ( ( ord_le3318621148231462513_mat_a @ A @ ( insert_mat_a @ X4 @ B ) )
= ( ord_le3318621148231462513_mat_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_347_subset__insertI,axiom,
! [B: set_nat,A3: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A3 @ B ) ) ).
% subset_insertI
thf(fact_348_subset__insertI,axiom,
! [B: set_mat_a,A3: mat_a] : ( ord_le3318621148231462513_mat_a @ B @ ( insert_mat_a @ A3 @ B ) ) ).
% subset_insertI
thf(fact_349_subset__insertI2,axiom,
! [A: set_nat,B: set_nat,B3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_350_subset__insertI2,axiom,
! [A: set_mat_a,B: set_mat_a,B3: mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( ord_le3318621148231462513_mat_a @ A @ ( insert_mat_a @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_351_singletonD,axiom,
! [B3: mat_nat,A3: mat_nat] :
( ( member_mat_nat @ B3 @ ( insert_mat_nat @ A3 @ bot_bot_set_mat_nat ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_352_singletonD,axiom,
! [B3: set_mat_a,A3: set_mat_a] :
( ( member_set_mat_a @ B3 @ ( insert_set_mat_a @ A3 @ bot_bo8661580253428394715_mat_a ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_353_singletonD,axiom,
! [B3: nat,A3: nat] :
( ( member_nat @ B3 @ ( insert_nat @ A3 @ bot_bot_set_nat ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_354_singletonD,axiom,
! [B3: mat_a,A3: mat_a] :
( ( member_mat_a @ B3 @ ( insert_mat_a @ A3 @ bot_bot_set_mat_a ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_355_singletonI,axiom,
! [A3: mat_nat] : ( member_mat_nat @ A3 @ ( insert_mat_nat @ A3 @ bot_bot_set_mat_nat ) ) ).
% singletonI
thf(fact_356_singletonI,axiom,
! [A3: set_mat_a] : ( member_set_mat_a @ A3 @ ( insert_set_mat_a @ A3 @ bot_bo8661580253428394715_mat_a ) ) ).
% singletonI
thf(fact_357_singletonI,axiom,
! [A3: nat] : ( member_nat @ A3 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_358_singletonI,axiom,
! [A3: mat_a] : ( member_mat_a @ A3 @ ( insert_mat_a @ A3 @ bot_bot_set_mat_a ) ) ).
% singletonI
thf(fact_359_singleton__iff,axiom,
! [B3: mat_nat,A3: mat_nat] :
( ( member_mat_nat @ B3 @ ( insert_mat_nat @ A3 @ bot_bot_set_mat_nat ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_360_singleton__iff,axiom,
! [B3: set_mat_a,A3: set_mat_a] :
( ( member_set_mat_a @ B3 @ ( insert_set_mat_a @ A3 @ bot_bo8661580253428394715_mat_a ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_361_singleton__iff,axiom,
! [B3: nat,A3: nat] :
( ( member_nat @ B3 @ ( insert_nat @ A3 @ bot_bot_set_nat ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_362_singleton__iff,axiom,
! [B3: mat_a,A3: mat_a] :
( ( member_mat_a @ B3 @ ( insert_mat_a @ A3 @ bot_bot_set_mat_a ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_363_doubleton__eq__iff,axiom,
! [A3: nat,B3: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A3 @ ( insert_nat @ B3 @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A3 = C )
& ( B3 = D2 ) )
| ( ( A3 = D2 )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_364_doubleton__eq__iff,axiom,
! [A3: set_mat_a,B3: set_mat_a,C: set_mat_a,D2: set_mat_a] :
( ( ( insert_set_mat_a @ A3 @ ( insert_set_mat_a @ B3 @ bot_bo8661580253428394715_mat_a ) )
= ( insert_set_mat_a @ C @ ( insert_set_mat_a @ D2 @ bot_bo8661580253428394715_mat_a ) ) )
= ( ( ( A3 = C )
& ( B3 = D2 ) )
| ( ( A3 = D2 )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_365_doubleton__eq__iff,axiom,
! [A3: mat_a,B3: mat_a,C: mat_a,D2: mat_a] :
( ( ( insert_mat_a @ A3 @ ( insert_mat_a @ B3 @ bot_bot_set_mat_a ) )
= ( insert_mat_a @ C @ ( insert_mat_a @ D2 @ bot_bot_set_mat_a ) ) )
= ( ( ( A3 = C )
& ( B3 = D2 ) )
| ( ( A3 = D2 )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_366_insert__subsetI,axiom,
! [X4: set_mat_a,A: set_set_mat_a,X3: set_set_mat_a] :
( ( member_set_mat_a @ X4 @ A )
=> ( ( ord_le2341747070211005607_mat_a @ X3 @ A )
=> ( ord_le2341747070211005607_mat_a @ ( insert_set_mat_a @ X4 @ X3 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_367_insert__subsetI,axiom,
! [X4: mat_nat,A: set_mat_nat,X3: set_mat_nat] :
( ( member_mat_nat @ X4 @ A )
=> ( ( ord_le7789122042438455497at_nat @ X3 @ A )
=> ( ord_le7789122042438455497at_nat @ ( insert_mat_nat @ X4 @ X3 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_368_insert__subsetI,axiom,
! [X4: nat,A: set_nat,X3: set_nat] :
( ( member_nat @ X4 @ A )
=> ( ( ord_less_eq_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X4 @ X3 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_369_insert__subsetI,axiom,
! [X4: mat_a,A: set_mat_a,X3: set_mat_a] :
( ( member_mat_a @ X4 @ A )
=> ( ( ord_le3318621148231462513_mat_a @ X3 @ A )
=> ( ord_le3318621148231462513_mat_a @ ( insert_mat_a @ X4 @ X3 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_370_subset__emptyI,axiom,
! [A: set_mat_nat] :
( ! [X2: mat_nat] :
~ ( member_mat_nat @ X2 @ A )
=> ( ord_le7789122042438455497at_nat @ A @ bot_bot_set_mat_nat ) ) ).
% subset_emptyI
thf(fact_371_subset__emptyI,axiom,
! [A: set_set_mat_a] :
( ! [X2: set_mat_a] :
~ ( member_set_mat_a @ X2 @ A )
=> ( ord_le2341747070211005607_mat_a @ A @ bot_bo8661580253428394715_mat_a ) ) ).
% subset_emptyI
thf(fact_372_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X2: nat] :
~ ( member_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_373_subset__emptyI,axiom,
! [A: set_mat_a] :
( ! [X2: mat_a] :
~ ( member_mat_a @ X2 @ A )
=> ( ord_le3318621148231462513_mat_a @ A @ bot_bot_set_mat_a ) ) ).
% subset_emptyI
thf(fact_374_le__bot,axiom,
! [A3: set_set_mat_a] :
( ( ord_le2341747070211005607_mat_a @ A3 @ bot_bo8661580253428394715_mat_a )
=> ( A3 = bot_bo8661580253428394715_mat_a ) ) ).
% le_bot
thf(fact_375_le__bot,axiom,
! [A3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
=> ( A3 = bot_bot_set_nat ) ) ).
% le_bot
thf(fact_376_le__bot,axiom,
! [A3: mat_a > $o] :
( ( ord_less_eq_mat_a_o @ A3 @ bot_bot_mat_a_o )
=> ( A3 = bot_bot_mat_a_o ) ) ).
% le_bot
thf(fact_377_le__bot,axiom,
! [A3: nat > $o] :
( ( ord_less_eq_nat_o @ A3 @ bot_bot_nat_o )
=> ( A3 = bot_bot_nat_o ) ) ).
% le_bot
thf(fact_378_le__bot,axiom,
! [A3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A3 @ bot_bot_set_mat_a )
=> ( A3 = bot_bot_set_mat_a ) ) ).
% le_bot
thf(fact_379_le__bot,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
=> ( A3 = bot_bot_nat ) ) ).
% le_bot
thf(fact_380_bot__least,axiom,
! [A3: set_set_mat_a] : ( ord_le2341747070211005607_mat_a @ bot_bo8661580253428394715_mat_a @ A3 ) ).
% bot_least
thf(fact_381_bot__least,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A3 ) ).
% bot_least
thf(fact_382_bot__least,axiom,
! [A3: mat_a > $o] : ( ord_less_eq_mat_a_o @ bot_bot_mat_a_o @ A3 ) ).
% bot_least
thf(fact_383_bot__least,axiom,
! [A3: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A3 ) ).
% bot_least
thf(fact_384_bot__least,axiom,
! [A3: set_mat_a] : ( ord_le3318621148231462513_mat_a @ bot_bot_set_mat_a @ A3 ) ).
% bot_least
thf(fact_385_bot__least,axiom,
! [A3: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A3 ) ).
% bot_least
thf(fact_386_bot__unique,axiom,
! [A3: set_set_mat_a] :
( ( ord_le2341747070211005607_mat_a @ A3 @ bot_bo8661580253428394715_mat_a )
= ( A3 = bot_bo8661580253428394715_mat_a ) ) ).
% bot_unique
thf(fact_387_bot__unique,axiom,
! [A3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
= ( A3 = bot_bot_set_nat ) ) ).
% bot_unique
thf(fact_388_bot__unique,axiom,
! [A3: mat_a > $o] :
( ( ord_less_eq_mat_a_o @ A3 @ bot_bot_mat_a_o )
= ( A3 = bot_bot_mat_a_o ) ) ).
% bot_unique
thf(fact_389_bot__unique,axiom,
! [A3: nat > $o] :
( ( ord_less_eq_nat_o @ A3 @ bot_bot_nat_o )
= ( A3 = bot_bot_nat_o ) ) ).
% bot_unique
thf(fact_390_bot__unique,axiom,
! [A3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A3 @ bot_bot_set_mat_a )
= ( A3 = bot_bot_set_mat_a ) ) ).
% bot_unique
thf(fact_391_bot__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
= ( A3 = bot_bot_nat ) ) ).
% bot_unique
thf(fact_392_unitary__is__corthogonal,axiom,
! [U2: mat_a,N: nat] :
( ( member_mat_a @ U2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_unitary_a @ U2 )
=> ( schur_4042290226164342457_mat_a @ U2 ) ) ) ).
% unitary_is_corthogonal
thf(fact_393_hermitian__mat__conj_H,axiom,
! [A: mat_a,N: nat,U2: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ U2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_hermitian_a @ A )
=> ( complex_hermitian_a @ ( spectr5828033140197310157conj_a @ ( schur_mat_adjoint_a @ U2 ) @ A ) ) ) ) ) ).
% hermitian_mat_conj'
thf(fact_394_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_395_bot__set__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat @ bot_bot_nat_nat_o ) ) ).
% bot_set_def
thf(fact_396_bot__set__def,axiom,
( bot_bo8661580253428394715_mat_a
= ( collect_set_mat_a @ bot_bot_set_mat_a_o ) ) ).
% bot_set_def
thf(fact_397_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_398_bot__set__def,axiom,
( bot_bot_set_mat_a
= ( collect_mat_a @ bot_bot_mat_a_o ) ) ).
% bot_set_def
thf(fact_399_hermitian__square__hermitian,axiom,
! [A: mat_a] :
( ( complex_hermitian_a @ A )
=> ( complex_hermitian_a @ ( times_times_mat_a @ A @ A ) ) ) ).
% hermitian_square_hermitian
thf(fact_400_hermitian__one,axiom,
! [N: nat] : ( complex_hermitian_a @ ( one_mat_a @ N ) ) ).
% hermitian_one
thf(fact_401_hermitian__def,axiom,
( complex_hermitian_a
= ( ^ [A2: mat_a] :
( ( schur_mat_adjoint_a @ A2 )
= A2 ) ) ) ).
% hermitian_def
thf(fact_402_hermitian__square,axiom,
! [M2: mat_a] :
( ( complex_hermitian_a @ M2 )
=> ( member_mat_a @ M2 @ ( carrier_mat_a @ ( dim_row_a @ M2 ) @ ( dim_row_a @ M2 ) ) ) ) ).
% hermitian_square
thf(fact_403_order__antisym__conv,axiom,
! [Y: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X4 )
=> ( ( ord_less_eq_set_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_404_order__antisym__conv,axiom,
! [Y: mat_a > $o,X4: mat_a > $o] :
( ( ord_less_eq_mat_a_o @ Y @ X4 )
=> ( ( ord_less_eq_mat_a_o @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_405_order__antisym__conv,axiom,
! [Y: nat > $o,X4: nat > $o] :
( ( ord_less_eq_nat_o @ Y @ X4 )
=> ( ( ord_less_eq_nat_o @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_406_order__antisym__conv,axiom,
! [Y: set_mat_a,X4: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ Y @ X4 )
=> ( ( ord_le3318621148231462513_mat_a @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_407_order__antisym__conv,axiom,
! [Y: nat,X4: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_408_linorder__le__cases,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ Y @ X4 ) ) ).
% linorder_le_cases
thf(fact_409_linorder__linear,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
| ( ord_less_eq_nat @ Y @ X4 ) ) ).
% linorder_linear
thf(fact_410_order__eq__refl,axiom,
! [X4: set_nat,Y: set_nat] :
( ( X4 = Y )
=> ( ord_less_eq_set_nat @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_411_order__eq__refl,axiom,
! [X4: mat_a > $o,Y: mat_a > $o] :
( ( X4 = Y )
=> ( ord_less_eq_mat_a_o @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_412_order__eq__refl,axiom,
! [X4: nat > $o,Y: nat > $o] :
( ( X4 = Y )
=> ( ord_less_eq_nat_o @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_413_order__eq__refl,axiom,
! [X4: set_mat_a,Y: set_mat_a] :
( ( X4 = Y )
=> ( ord_le3318621148231462513_mat_a @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_414_order__eq__refl,axiom,
! [X4: nat,Y: nat] :
( ( X4 = Y )
=> ( ord_less_eq_nat @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_415_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_416_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: mat_a > $o,Z2: mat_a > $o] : ( Y3 = Z2 ) )
= ( ^ [A4: mat_a > $o,B4: mat_a > $o] :
( ( ord_less_eq_mat_a_o @ A4 @ B4 )
& ( ord_less_eq_mat_a_o @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_417_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat > $o,Z2: nat > $o] : ( Y3 = Z2 ) )
= ( ^ [A4: nat > $o,B4: nat > $o] :
( ( ord_less_eq_nat_o @ A4 @ B4 )
& ( ord_less_eq_nat_o @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_418_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_mat_a,Z2: set_mat_a] : ( Y3 = Z2 ) )
= ( ^ [A4: set_mat_a,B4: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A4 @ B4 )
& ( ord_le3318621148231462513_mat_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_419_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_420_le__fun__def,axiom,
( ord_less_eq_mat_a_o
= ( ^ [F2: mat_a > $o,G: mat_a > $o] :
! [X: mat_a] : ( ord_less_eq_o @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_fun_def
thf(fact_421_le__fun__def,axiom,
( ord_less_eq_nat_o
= ( ^ [F2: nat > $o,G: nat > $o] :
! [X: nat] : ( ord_less_eq_o @ ( F2 @ X ) @ ( G @ X ) ) ) ) ).
% le_fun_def
thf(fact_422_le__funI,axiom,
! [F: mat_a > $o,G2: mat_a > $o] :
( ! [X2: mat_a] : ( ord_less_eq_o @ ( F @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq_mat_a_o @ F @ G2 ) ) ).
% le_funI
thf(fact_423_le__funI,axiom,
! [F: nat > $o,G2: nat > $o] :
( ! [X2: nat] : ( ord_less_eq_o @ ( F @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq_nat_o @ F @ G2 ) ) ).
% le_funI
thf(fact_424_le__funE,axiom,
! [F: mat_a > $o,G2: mat_a > $o,X4: mat_a] :
( ( ord_less_eq_mat_a_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G2 @ X4 ) ) ) ).
% le_funE
thf(fact_425_le__funE,axiom,
! [F: nat > $o,G2: nat > $o,X4: nat] :
( ( ord_less_eq_nat_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G2 @ X4 ) ) ) ).
% le_funE
thf(fact_426_le__funD,axiom,
! [F: mat_a > $o,G2: mat_a > $o,X4: mat_a] :
( ( ord_less_eq_mat_a_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G2 @ X4 ) ) ) ).
% le_funD
thf(fact_427_le__funD,axiom,
! [F: nat > $o,G2: nat > $o,X4: nat] :
( ( ord_less_eq_nat_o @ F @ G2 )
=> ( ord_less_eq_o @ ( F @ X4 ) @ ( G2 @ X4 ) ) ) ).
% le_funD
thf(fact_428_dual__order_Otrans,axiom,
! [B3: set_nat,A3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( ( ord_less_eq_set_nat @ C @ B3 )
=> ( ord_less_eq_set_nat @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_429_dual__order_Otrans,axiom,
! [B3: mat_a > $o,A3: mat_a > $o,C: mat_a > $o] :
( ( ord_less_eq_mat_a_o @ B3 @ A3 )
=> ( ( ord_less_eq_mat_a_o @ C @ B3 )
=> ( ord_less_eq_mat_a_o @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_430_dual__order_Otrans,axiom,
! [B3: nat > $o,A3: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat_o @ B3 @ A3 )
=> ( ( ord_less_eq_nat_o @ C @ B3 )
=> ( ord_less_eq_nat_o @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_431_dual__order_Otrans,axiom,
! [B3: set_mat_a,A3: set_mat_a,C: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ B3 @ A3 )
=> ( ( ord_le3318621148231462513_mat_a @ C @ B3 )
=> ( ord_le3318621148231462513_mat_a @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_432_dual__order_Otrans,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_433_dual__order_Orefl,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_434_dual__order_Orefl,axiom,
! [A3: mat_a > $o] : ( ord_less_eq_mat_a_o @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_435_dual__order_Orefl,axiom,
! [A3: nat > $o] : ( ord_less_eq_nat_o @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_436_dual__order_Orefl,axiom,
! [A3: set_mat_a] : ( ord_le3318621148231462513_mat_a @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_437_dual__order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% dual_order.refl
thf(fact_438_dual__order_Oantisym,axiom,
! [B3: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
=> ( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_439_dual__order_Oantisym,axiom,
! [B3: mat_a > $o,A3: mat_a > $o] :
( ( ord_less_eq_mat_a_o @ B3 @ A3 )
=> ( ( ord_less_eq_mat_a_o @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_440_dual__order_Oantisym,axiom,
! [B3: nat > $o,A3: nat > $o] :
( ( ord_less_eq_nat_o @ B3 @ A3 )
=> ( ( ord_less_eq_nat_o @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_441_dual__order_Oantisym,axiom,
! [B3: set_mat_a,A3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ B3 @ A3 )
=> ( ( ord_le3318621148231462513_mat_a @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_442_dual__order_Oantisym,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_443_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_444_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: mat_a > $o,Z2: mat_a > $o] : ( Y3 = Z2 ) )
= ( ^ [A4: mat_a > $o,B4: mat_a > $o] :
( ( ord_less_eq_mat_a_o @ B4 @ A4 )
& ( ord_less_eq_mat_a_o @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_445_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat > $o,Z2: nat > $o] : ( Y3 = Z2 ) )
= ( ^ [A4: nat > $o,B4: nat > $o] :
( ( ord_less_eq_nat_o @ B4 @ A4 )
& ( ord_less_eq_nat_o @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_446_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_mat_a,Z2: set_mat_a] : ( Y3 = Z2 ) )
= ( ^ [A4: set_mat_a,B4: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ B4 @ A4 )
& ( ord_le3318621148231462513_mat_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_447_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_448_linorder__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: nat,B6: nat] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_449_order_Otrans,axiom,
! [A3: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A3 @ C ) ) ) ).
% order.trans
thf(fact_450_order_Otrans,axiom,
! [A3: mat_a > $o,B3: mat_a > $o,C: mat_a > $o] :
( ( ord_less_eq_mat_a_o @ A3 @ B3 )
=> ( ( ord_less_eq_mat_a_o @ B3 @ C )
=> ( ord_less_eq_mat_a_o @ A3 @ C ) ) ) ).
% order.trans
thf(fact_451_order_Otrans,axiom,
! [A3: nat > $o,B3: nat > $o,C: nat > $o] :
( ( ord_less_eq_nat_o @ A3 @ B3 )
=> ( ( ord_less_eq_nat_o @ B3 @ C )
=> ( ord_less_eq_nat_o @ A3 @ C ) ) ) ).
% order.trans
thf(fact_452_order_Otrans,axiom,
! [A3: set_mat_a,B3: set_mat_a,C: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A3 @ B3 )
=> ( ( ord_le3318621148231462513_mat_a @ B3 @ C )
=> ( ord_le3318621148231462513_mat_a @ A3 @ C ) ) ) ).
% order.trans
thf(fact_453_order_Otrans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% order.trans
thf(fact_454_order__refl,axiom,
! [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_455_order__refl,axiom,
! [X4: mat_a > $o] : ( ord_less_eq_mat_a_o @ X4 @ X4 ) ).
% order_refl
thf(fact_456_order__refl,axiom,
! [X4: nat > $o] : ( ord_less_eq_nat_o @ X4 @ X4 ) ).
% order_refl
thf(fact_457_order__refl,axiom,
! [X4: set_mat_a] : ( ord_le3318621148231462513_mat_a @ X4 @ X4 ) ).
% order_refl
thf(fact_458_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_459_order__antisym,axiom,
! [X4: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_460_order__antisym,axiom,
! [X4: mat_a > $o,Y: mat_a > $o] :
( ( ord_less_eq_mat_a_o @ X4 @ Y )
=> ( ( ord_less_eq_mat_a_o @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_461_order__antisym,axiom,
! [X4: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X4 @ Y )
=> ( ( ord_less_eq_nat_o @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_462_order__antisym,axiom,
! [X4: set_mat_a,Y: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X4 @ Y )
=> ( ( ord_le3318621148231462513_mat_a @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_463_order__antisym,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_464_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_465_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: mat_a > $o,Z2: mat_a > $o] : ( Y3 = Z2 ) )
= ( ^ [X: mat_a > $o,Y4: mat_a > $o] :
( ( ord_less_eq_mat_a_o @ X @ Y4 )
& ( ord_less_eq_mat_a_o @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_466_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat > $o,Z2: nat > $o] : ( Y3 = Z2 ) )
= ( ^ [X: nat > $o,Y4: nat > $o] :
( ( ord_less_eq_nat_o @ X @ Y4 )
& ( ord_less_eq_nat_o @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_467_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_mat_a,Z2: set_mat_a] : ( Y3 = Z2 ) )
= ( ^ [X: set_mat_a,Y4: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X @ Y4 )
& ( ord_le3318621148231462513_mat_a @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_468_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_469_le__cases3,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_470_nle__le,axiom,
! [A3: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A3 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A3 )
& ( B3 != A3 ) ) ) ).
% nle_le
thf(fact_471_bot__fun__def,axiom,
( bot_bot_mat_a_o
= ( ^ [X: mat_a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_472_bot__fun__def,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_473_bot__apply,axiom,
( bot_bot_mat_a_o
= ( ^ [X: mat_a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_474_bot__apply,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_475_hermitian__is__normal,axiom,
! [A: mat_a] :
( ( complex_hermitian_a @ A )
=> ( ( times_times_mat_a @ A @ ( schur_mat_adjoint_a @ A ) )
= ( times_times_mat_a @ ( schur_mat_adjoint_a @ A ) @ A ) ) ) ).
% hermitian_is_normal
thf(fact_476_hermitian__mat__conj,axiom,
! [A: mat_a,N: nat,U2: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ U2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_hermitian_a @ A )
=> ( complex_hermitian_a @ ( spectr5828033140197310157conj_a @ U2 @ A ) ) ) ) ) ).
% hermitian_mat_conj
thf(fact_477_mult__adjoint__hermitian,axiom,
! [A: mat_a,N: nat,M: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( complex_hermitian_a @ ( times_times_mat_a @ ( schur_mat_adjoint_a @ A ) @ A ) ) ) ).
% mult_adjoint_hermitian
thf(fact_478_projector__def,axiom,
( linear2821214051344812439ctor_a
= ( ^ [M3: mat_a] :
( ( complex_hermitian_a @ M3 )
& ( ( times_times_mat_a @ M3 @ M3 )
= M3 ) ) ) ) ).
% projector_def
thf(fact_479_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A2: set_nat] : ( A2 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_480_Set_Ois__empty__def,axiom,
( is_empty_set_mat_a
= ( ^ [A2: set_set_mat_a] : ( A2 = bot_bo8661580253428394715_mat_a ) ) ) ).
% Set.is_empty_def
thf(fact_481_Set_Ois__empty__def,axiom,
( is_empty_mat_a
= ( ^ [A2: set_mat_a] : ( A2 = bot_bot_set_mat_a ) ) ) ).
% Set.is_empty_def
thf(fact_482_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A2: set_nat] :
? [X: nat] :
( A2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_483_is__singleton__def,axiom,
( is_sin4571450623289582109_mat_a
= ( ^ [A2: set_set_mat_a] :
? [X: set_mat_a] :
( A2
= ( insert_set_mat_a @ X @ bot_bo8661580253428394715_mat_a ) ) ) ) ).
% is_singleton_def
thf(fact_484_is__singleton__def,axiom,
( is_singleton_mat_a
= ( ^ [A2: set_mat_a] :
? [X: mat_a] :
( A2
= ( insert_mat_a @ X @ bot_bot_set_mat_a ) ) ) ) ).
% is_singleton_def
thf(fact_485_is__singletonI,axiom,
! [X4: nat] : ( is_singleton_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_486_is__singletonI,axiom,
! [X4: set_mat_a] : ( is_sin4571450623289582109_mat_a @ ( insert_set_mat_a @ X4 @ bot_bo8661580253428394715_mat_a ) ) ).
% is_singletonI
thf(fact_487_is__singletonI,axiom,
! [X4: mat_a] : ( is_singleton_mat_a @ ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) ) ).
% is_singletonI
thf(fact_488_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X2: nat] :
( A
!= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_489_is__singletonE,axiom,
! [A: set_set_mat_a] :
( ( is_sin4571450623289582109_mat_a @ A )
=> ~ ! [X2: set_mat_a] :
( A
!= ( insert_set_mat_a @ X2 @ bot_bo8661580253428394715_mat_a ) ) ) ).
% is_singletonE
thf(fact_490_is__singletonE,axiom,
! [A: set_mat_a] :
( ( is_singleton_mat_a @ A )
=> ~ ! [X2: mat_a] :
( A
!= ( insert_mat_a @ X2 @ bot_bot_set_mat_a ) ) ) ).
% is_singletonE
thf(fact_491_similar__mat__wit__def,axiom,
( similar_mat_wit_nat
= ( ^ [A2: mat_nat,B2: mat_nat,P4: mat_nat,Q4: mat_nat] :
( ( ord_le7789122042438455497at_nat @ ( insert_mat_nat @ A2 @ ( insert_mat_nat @ B2 @ ( insert_mat_nat @ P4 @ ( insert_mat_nat @ Q4 @ bot_bot_set_mat_nat ) ) ) ) @ ( carrier_mat_nat @ ( dim_row_nat @ A2 ) @ ( dim_row_nat @ A2 ) ) )
& ( ( times_times_mat_nat @ P4 @ Q4 )
= ( one_mat_nat @ ( dim_row_nat @ A2 ) ) )
& ( ( times_times_mat_nat @ Q4 @ P4 )
= ( one_mat_nat @ ( dim_row_nat @ A2 ) ) )
& ( A2
= ( times_times_mat_nat @ ( times_times_mat_nat @ P4 @ B2 ) @ Q4 ) ) ) ) ) ).
% similar_mat_wit_def
thf(fact_492_similar__mat__wit__def,axiom,
( similar_mat_wit_a
= ( ^ [A2: mat_a,B2: mat_a,P4: mat_a,Q4: mat_a] :
( ( ord_le3318621148231462513_mat_a @ ( insert_mat_a @ A2 @ ( insert_mat_a @ B2 @ ( insert_mat_a @ P4 @ ( insert_mat_a @ Q4 @ bot_bot_set_mat_a ) ) ) ) @ ( carrier_mat_a @ ( dim_row_a @ A2 ) @ ( dim_row_a @ A2 ) ) )
& ( ( times_times_mat_a @ P4 @ Q4 )
= ( one_mat_a @ ( dim_row_a @ A2 ) ) )
& ( ( times_times_mat_a @ Q4 @ P4 )
= ( one_mat_a @ ( dim_row_a @ A2 ) ) )
& ( A2
= ( times_times_mat_a @ ( times_times_mat_a @ P4 @ B2 ) @ Q4 ) ) ) ) ) ).
% similar_mat_wit_def
thf(fact_493_gauss__jordan__single_I4_J,axiom,
! [A: mat_a,Nr: nat,Nc: nat,C2: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( ( gauss_4684855476144371464ngle_a @ A )
= C2 )
=> ? [P3: mat_a,Q3: mat_a] :
( ( C2
= ( times_times_mat_a @ P3 @ A ) )
& ( member_mat_a @ P3 @ ( carrier_mat_a @ Nr @ Nr ) )
& ( member_mat_a @ Q3 @ ( carrier_mat_a @ Nr @ Nr ) )
& ( ( times_times_mat_a @ P3 @ Q3 )
= ( one_mat_a @ Nr ) )
& ( ( times_times_mat_a @ Q3 @ P3 )
= ( one_mat_a @ Nr ) ) ) ) ) ).
% gauss_jordan_single(4)
thf(fact_494_projector__hermitian,axiom,
! [M2: mat_a] :
( ( linear2821214051344812439ctor_a @ M2 )
=> ( complex_hermitian_a @ M2 ) ) ).
% projector_hermitian
thf(fact_495_Collect__restrict,axiom,
! [X3: set_mat_nat,P: mat_nat > $o] :
( ord_le7789122042438455497at_nat
@ ( collect_mat_nat
@ ^ [X: mat_nat] :
( ( member_mat_nat @ X @ X3 )
& ( P @ X ) ) )
@ X3 ) ).
% Collect_restrict
thf(fact_496_Collect__restrict,axiom,
! [X3: set_set_mat_a,P: set_mat_a > $o] :
( ord_le2341747070211005607_mat_a
@ ( collect_set_mat_a
@ ^ [X: set_mat_a] :
( ( member_set_mat_a @ X @ X3 )
& ( P @ X ) ) )
@ X3 ) ).
% Collect_restrict
thf(fact_497_Collect__restrict,axiom,
! [X3: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ X3 )
& ( P @ X ) ) )
@ X3 ) ).
% Collect_restrict
thf(fact_498_Collect__restrict,axiom,
! [X3: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ X3 )
& ( P @ X ) ) )
@ X3 ) ).
% Collect_restrict
thf(fact_499_Collect__restrict,axiom,
! [X3: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X3 )
& ( P @ X ) ) )
@ X3 ) ).
% Collect_restrict
thf(fact_500_Collect__restrict,axiom,
! [X3: set_mat_a,P: mat_a > $o] :
( ord_le3318621148231462513_mat_a
@ ( collect_mat_a
@ ^ [X: mat_a] :
( ( member_mat_a @ X @ X3 )
& ( P @ X ) ) )
@ X3 ) ).
% Collect_restrict
thf(fact_501_prop__restrict,axiom,
! [X4: mat_nat,Z3: set_mat_nat,X3: set_mat_nat,P: mat_nat > $o] :
( ( member_mat_nat @ X4 @ Z3 )
=> ( ( ord_le7789122042438455497at_nat @ Z3
@ ( collect_mat_nat
@ ^ [X: mat_nat] :
( ( member_mat_nat @ X @ X3 )
& ( P @ X ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_502_prop__restrict,axiom,
! [X4: set_mat_a,Z3: set_set_mat_a,X3: set_set_mat_a,P: set_mat_a > $o] :
( ( member_set_mat_a @ X4 @ Z3 )
=> ( ( ord_le2341747070211005607_mat_a @ Z3
@ ( collect_set_mat_a
@ ^ [X: set_mat_a] :
( ( member_set_mat_a @ X @ X3 )
& ( P @ X ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_503_prop__restrict,axiom,
! [X4: set_nat,Z3: set_set_nat,X3: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X4 @ Z3 )
=> ( ( ord_le6893508408891458716et_nat @ Z3
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ X3 )
& ( P @ X ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_504_prop__restrict,axiom,
! [X4: nat > nat,Z3: set_nat_nat,X3: set_nat_nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ X4 @ Z3 )
=> ( ( ord_le9059583361652607317at_nat @ Z3
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ X3 )
& ( P @ X ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_505_prop__restrict,axiom,
! [X4: nat,Z3: set_nat,X3: set_nat,P: nat > $o] :
( ( member_nat @ X4 @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X3 )
& ( P @ X ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_506_prop__restrict,axiom,
! [X4: mat_a,Z3: set_mat_a,X3: set_mat_a,P: mat_a > $o] :
( ( member_mat_a @ X4 @ Z3 )
=> ( ( ord_le3318621148231462513_mat_a @ Z3
@ ( collect_mat_a
@ ^ [X: mat_a] :
( ( member_mat_a @ X @ X3 )
& ( P @ X ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_507_less__eq__set__def,axiom,
( ord_le2341747070211005607_mat_a
= ( ^ [A2: set_set_mat_a,B2: set_set_mat_a] :
( ord_le2661774091922174110at_a_o
@ ^ [X: set_mat_a] : ( member_set_mat_a @ X @ A2 )
@ ^ [X: set_mat_a] : ( member_set_mat_a @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_508_less__eq__set__def,axiom,
( ord_le7789122042438455497at_nat
= ( ^ [A2: set_mat_nat,B2: set_mat_nat] :
( ord_le1720399365423063892_nat_o
@ ^ [X: mat_nat] : ( member_mat_nat @ X @ A2 )
@ ^ [X: mat_nat] : ( member_mat_nat @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_509_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A2 )
@ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_510_less__eq__set__def,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A2: set_mat_a,B2: set_mat_a] :
( ord_less_eq_mat_a_o
@ ^ [X: mat_a] : ( member_mat_a @ X @ A2 )
@ ^ [X: mat_a] : ( member_mat_a @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_511_Collect__subset,axiom,
! [A: set_mat_nat,P: mat_nat > $o] :
( ord_le7789122042438455497at_nat
@ ( collect_mat_nat
@ ^ [X: mat_nat] :
( ( member_mat_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_512_Collect__subset,axiom,
! [A: set_set_mat_a,P: set_mat_a > $o] :
( ord_le2341747070211005607_mat_a
@ ( collect_set_mat_a
@ ^ [X: set_mat_a] :
( ( member_set_mat_a @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_513_Collect__subset,axiom,
! [A: set_set_nat,P: set_nat > $o] :
( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_514_Collect__subset,axiom,
! [A: set_nat_nat,P: ( nat > nat ) > $o] :
( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_515_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_516_Collect__subset,axiom,
! [A: set_mat_a,P: mat_a > $o] :
( ord_le3318621148231462513_mat_a
@ ( collect_mat_a
@ ^ [X: mat_a] :
( ( member_mat_a @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_517_Set_Oempty__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat
@ ^ [X: set_nat] : $false ) ) ).
% Set.empty_def
thf(fact_518_Set_Oempty__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat
@ ^ [X: nat > nat] : $false ) ) ).
% Set.empty_def
thf(fact_519_Set_Oempty__def,axiom,
( bot_bo8661580253428394715_mat_a
= ( collect_set_mat_a
@ ^ [X: set_mat_a] : $false ) ) ).
% Set.empty_def
thf(fact_520_Set_Oempty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% Set.empty_def
thf(fact_521_Set_Oempty__def,axiom,
( bot_bot_set_mat_a
= ( collect_mat_a
@ ^ [X: mat_a] : $false ) ) ).
% Set.empty_def
thf(fact_522_insert__compr,axiom,
( insert_mat_nat
= ( ^ [A4: mat_nat,B2: set_mat_nat] :
( collect_mat_nat
@ ^ [X: mat_nat] :
( ( X = A4 )
| ( member_mat_nat @ X @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_523_insert__compr,axiom,
( insert_set_mat_a
= ( ^ [A4: set_mat_a,B2: set_set_mat_a] :
( collect_set_mat_a
@ ^ [X: set_mat_a] :
( ( X = A4 )
| ( member_set_mat_a @ X @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_524_insert__compr,axiom,
( insert_set_nat
= ( ^ [A4: set_nat,B2: set_set_nat] :
( collect_set_nat
@ ^ [X: set_nat] :
( ( X = A4 )
| ( member_set_nat @ X @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_525_insert__compr,axiom,
( insert_nat_nat
= ( ^ [A4: nat > nat,B2: set_nat_nat] :
( collect_nat_nat
@ ^ [X: nat > nat] :
( ( X = A4 )
| ( member_nat_nat @ X @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_526_insert__compr,axiom,
( insert_mat_a
= ( ^ [A4: mat_a,B2: set_mat_a] :
( collect_mat_a
@ ^ [X: mat_a] :
( ( X = A4 )
| ( member_mat_a @ X @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_527_insert__compr,axiom,
( insert_nat
= ( ^ [A4: nat,B2: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( X = A4 )
| ( member_nat @ X @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_528_insert__Collect,axiom,
! [A3: set_mat_a,P: set_mat_a > $o] :
( ( insert_set_mat_a @ A3 @ ( collect_set_mat_a @ P ) )
= ( collect_set_mat_a
@ ^ [U3: set_mat_a] :
( ( U3 != A3 )
=> ( P @ U3 ) ) ) ) ).
% insert_Collect
thf(fact_529_insert__Collect,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( insert_set_nat @ A3 @ ( collect_set_nat @ P ) )
= ( collect_set_nat
@ ^ [U3: set_nat] :
( ( U3 != A3 )
=> ( P @ U3 ) ) ) ) ).
% insert_Collect
thf(fact_530_insert__Collect,axiom,
! [A3: nat > nat,P: ( nat > nat ) > $o] :
( ( insert_nat_nat @ A3 @ ( collect_nat_nat @ P ) )
= ( collect_nat_nat
@ ^ [U3: nat > nat] :
( ( U3 != A3 )
=> ( P @ U3 ) ) ) ) ).
% insert_Collect
thf(fact_531_insert__Collect,axiom,
! [A3: mat_a,P: mat_a > $o] :
( ( insert_mat_a @ A3 @ ( collect_mat_a @ P ) )
= ( collect_mat_a
@ ^ [U3: mat_a] :
( ( U3 != A3 )
=> ( P @ U3 ) ) ) ) ).
% insert_Collect
thf(fact_532_insert__Collect,axiom,
! [A3: nat,P: nat > $o] :
( ( insert_nat @ A3 @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U3: nat] :
( ( U3 != A3 )
=> ( P @ U3 ) ) ) ) ).
% insert_Collect
thf(fact_533_singleton__conv,axiom,
! [A3: set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( X = A3 ) )
= ( insert_set_nat @ A3 @ bot_bot_set_set_nat ) ) ).
% singleton_conv
thf(fact_534_singleton__conv,axiom,
! [A3: nat > nat] :
( ( collect_nat_nat
@ ^ [X: nat > nat] : ( X = A3 ) )
= ( insert_nat_nat @ A3 @ bot_bot_set_nat_nat ) ) ).
% singleton_conv
thf(fact_535_singleton__conv,axiom,
! [A3: set_mat_a] :
( ( collect_set_mat_a
@ ^ [X: set_mat_a] : ( X = A3 ) )
= ( insert_set_mat_a @ A3 @ bot_bo8661580253428394715_mat_a ) ) ).
% singleton_conv
thf(fact_536_singleton__conv,axiom,
! [A3: nat] :
( ( collect_nat
@ ^ [X: nat] : ( X = A3 ) )
= ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_537_singleton__conv,axiom,
! [A3: mat_a] :
( ( collect_mat_a
@ ^ [X: mat_a] : ( X = A3 ) )
= ( insert_mat_a @ A3 @ bot_bot_set_mat_a ) ) ).
% singleton_conv
thf(fact_538_Collect__conv__if,axiom,
! [P: set_nat > $o,A3: set_nat] :
( ( ( P @ A3 )
=> ( ( collect_set_nat
@ ^ [X: set_nat] :
( ( X = A3 )
& ( P @ X ) ) )
= ( insert_set_nat @ A3 @ bot_bot_set_set_nat ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect_set_nat
@ ^ [X: set_nat] :
( ( X = A3 )
& ( P @ X ) ) )
= bot_bot_set_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_539_Collect__conv__if,axiom,
! [P: ( nat > nat ) > $o,A3: nat > nat] :
( ( ( P @ A3 )
=> ( ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( X = A3 )
& ( P @ X ) ) )
= ( insert_nat_nat @ A3 @ bot_bot_set_nat_nat ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( X = A3 )
& ( P @ X ) ) )
= bot_bot_set_nat_nat ) ) ) ).
% Collect_conv_if
thf(fact_540_Collect__conv__if,axiom,
! [P: set_mat_a > $o,A3: set_mat_a] :
( ( ( P @ A3 )
=> ( ( collect_set_mat_a
@ ^ [X: set_mat_a] :
( ( X = A3 )
& ( P @ X ) ) )
= ( insert_set_mat_a @ A3 @ bot_bo8661580253428394715_mat_a ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect_set_mat_a
@ ^ [X: set_mat_a] :
( ( X = A3 )
& ( P @ X ) ) )
= bot_bo8661580253428394715_mat_a ) ) ) ).
% Collect_conv_if
thf(fact_541_Collect__conv__if,axiom,
! [P: nat > $o,A3: nat] :
( ( ( P @ A3 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A3 )
& ( P @ X ) ) )
= ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A3 )
& ( P @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_542_Collect__conv__if,axiom,
! [P: mat_a > $o,A3: mat_a] :
( ( ( P @ A3 )
=> ( ( collect_mat_a
@ ^ [X: mat_a] :
( ( X = A3 )
& ( P @ X ) ) )
= ( insert_mat_a @ A3 @ bot_bot_set_mat_a ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect_mat_a
@ ^ [X: mat_a] :
( ( X = A3 )
& ( P @ X ) ) )
= bot_bot_set_mat_a ) ) ) ).
% Collect_conv_if
thf(fact_543_singleton__conv2,axiom,
! [A3: set_nat] :
( ( collect_set_nat
@ ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 )
@ A3 ) )
= ( insert_set_nat @ A3 @ bot_bot_set_set_nat ) ) ).
% singleton_conv2
thf(fact_544_singleton__conv2,axiom,
! [A3: nat > nat] :
( ( collect_nat_nat
@ ( ^ [Y3: nat > nat,Z2: nat > nat] : ( Y3 = Z2 )
@ A3 ) )
= ( insert_nat_nat @ A3 @ bot_bot_set_nat_nat ) ) ).
% singleton_conv2
thf(fact_545_singleton__conv2,axiom,
! [A3: set_mat_a] :
( ( collect_set_mat_a
@ ( ^ [Y3: set_mat_a,Z2: set_mat_a] : ( Y3 = Z2 )
@ A3 ) )
= ( insert_set_mat_a @ A3 @ bot_bo8661580253428394715_mat_a ) ) ).
% singleton_conv2
thf(fact_546_singleton__conv2,axiom,
! [A3: nat] :
( ( collect_nat
@ ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 )
@ A3 ) )
= ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_547_singleton__conv2,axiom,
! [A3: mat_a] :
( ( collect_mat_a
@ ( ^ [Y3: mat_a,Z2: mat_a] : ( Y3 = Z2 )
@ A3 ) )
= ( insert_mat_a @ A3 @ bot_bot_set_mat_a ) ) ).
% singleton_conv2
thf(fact_548_Collect__conv__if2,axiom,
! [P: set_nat > $o,A3: set_nat] :
( ( ( P @ A3 )
=> ( ( collect_set_nat
@ ^ [X: set_nat] :
( ( A3 = X )
& ( P @ X ) ) )
= ( insert_set_nat @ A3 @ bot_bot_set_set_nat ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect_set_nat
@ ^ [X: set_nat] :
( ( A3 = X )
& ( P @ X ) ) )
= bot_bot_set_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_549_Collect__conv__if2,axiom,
! [P: ( nat > nat ) > $o,A3: nat > nat] :
( ( ( P @ A3 )
=> ( ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( A3 = X )
& ( P @ X ) ) )
= ( insert_nat_nat @ A3 @ bot_bot_set_nat_nat ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( A3 = X )
& ( P @ X ) ) )
= bot_bot_set_nat_nat ) ) ) ).
% Collect_conv_if2
thf(fact_550_Collect__conv__if2,axiom,
! [P: set_mat_a > $o,A3: set_mat_a] :
( ( ( P @ A3 )
=> ( ( collect_set_mat_a
@ ^ [X: set_mat_a] :
( ( A3 = X )
& ( P @ X ) ) )
= ( insert_set_mat_a @ A3 @ bot_bo8661580253428394715_mat_a ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect_set_mat_a
@ ^ [X: set_mat_a] :
( ( A3 = X )
& ( P @ X ) ) )
= bot_bo8661580253428394715_mat_a ) ) ) ).
% Collect_conv_if2
thf(fact_551_Collect__conv__if2,axiom,
! [P: nat > $o,A3: nat] :
( ( ( P @ A3 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A3 = X )
& ( P @ X ) ) )
= ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A3 = X )
& ( P @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_552_Collect__conv__if2,axiom,
! [P: mat_a > $o,A3: mat_a] :
( ( ( P @ A3 )
=> ( ( collect_mat_a
@ ^ [X: mat_a] :
( ( A3 = X )
& ( P @ X ) ) )
= ( insert_mat_a @ A3 @ bot_bot_set_mat_a ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect_mat_a
@ ^ [X: mat_a] :
( ( A3 = X )
& ( P @ X ) ) )
= bot_bot_set_mat_a ) ) ) ).
% Collect_conv_if2
thf(fact_553_gauss__jordan__single_I2_J,axiom,
! [A: mat_a,Nr: nat,Nc: nat,C2: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( ( gauss_4684855476144371464ngle_a @ A )
= C2 )
=> ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% gauss_jordan_single(2)
thf(fact_554_is__singletonI_H,axiom,
! [A: set_mat_nat] :
( ( A != bot_bot_set_mat_nat )
=> ( ! [X2: mat_nat,Y2: mat_nat] :
( ( member_mat_nat @ X2 @ A )
=> ( ( member_mat_nat @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_mat_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_555_is__singletonI_H,axiom,
! [A: set_set_mat_a] :
( ( A != bot_bo8661580253428394715_mat_a )
=> ( ! [X2: set_mat_a,Y2: set_mat_a] :
( ( member_set_mat_a @ X2 @ A )
=> ( ( member_set_mat_a @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_sin4571450623289582109_mat_a @ A ) ) ) ).
% is_singletonI'
thf(fact_556_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_557_is__singletonI_H,axiom,
! [A: set_mat_a] :
( ( A != bot_bot_set_mat_a )
=> ( ! [X2: mat_a,Y2: mat_a] :
( ( member_mat_a @ X2 @ A )
=> ( ( member_mat_a @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_mat_a @ A ) ) ) ).
% is_singletonI'
thf(fact_558_projector__square__eq,axiom,
! [M2: mat_a] :
( ( linear2821214051344812439ctor_a @ M2 )
=> ( ( times_times_mat_a @ M2 @ M2 )
= M2 ) ) ).
% projector_square_eq
thf(fact_559_bot__empty__eq,axiom,
( bot_bot_mat_nat_o
= ( ^ [X: mat_nat] : ( member_mat_nat @ X @ bot_bot_set_mat_nat ) ) ) ).
% bot_empty_eq
thf(fact_560_bot__empty__eq,axiom,
( bot_bot_set_mat_a_o
= ( ^ [X: set_mat_a] : ( member_set_mat_a @ X @ bot_bo8661580253428394715_mat_a ) ) ) ).
% bot_empty_eq
thf(fact_561_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_562_bot__empty__eq,axiom,
( bot_bot_mat_a_o
= ( ^ [X: mat_a] : ( member_mat_a @ X @ bot_bot_set_mat_a ) ) ) ).
% bot_empty_eq
thf(fact_563_Collect__empty__eq__bot,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( P = bot_bot_set_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_564_Collect__empty__eq__bot,axiom,
! [P: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P )
= bot_bot_set_nat_nat )
= ( P = bot_bot_nat_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_565_Collect__empty__eq__bot,axiom,
! [P: set_mat_a > $o] :
( ( ( collect_set_mat_a @ P )
= bot_bo8661580253428394715_mat_a )
= ( P = bot_bot_set_mat_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_566_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_567_Collect__empty__eq__bot,axiom,
! [P: mat_a > $o] :
( ( ( collect_mat_a @ P )
= bot_bot_set_mat_a )
= ( P = bot_bot_mat_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_568_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A2: set_nat] :
( A2
= ( insert_nat @ ( the_elem_nat @ A2 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_569_is__singleton__the__elem,axiom,
( is_sin4571450623289582109_mat_a
= ( ^ [A2: set_set_mat_a] :
( A2
= ( insert_set_mat_a @ ( the_elem_set_mat_a @ A2 ) @ bot_bo8661580253428394715_mat_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_570_is__singleton__the__elem,axiom,
( is_singleton_mat_a
= ( ^ [A2: set_mat_a] :
( A2
= ( insert_mat_a @ ( the_elem_mat_a @ A2 ) @ bot_bot_set_mat_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_571_mat__diag__diag,axiom,
! [N: nat,F: nat > nat,G2: nat > nat] :
( ( times_times_mat_nat @ ( mat_diag_nat @ N @ F ) @ ( mat_diag_nat @ N @ G2 ) )
= ( mat_diag_nat @ N
@ ^ [I: nat] : ( times_times_nat @ ( F @ I ) @ ( G2 @ I ) ) ) ) ).
% mat_diag_diag
thf(fact_572_mat__diag__diag,axiom,
! [N: nat,F: nat > a,G2: nat > a] :
( ( times_times_mat_a @ ( mat_diag_a @ N @ F ) @ ( mat_diag_a @ N @ G2 ) )
= ( mat_diag_a @ N
@ ^ [I: nat] : ( times_times_a @ ( F @ I ) @ ( G2 @ I ) ) ) ) ).
% mat_diag_diag
thf(fact_573_subset__singleton__iff__Uniq,axiom,
! [A: set_mat_nat] :
( ( ? [A4: mat_nat] : ( ord_le7789122042438455497at_nat @ A @ ( insert_mat_nat @ A4 @ bot_bot_set_mat_nat ) ) )
= ( uniq_mat_nat
@ ^ [X: mat_nat] : ( member_mat_nat @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_574_subset__singleton__iff__Uniq,axiom,
! [A: set_set_mat_a] :
( ( ? [A4: set_mat_a] : ( ord_le2341747070211005607_mat_a @ A @ ( insert_set_mat_a @ A4 @ bot_bo8661580253428394715_mat_a ) ) )
= ( uniq_set_mat_a
@ ^ [X: set_mat_a] : ( member_set_mat_a @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_575_subset__singleton__iff__Uniq,axiom,
! [A: set_nat] :
( ( ? [A4: nat] : ( ord_less_eq_set_nat @ A @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) )
= ( uniq_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_576_subset__singleton__iff__Uniq,axiom,
! [A: set_mat_a] :
( ( ? [A4: mat_a] : ( ord_le3318621148231462513_mat_a @ A @ ( insert_mat_a @ A4 @ bot_bot_set_mat_a ) ) )
= ( uniq_mat_a
@ ^ [X: mat_a] : ( member_mat_a @ X @ A ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_577_subset__Collect__iff,axiom,
! [B: set_mat_nat,A: set_mat_nat,P: mat_nat > $o] :
( ( ord_le7789122042438455497at_nat @ B @ A )
=> ( ( ord_le7789122042438455497at_nat @ B
@ ( collect_mat_nat
@ ^ [X: mat_nat] :
( ( member_mat_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: mat_nat] :
( ( member_mat_nat @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_578_subset__Collect__iff,axiom,
! [B: set_set_mat_a,A: set_set_mat_a,P: set_mat_a > $o] :
( ( ord_le2341747070211005607_mat_a @ B @ A )
=> ( ( ord_le2341747070211005607_mat_a @ B
@ ( collect_set_mat_a
@ ^ [X: set_mat_a] :
( ( member_set_mat_a @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: set_mat_a] :
( ( member_set_mat_a @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_579_subset__Collect__iff,axiom,
! [B: set_set_nat,A: set_set_nat,P: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( ord_le6893508408891458716et_nat @ B
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_580_subset__Collect__iff,axiom,
! [B: set_nat_nat,A: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( ord_le9059583361652607317at_nat @ B
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_581_subset__Collect__iff,axiom,
! [B: set_nat,A: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_582_subset__Collect__iff,axiom,
! [B: set_mat_a,A: set_mat_a,P: mat_a > $o] :
( ( ord_le3318621148231462513_mat_a @ B @ A )
=> ( ( ord_le3318621148231462513_mat_a @ B
@ ( collect_mat_a
@ ^ [X: mat_a] :
( ( member_mat_a @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: mat_a] :
( ( member_mat_a @ X @ B )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_583_subset__CollectI,axiom,
! [B: set_mat_nat,A: set_mat_nat,Q: mat_nat > $o,P: mat_nat > $o] :
( ( ord_le7789122042438455497at_nat @ B @ A )
=> ( ! [X2: mat_nat] :
( ( member_mat_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le7789122042438455497at_nat
@ ( collect_mat_nat
@ ^ [X: mat_nat] :
( ( member_mat_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collect_mat_nat
@ ^ [X: mat_nat] :
( ( member_mat_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_584_subset__CollectI,axiom,
! [B: set_set_mat_a,A: set_set_mat_a,Q: set_mat_a > $o,P: set_mat_a > $o] :
( ( ord_le2341747070211005607_mat_a @ B @ A )
=> ( ! [X2: set_mat_a] :
( ( member_set_mat_a @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le2341747070211005607_mat_a
@ ( collect_set_mat_a
@ ^ [X: set_mat_a] :
( ( member_set_mat_a @ X @ B )
& ( Q @ X ) ) )
@ ( collect_set_mat_a
@ ^ [X: set_mat_a] :
( ( member_set_mat_a @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_585_subset__CollectI,axiom,
! [B: set_set_nat,A: set_set_nat,Q: set_nat > $o,P: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le6893508408891458716et_nat
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_586_subset__CollectI,axiom,
! [B: set_nat_nat,A: set_nat_nat,Q: ( nat > nat ) > $o,P: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le9059583361652607317at_nat
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_587_subset__CollectI,axiom,
! [B: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ B )
& ( Q @ X ) ) )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_588_subset__CollectI,axiom,
! [B: set_mat_a,A: set_mat_a,Q: mat_a > $o,P: mat_a > $o] :
( ( ord_le3318621148231462513_mat_a @ B @ A )
=> ( ! [X2: mat_a] :
( ( member_mat_a @ X2 @ B )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le3318621148231462513_mat_a
@ ( collect_mat_a
@ ^ [X: mat_a] :
( ( member_mat_a @ X @ B )
& ( Q @ X ) ) )
@ ( collect_mat_a
@ ^ [X: mat_a] :
( ( member_mat_a @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_589_rev__predicate1D,axiom,
! [P: mat_a > $o,X4: mat_a,Q: mat_a > $o] :
( ( P @ X4 )
=> ( ( ord_less_eq_mat_a_o @ P @ Q )
=> ( Q @ X4 ) ) ) ).
% rev_predicate1D
thf(fact_590_rev__predicate1D,axiom,
! [P: nat > $o,X4: nat,Q: nat > $o] :
( ( P @ X4 )
=> ( ( ord_less_eq_nat_o @ P @ Q )
=> ( Q @ X4 ) ) ) ).
% rev_predicate1D
thf(fact_591_predicate1I,axiom,
! [P: mat_a > $o,Q: mat_a > $o] :
( ! [X2: mat_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_mat_a_o @ P @ Q ) ) ).
% predicate1I
thf(fact_592_predicate1I,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_nat_o @ P @ Q ) ) ).
% predicate1I
thf(fact_593_predicate1D,axiom,
! [P: mat_a > $o,Q: mat_a > $o,X4: mat_a] :
( ( ord_less_eq_mat_a_o @ P @ Q )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) ) ).
% predicate1D
thf(fact_594_predicate1D,axiom,
! [P: nat > $o,Q: nat > $o,X4: nat] :
( ( ord_less_eq_nat_o @ P @ Q )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) ) ).
% predicate1D
thf(fact_595_mat__diag__dim,axiom,
! [N: nat,F: nat > nat] : ( member_mat_nat @ ( mat_diag_nat @ N @ F ) @ ( carrier_mat_nat @ N @ N ) ) ).
% mat_diag_dim
thf(fact_596_mat__diag__dim,axiom,
! [N: nat,F: nat > a] : ( member_mat_a @ ( mat_diag_a @ N @ F ) @ ( carrier_mat_a @ N @ N ) ) ).
% mat_diag_dim
thf(fact_597_the__elem__eq,axiom,
! [X4: nat] :
( ( the_elem_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= X4 ) ).
% the_elem_eq
thf(fact_598_the__elem__eq,axiom,
! [X4: set_mat_a] :
( ( the_elem_set_mat_a @ ( insert_set_mat_a @ X4 @ bot_bo8661580253428394715_mat_a ) )
= X4 ) ).
% the_elem_eq
thf(fact_599_the__elem__eq,axiom,
! [X4: mat_a] :
( ( the_elem_mat_a @ ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) )
= X4 ) ).
% the_elem_eq
thf(fact_600_pred__subset__eq,axiom,
! [R: set_set_mat_a,S: set_set_mat_a] :
( ( ord_le2661774091922174110at_a_o
@ ^ [X: set_mat_a] : ( member_set_mat_a @ X @ R )
@ ^ [X: set_mat_a] : ( member_set_mat_a @ X @ S ) )
= ( ord_le2341747070211005607_mat_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_601_pred__subset__eq,axiom,
! [R: set_mat_nat,S: set_mat_nat] :
( ( ord_le1720399365423063892_nat_o
@ ^ [X: mat_nat] : ( member_mat_nat @ X @ R )
@ ^ [X: mat_nat] : ( member_mat_nat @ X @ S ) )
= ( ord_le7789122042438455497at_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_602_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_603_pred__subset__eq,axiom,
! [R: set_mat_a,S: set_mat_a] :
( ( ord_less_eq_mat_a_o
@ ^ [X: mat_a] : ( member_mat_a @ X @ R )
@ ^ [X: mat_a] : ( member_mat_a @ X @ S ) )
= ( ord_le3318621148231462513_mat_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_604_alt__ex1E_H,axiom,
! [P: mat_a > $o] :
( ? [X6: mat_a] :
( ( P @ X6 )
& ! [Y2: mat_a] :
( ( P @ Y2 )
=> ( Y2 = X6 ) ) )
=> ~ ( ? [X_1: mat_a] : ( P @ X_1 )
=> ~ ( uniq_mat_a @ P ) ) ) ).
% alt_ex1E'
thf(fact_605_alt__ex1E_H,axiom,
! [P: nat > $o] :
( ? [X6: nat] :
( ( P @ X6 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( Y2 = X6 ) ) )
=> ~ ( ? [X_1: nat] : ( P @ X_1 )
=> ~ ( uniq_nat @ P ) ) ) ).
% alt_ex1E'
thf(fact_606_ex1__iff__ex__Uniq,axiom,
( ex1_mat_a
= ( ^ [P4: mat_a > $o] :
( ? [X5: mat_a] : ( P4 @ X5 )
& ( uniq_mat_a @ P4 ) ) ) ) ).
% ex1_iff_ex_Uniq
thf(fact_607_ex1__iff__ex__Uniq,axiom,
( ex1_nat
= ( ^ [P4: nat > $o] :
( ? [X5: nat] : ( P4 @ X5 )
& ( uniq_nat @ P4 ) ) ) ) ).
% ex1_iff_ex_Uniq
thf(fact_608_conj__subset__def,axiom,
! [A: set_set_mat_a,P: set_mat_a > $o,Q: set_mat_a > $o] :
( ( ord_le2341747070211005607_mat_a @ A
@ ( collect_set_mat_a
@ ^ [X: set_mat_a] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le2341747070211005607_mat_a @ A @ ( collect_set_mat_a @ P ) )
& ( ord_le2341747070211005607_mat_a @ A @ ( collect_set_mat_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_609_conj__subset__def,axiom,
! [A: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A
@ ( collect_set_nat
@ ^ [X: set_nat] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le6893508408891458716et_nat @ A @ ( collect_set_nat @ P ) )
& ( ord_le6893508408891458716et_nat @ A @ ( collect_set_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_610_conj__subset__def,axiom,
! [A: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ A
@ ( collect_nat_nat
@ ^ [X: nat > nat] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le9059583361652607317at_nat @ A @ ( collect_nat_nat @ P ) )
& ( ord_le9059583361652607317at_nat @ A @ ( collect_nat_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_611_conj__subset__def,axiom,
! [A: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_612_conj__subset__def,axiom,
! [A: set_mat_a,P: mat_a > $o,Q: mat_a > $o] :
( ( ord_le3318621148231462513_mat_a @ A
@ ( collect_mat_a
@ ^ [X: mat_a] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le3318621148231462513_mat_a @ A @ ( collect_mat_a @ P ) )
& ( ord_le3318621148231462513_mat_a @ A @ ( collect_mat_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_613_mult__commute__abs,axiom,
! [C: a] :
( ( ^ [X: a] : ( times_times_a @ X @ C ) )
= ( times_times_a @ C ) ) ).
% mult_commute_abs
thf(fact_614_mult__commute__abs,axiom,
! [C: set_nat] :
( ( ^ [X: set_nat] : ( times_times_set_nat @ X @ C ) )
= ( times_times_set_nat @ C ) ) ).
% mult_commute_abs
thf(fact_615_mult__commute__abs,axiom,
! [C: nat] :
( ( ^ [X: nat] : ( times_times_nat @ X @ C ) )
= ( times_times_nat @ C ) ) ).
% mult_commute_abs
thf(fact_616_gauss__jordan__single_I3_J,axiom,
! [A: mat_a,Nr: nat,Nc: nat,C2: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( ( gauss_4684855476144371464ngle_a @ A )
= C2 )
=> ( gauss_5855338539171749649form_a @ C2 ) ) ) ).
% gauss_jordan_single(3)
thf(fact_617_the__elem__def,axiom,
( the_elem_nat
= ( ^ [X5: set_nat] :
( the_nat
@ ^ [X: nat] :
( X5
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% the_elem_def
thf(fact_618_the__elem__def,axiom,
( the_elem_set_mat_a
= ( ^ [X5: set_set_mat_a] :
( the_set_mat_a
@ ^ [X: set_mat_a] :
( X5
= ( insert_set_mat_a @ X @ bot_bo8661580253428394715_mat_a ) ) ) ) ) ).
% the_elem_def
thf(fact_619_the__elem__def,axiom,
( the_elem_mat_a
= ( ^ [X5: set_mat_a] :
( the_mat_a
@ ^ [X: mat_a] :
( X5
= ( insert_mat_a @ X @ bot_bot_set_mat_a ) ) ) ) ) ).
% the_elem_def
thf(fact_620_Greatest__equality,axiom,
! [P: set_nat > $o,X4: set_nat] :
( ( P @ X4 )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X4 ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_621_Greatest__equality,axiom,
! [P: ( mat_a > $o ) > $o,X4: mat_a > $o] :
( ( P @ X4 )
=> ( ! [Y2: mat_a > $o] :
( ( P @ Y2 )
=> ( ord_less_eq_mat_a_o @ Y2 @ X4 ) )
=> ( ( order_1137619110770037133at_a_o @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_622_Greatest__equality,axiom,
! [P: ( nat > $o ) > $o,X4: nat > $o] :
( ( P @ X4 )
=> ( ! [Y2: nat > $o] :
( ( P @ Y2 )
=> ( ord_less_eq_nat_o @ Y2 @ X4 ) )
=> ( ( order_Greatest_nat_o @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_623_Greatest__equality,axiom,
! [P: set_mat_a > $o,X4: set_mat_a] :
( ( P @ X4 )
=> ( ! [Y2: set_mat_a] :
( ( P @ Y2 )
=> ( ord_le3318621148231462513_mat_a @ Y2 @ X4 ) )
=> ( ( order_3466378972280292088_mat_a @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_624_Greatest__equality,axiom,
! [P: nat > $o,X4: nat] :
( ( P @ X4 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X4 ) )
=> ( ( order_Greatest_nat @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_625_the__sym__eq__trivial,axiom,
! [X4: mat_a] :
( ( the_mat_a
@ ( ^ [Y3: mat_a,Z2: mat_a] : ( Y3 = Z2 )
@ X4 ) )
= X4 ) ).
% the_sym_eq_trivial
thf(fact_626_the__sym__eq__trivial,axiom,
! [X4: nat] :
( ( the_nat
@ ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 )
@ X4 ) )
= X4 ) ).
% the_sym_eq_trivial
thf(fact_627_the__sym__eq__trivial,axiom,
! [X4: set_mat_a] :
( ( the_set_mat_a
@ ( ^ [Y3: set_mat_a,Z2: set_mat_a] : ( Y3 = Z2 )
@ X4 ) )
= X4 ) ).
% the_sym_eq_trivial
thf(fact_628_the__eq__trivial,axiom,
! [A3: mat_a] :
( ( the_mat_a
@ ^ [X: mat_a] : ( X = A3 ) )
= A3 ) ).
% the_eq_trivial
thf(fact_629_the__eq__trivial,axiom,
! [A3: nat] :
( ( the_nat
@ ^ [X: nat] : ( X = A3 ) )
= A3 ) ).
% the_eq_trivial
thf(fact_630_the__eq__trivial,axiom,
! [A3: set_mat_a] :
( ( the_set_mat_a
@ ^ [X: set_mat_a] : ( X = A3 ) )
= A3 ) ).
% the_eq_trivial
thf(fact_631_the1__equality,axiom,
! [P: mat_a > $o,A3: mat_a] :
( ? [X6: mat_a] :
( ( P @ X6 )
& ! [Y2: mat_a] :
( ( P @ Y2 )
=> ( Y2 = X6 ) ) )
=> ( ( P @ A3 )
=> ( ( the_mat_a @ P )
= A3 ) ) ) ).
% the1_equality
thf(fact_632_the1__equality,axiom,
! [P: nat > $o,A3: nat] :
( ? [X6: nat] :
( ( P @ X6 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( Y2 = X6 ) ) )
=> ( ( P @ A3 )
=> ( ( the_nat @ P )
= A3 ) ) ) ).
% the1_equality
thf(fact_633_the1__equality,axiom,
! [P: set_mat_a > $o,A3: set_mat_a] :
( ? [X6: set_mat_a] :
( ( P @ X6 )
& ! [Y2: set_mat_a] :
( ( P @ Y2 )
=> ( Y2 = X6 ) ) )
=> ( ( P @ A3 )
=> ( ( the_set_mat_a @ P )
= A3 ) ) ) ).
% the1_equality
thf(fact_634_the__equality,axiom,
! [P: mat_a > $o,A3: mat_a] :
( ( P @ A3 )
=> ( ! [X2: mat_a] :
( ( P @ X2 )
=> ( X2 = A3 ) )
=> ( ( the_mat_a @ P )
= A3 ) ) ) ).
% the_equality
thf(fact_635_the__equality,axiom,
! [P: nat > $o,A3: nat] :
( ( P @ A3 )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( X2 = A3 ) )
=> ( ( the_nat @ P )
= A3 ) ) ) ).
% the_equality
thf(fact_636_the__equality,axiom,
! [P: set_mat_a > $o,A3: set_mat_a] :
( ( P @ A3 )
=> ( ! [X2: set_mat_a] :
( ( P @ X2 )
=> ( X2 = A3 ) )
=> ( ( the_set_mat_a @ P )
= A3 ) ) ) ).
% the_equality
thf(fact_637_the1I2,axiom,
! [P: mat_a > $o,Q: mat_a > $o] :
( ? [X6: mat_a] :
( ( P @ X6 )
& ! [Y2: mat_a] :
( ( P @ Y2 )
=> ( Y2 = X6 ) ) )
=> ( ! [X2: mat_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( the_mat_a @ P ) ) ) ) ).
% the1I2
thf(fact_638_the1I2,axiom,
! [P: nat > $o,Q: nat > $o] :
( ? [X6: nat] :
( ( P @ X6 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( Y2 = X6 ) ) )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( the_nat @ P ) ) ) ) ).
% the1I2
thf(fact_639_the1I2,axiom,
! [P: set_mat_a > $o,Q: set_mat_a > $o] :
( ? [X6: set_mat_a] :
( ( P @ X6 )
& ! [Y2: set_mat_a] :
( ( P @ Y2 )
=> ( Y2 = X6 ) ) )
=> ( ! [X2: set_mat_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( the_set_mat_a @ P ) ) ) ) ).
% the1I2
thf(fact_640_If__def,axiom,
( if_mat_a
= ( ^ [P4: $o,X: mat_a,Y4: mat_a] :
( the_mat_a
@ ^ [Z4: mat_a] :
( ( P4
=> ( Z4 = X ) )
& ( ~ P4
=> ( Z4 = Y4 ) ) ) ) ) ) ).
% If_def
thf(fact_641_If__def,axiom,
( if_nat
= ( ^ [P4: $o,X: nat,Y4: nat] :
( the_nat
@ ^ [Z4: nat] :
( ( P4
=> ( Z4 = X ) )
& ( ~ P4
=> ( Z4 = Y4 ) ) ) ) ) ) ).
% If_def
thf(fact_642_If__def,axiom,
( if_set_mat_a
= ( ^ [P4: $o,X: set_mat_a,Y4: set_mat_a] :
( the_set_mat_a
@ ^ [Z4: set_mat_a] :
( ( P4
=> ( Z4 = X ) )
& ( ~ P4
=> ( Z4 = Y4 ) ) ) ) ) ) ).
% If_def
thf(fact_643_theI2,axiom,
! [P: mat_a > $o,A3: mat_a,Q: mat_a > $o] :
( ( P @ A3 )
=> ( ! [X2: mat_a] :
( ( P @ X2 )
=> ( X2 = A3 ) )
=> ( ! [X2: mat_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( the_mat_a @ P ) ) ) ) ) ).
% theI2
thf(fact_644_theI2,axiom,
! [P: nat > $o,A3: nat,Q: nat > $o] :
( ( P @ A3 )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( X2 = A3 ) )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( the_nat @ P ) ) ) ) ) ).
% theI2
thf(fact_645_theI2,axiom,
! [P: set_mat_a > $o,A3: set_mat_a,Q: set_mat_a > $o] :
( ( P @ A3 )
=> ( ! [X2: set_mat_a] :
( ( P @ X2 )
=> ( X2 = A3 ) )
=> ( ! [X2: set_mat_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( the_set_mat_a @ P ) ) ) ) ) ).
% theI2
thf(fact_646_theI_H,axiom,
! [P: mat_a > $o] :
( ? [X6: mat_a] :
( ( P @ X6 )
& ! [Y2: mat_a] :
( ( P @ Y2 )
=> ( Y2 = X6 ) ) )
=> ( P @ ( the_mat_a @ P ) ) ) ).
% theI'
thf(fact_647_theI_H,axiom,
! [P: nat > $o] :
( ? [X6: nat] :
( ( P @ X6 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( Y2 = X6 ) ) )
=> ( P @ ( the_nat @ P ) ) ) ).
% theI'
thf(fact_648_theI_H,axiom,
! [P: set_mat_a > $o] :
( ? [X6: set_mat_a] :
( ( P @ X6 )
& ! [Y2: set_mat_a] :
( ( P @ Y2 )
=> ( Y2 = X6 ) ) )
=> ( P @ ( the_set_mat_a @ P ) ) ) ).
% theI'
thf(fact_649_theI,axiom,
! [P: mat_a > $o,A3: mat_a] :
( ( P @ A3 )
=> ( ! [X2: mat_a] :
( ( P @ X2 )
=> ( X2 = A3 ) )
=> ( P @ ( the_mat_a @ P ) ) ) ) ).
% theI
thf(fact_650_theI,axiom,
! [P: nat > $o,A3: nat] :
( ( P @ A3 )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( X2 = A3 ) )
=> ( P @ ( the_nat @ P ) ) ) ) ).
% theI
thf(fact_651_theI,axiom,
! [P: set_mat_a > $o,A3: set_mat_a] :
( ( P @ A3 )
=> ( ! [X2: set_mat_a] :
( ( P @ X2 )
=> ( X2 = A3 ) )
=> ( P @ ( the_set_mat_a @ P ) ) ) ) ).
% theI
thf(fact_652_the1__equality_H,axiom,
! [P: set_mat_a > $o,A3: set_mat_a] :
( ( uniq_set_mat_a @ P )
=> ( ( P @ A3 )
=> ( ( the_set_mat_a @ P )
= A3 ) ) ) ).
% the1_equality'
thf(fact_653_the1__equality_H,axiom,
! [P: mat_a > $o,A3: mat_a] :
( ( uniq_mat_a @ P )
=> ( ( P @ A3 )
=> ( ( the_mat_a @ P )
= A3 ) ) ) ).
% the1_equality'
thf(fact_654_the1__equality_H,axiom,
! [P: nat > $o,A3: nat] :
( ( uniq_nat @ P )
=> ( ( P @ A3 )
=> ( ( the_nat @ P )
= A3 ) ) ) ).
% the1_equality'
thf(fact_655_GreatestI2__order,axiom,
! [P: set_nat > $o,X4: set_nat,Q: set_nat > $o] :
( ( P @ X4 )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X4 ) )
=> ( ! [X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y5: set_nat] :
( ( P @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_656_GreatestI2__order,axiom,
! [P: ( mat_a > $o ) > $o,X4: mat_a > $o,Q: ( mat_a > $o ) > $o] :
( ( P @ X4 )
=> ( ! [Y2: mat_a > $o] :
( ( P @ Y2 )
=> ( ord_less_eq_mat_a_o @ Y2 @ X4 ) )
=> ( ! [X2: mat_a > $o] :
( ( P @ X2 )
=> ( ! [Y5: mat_a > $o] :
( ( P @ Y5 )
=> ( ord_less_eq_mat_a_o @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_1137619110770037133at_a_o @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_657_GreatestI2__order,axiom,
! [P: ( nat > $o ) > $o,X4: nat > $o,Q: ( nat > $o ) > $o] :
( ( P @ X4 )
=> ( ! [Y2: nat > $o] :
( ( P @ Y2 )
=> ( ord_less_eq_nat_o @ Y2 @ X4 ) )
=> ( ! [X2: nat > $o] :
( ( P @ X2 )
=> ( ! [Y5: nat > $o] :
( ( P @ Y5 )
=> ( ord_less_eq_nat_o @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_nat_o @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_658_GreatestI2__order,axiom,
! [P: set_mat_a > $o,X4: set_mat_a,Q: set_mat_a > $o] :
( ( P @ X4 )
=> ( ! [Y2: set_mat_a] :
( ( P @ Y2 )
=> ( ord_le3318621148231462513_mat_a @ Y2 @ X4 ) )
=> ( ! [X2: set_mat_a] :
( ( P @ X2 )
=> ( ! [Y5: set_mat_a] :
( ( P @ Y5 )
=> ( ord_le3318621148231462513_mat_a @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_3466378972280292088_mat_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_659_GreatestI2__order,axiom,
! [P: nat > $o,X4: nat,Q: nat > $o] :
( ( P @ X4 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X4 ) )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_660_Greatest__def,axiom,
( order_3466378972280292088_mat_a
= ( ^ [P4: set_mat_a > $o] :
( the_set_mat_a
@ ^ [X: set_mat_a] :
( ( P4 @ X )
& ! [Y4: set_mat_a] :
( ( P4 @ Y4 )
=> ( ord_le3318621148231462513_mat_a @ Y4 @ X ) ) ) ) ) ) ).
% Greatest_def
thf(fact_661_Greatest__def,axiom,
( order_Greatest_nat
= ( ^ [P4: nat > $o] :
( the_nat
@ ^ [X: nat] :
( ( P4 @ X )
& ! [Y4: nat] :
( ( P4 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ) ) ).
% Greatest_def
thf(fact_662_mat__diag__one,axiom,
! [N: nat] :
( ( mat_diag_a @ N
@ ^ [X: nat] : one_one_a )
= ( one_mat_a @ N ) ) ).
% mat_diag_one
thf(fact_663_mat__conj__smult,axiom,
! [A: mat_a,N: nat,U2: mat_a,B: mat_a,X4: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ U2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( A
= ( times_times_mat_a @ ( times_times_mat_a @ U2 @ B ) @ ( schur_mat_adjoint_a @ U2 ) ) )
=> ( ( smult_mat_a @ X4 @ A )
= ( times_times_mat_a @ ( times_times_mat_a @ U2 @ ( smult_mat_a @ X4 @ B ) ) @ ( schur_mat_adjoint_a @ U2 ) ) ) ) ) ) ) ).
% mat_conj_smult
thf(fact_664_carrier__matD_I2_J,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( dim_col_a @ A )
= Nc ) ) ).
% carrier_matD(2)
thf(fact_665_smult__smult__times,axiom,
! [A3: nat,K: nat,A: mat_nat] :
( ( smult_mat_nat @ A3 @ ( smult_mat_nat @ K @ A ) )
= ( smult_mat_nat @ ( times_times_nat @ A3 @ K ) @ A ) ) ).
% smult_smult_times
thf(fact_666_index__mult__mat_I3_J,axiom,
! [A: mat_a,B: mat_a] :
( ( dim_col_a @ ( times_times_mat_a @ A @ B ) )
= ( dim_col_a @ B ) ) ).
% index_mult_mat(3)
thf(fact_667_index__one__mat_I3_J,axiom,
! [N: nat] :
( ( dim_col_a @ ( one_mat_a @ N ) )
= N ) ).
% index_one_mat(3)
thf(fact_668_smult__carrier__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat,K: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( smult_mat_a @ K @ A ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% smult_carrier_mat
thf(fact_669_index__smult__mat_I2_J,axiom,
! [A3: a,A: mat_a] :
( ( dim_row_a @ ( smult_mat_a @ A3 @ A ) )
= ( dim_row_a @ A ) ) ).
% index_smult_mat(2)
thf(fact_670_comm__monoid__mult__class_Omult__1,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_671_mult__1__left,axiom,
! [A3: nat] :
( ( times_times_nat @ one_one_nat @ A3 )
= A3 ) ).
% mult_1_left
thf(fact_672_mult__1__right,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult_1_right
thf(fact_673_mult_Ocomm__neutral,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ one_one_nat )
= A3 ) ).
% mult.comm_neutral
thf(fact_674_carrier__mat__triv,axiom,
! [M: mat_a] : ( member_mat_a @ M @ ( carrier_mat_a @ ( dim_row_a @ M ) @ ( dim_col_a @ M ) ) ) ).
% carrier_mat_triv
thf(fact_675_carrier__matI,axiom,
! [A: mat_a,Nr: nat,Nc: nat] :
( ( ( dim_row_a @ A )
= Nr )
=> ( ( ( dim_col_a @ A )
= Nc )
=> ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% carrier_matI
thf(fact_676_right__mult__one__mat_H,axiom,
! [A: mat_a,N: nat] :
( ( ( dim_col_a @ A )
= N )
=> ( ( times_times_mat_a @ A @ ( one_mat_a @ N ) )
= A ) ) ).
% right_mult_one_mat'
thf(fact_677_adjoint__dim__col,axiom,
! [A: mat_a] :
( ( dim_col_a @ ( schur_mat_adjoint_a @ A ) )
= ( dim_row_a @ A ) ) ).
% adjoint_dim_col
thf(fact_678_adjoint__dim__row,axiom,
! [A: mat_a] :
( ( dim_row_a @ ( schur_mat_adjoint_a @ A ) )
= ( dim_col_a @ A ) ) ).
% adjoint_dim_row
thf(fact_679_mult__smult__assoc__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,B: mat_a,Nc: nat,K: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ ( smult_mat_a @ K @ A ) @ B )
= ( smult_mat_a @ K @ ( times_times_mat_a @ A @ B ) ) ) ) ) ).
% mult_smult_assoc_mat
thf(fact_680_mult__smult__distrib,axiom,
! [A: mat_a,Nr: nat,N: nat,B: mat_a,Nc: nat,K: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ A @ ( smult_mat_a @ K @ B ) )
= ( smult_mat_a @ K @ ( times_times_mat_a @ A @ B ) ) ) ) ) ).
% mult_smult_distrib
thf(fact_681_carrier__mat__def,axiom,
( carrier_mat_a
= ( ^ [Nr2: nat,Nc2: nat] :
( collect_mat_a
@ ^ [M4: mat_a] :
( ( ( dim_row_a @ M4 )
= Nr2 )
& ( ( dim_col_a @ M4 )
= Nc2 ) ) ) ) ) ).
% carrier_mat_def
thf(fact_682_unitarily__equiv__smult,axiom,
! [A: mat_a,N: nat,B: mat_a,U2: mat_a,X4: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( spectr4825054497075562704quiv_a @ A @ B @ U2 )
=> ( spectr4825054497075562704quiv_a @ ( smult_mat_a @ X4 @ A ) @ ( smult_mat_a @ X4 @ B ) @ U2 ) ) ) ).
% unitarily_equiv_smult
thf(fact_683_lambda__one,axiom,
( ( ^ [X: nat] : X )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_684_set__times__intro,axiom,
! [A3: mat_a,C2: set_mat_a,B3: mat_a,D: set_mat_a] :
( ( member_mat_a @ A3 @ C2 )
=> ( ( member_mat_a @ B3 @ D )
=> ( member_mat_a @ ( times_times_mat_a @ A3 @ B3 ) @ ( times_1230744552615602198_mat_a @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_685_set__times__intro,axiom,
! [A3: nat,C2: set_nat,B3: nat,D: set_nat] :
( ( member_nat @ A3 @ C2 )
=> ( ( member_nat @ B3 @ D )
=> ( member_nat @ ( times_times_nat @ A3 @ B3 ) @ ( times_times_set_nat @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_686_set__times__elim,axiom,
! [X4: mat_a,A: set_mat_a,B: set_mat_a] :
( ( member_mat_a @ X4 @ ( times_1230744552615602198_mat_a @ A @ B ) )
=> ~ ! [A5: mat_a,B6: mat_a] :
( ( X4
= ( times_times_mat_a @ A5 @ B6 ) )
=> ( ( member_mat_a @ A5 @ A )
=> ~ ( member_mat_a @ B6 @ B ) ) ) ) ).
% set_times_elim
thf(fact_687_set__times__elim,axiom,
! [X4: nat,A: set_nat,B: set_nat] :
( ( member_nat @ X4 @ ( times_times_set_nat @ A @ B ) )
=> ~ ! [A5: nat,B6: nat] :
( ( X4
= ( times_times_nat @ A5 @ B6 ) )
=> ( ( member_nat @ A5 @ A )
=> ~ ( member_nat @ B6 @ B ) ) ) ) ).
% set_times_elim
thf(fact_688_set__times__mono2,axiom,
! [C2: set_mat_a,D: set_mat_a,E: set_mat_a,F3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ C2 @ D )
=> ( ( ord_le3318621148231462513_mat_a @ E @ F3 )
=> ( ord_le3318621148231462513_mat_a @ ( times_1230744552615602198_mat_a @ C2 @ E ) @ ( times_1230744552615602198_mat_a @ D @ F3 ) ) ) ) ).
% set_times_mono2
thf(fact_689_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_690_verit__la__disequality,axiom,
! [A3: nat,B3: nat] :
( ( A3 = B3 )
| ~ ( ord_less_eq_nat @ A3 @ B3 )
| ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% verit_la_disequality
thf(fact_691_verit__comp__simplify1_I2_J,axiom,
! [A3: set_mat_a] : ( ord_le3318621148231462513_mat_a @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_692_verit__comp__simplify1_I2_J,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% verit_comp_simplify1(2)
thf(fact_693_trace__smult,axiom,
! [A: mat_a,N: nat,C: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_trace_a @ ( smult_mat_a @ C @ A ) )
= ( times_times_a @ C @ ( complex_trace_a @ A ) ) ) ) ).
% trace_smult
thf(fact_694_Least__def,axiom,
( ord_Least_set_mat_a
= ( ^ [P4: set_mat_a > $o] :
( the_set_mat_a
@ ^ [X: set_mat_a] :
( ( P4 @ X )
& ! [Y4: set_mat_a] :
( ( P4 @ Y4 )
=> ( ord_le3318621148231462513_mat_a @ X @ Y4 ) ) ) ) ) ) ).
% Least_def
thf(fact_695_Least__def,axiom,
( ord_Least_nat
= ( ^ [P4: nat > $o] :
( the_nat
@ ^ [X: nat] :
( ( P4 @ X )
& ! [Y4: nat] :
( ( P4 @ Y4 )
=> ( ord_less_eq_nat @ X @ Y4 ) ) ) ) ) ) ).
% Least_def
thf(fact_696_Least__ex1_I2_J,axiom,
! [P: set_mat_a > $o,Z: set_mat_a] :
( ? [X6: set_mat_a] :
( ( P @ X6 )
& ! [Y2: set_mat_a] :
( ( P @ Y2 )
=> ( ord_le3318621148231462513_mat_a @ X6 @ Y2 ) )
& ! [Y2: set_mat_a] :
( ( ( P @ Y2 )
& ! [Ya: set_mat_a] :
( ( P @ Ya )
=> ( ord_le3318621148231462513_mat_a @ Y2 @ Ya ) ) )
=> ( Y2 = X6 ) ) )
=> ( ( P @ Z )
=> ( ord_le3318621148231462513_mat_a @ ( ord_Least_set_mat_a @ P ) @ Z ) ) ) ).
% Least_ex1(2)
thf(fact_697_Least__ex1_I2_J,axiom,
! [P: nat > $o,Z: nat] :
( ? [X6: nat] :
( ( P @ X6 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ X6 @ Y2 ) )
& ! [Y2: nat] :
( ( ( P @ Y2 )
& ! [Ya: nat] :
( ( P @ Ya )
=> ( ord_less_eq_nat @ Y2 @ Ya ) ) )
=> ( Y2 = X6 ) ) )
=> ( ( P @ Z )
=> ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ Z ) ) ) ).
% Least_ex1(2)
thf(fact_698_Least__ex1_I1_J,axiom,
! [P: set_mat_a > $o] :
( ? [X6: set_mat_a] :
( ( P @ X6 )
& ! [Y2: set_mat_a] :
( ( P @ Y2 )
=> ( ord_le3318621148231462513_mat_a @ X6 @ Y2 ) )
& ! [Y2: set_mat_a] :
( ( ( P @ Y2 )
& ! [Ya: set_mat_a] :
( ( P @ Ya )
=> ( ord_le3318621148231462513_mat_a @ Y2 @ Ya ) ) )
=> ( Y2 = X6 ) ) )
=> ( P @ ( ord_Least_set_mat_a @ P ) ) ) ).
% Least_ex1(1)
thf(fact_699_Least__ex1_I1_J,axiom,
! [P: nat > $o] :
( ? [X6: nat] :
( ( P @ X6 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ X6 @ Y2 ) )
& ! [Y2: nat] :
( ( ( P @ Y2 )
& ! [Ya: nat] :
( ( P @ Ya )
=> ( ord_less_eq_nat @ Y2 @ Ya ) ) )
=> ( Y2 = X6 ) ) )
=> ( P @ ( ord_Least_nat @ P ) ) ) ).
% Least_ex1(1)
thf(fact_700_LeastI2__order,axiom,
! [P: set_mat_a > $o,X4: set_mat_a,Q: set_mat_a > $o] :
( ( P @ X4 )
=> ( ! [Y2: set_mat_a] :
( ( P @ Y2 )
=> ( ord_le3318621148231462513_mat_a @ X4 @ Y2 ) )
=> ( ! [X2: set_mat_a] :
( ( P @ X2 )
=> ( ! [Y5: set_mat_a] :
( ( P @ Y5 )
=> ( ord_le3318621148231462513_mat_a @ X2 @ Y5 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( ord_Least_set_mat_a @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_701_LeastI2__order,axiom,
! [P: nat > $o,X4: nat,Q: nat > $o] :
( ( P @ X4 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ X4 @ Y2 ) )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ X2 @ Y5 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_702_Least__equality,axiom,
! [P: set_mat_a > $o,X4: set_mat_a] :
( ( P @ X4 )
=> ( ! [Y2: set_mat_a] :
( ( P @ Y2 )
=> ( ord_le3318621148231462513_mat_a @ X4 @ Y2 ) )
=> ( ( ord_Least_set_mat_a @ P )
= X4 ) ) ) ).
% Least_equality
thf(fact_703_Least__equality,axiom,
! [P: nat > $o,X4: nat] :
( ( P @ X4 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ X4 @ Y2 ) )
=> ( ( ord_Least_nat @ P )
= X4 ) ) ) ).
% Least_equality
thf(fact_704_LeastI2__wellorder,axiom,
! [P: nat > $o,A3: nat,Q: nat > $o] :
( ( P @ A3 )
=> ( ! [A5: nat] :
( ( P @ A5 )
=> ( ! [B7: nat] :
( ( P @ B7 )
=> ( ord_less_eq_nat @ A5 @ B7 ) )
=> ( Q @ A5 ) ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ).
% LeastI2_wellorder
thf(fact_705_LeastI2__wellorder__ex,axiom,
! [P: nat > $o,Q: nat > $o] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [A5: nat] :
( ( P @ A5 )
=> ( ! [B7: nat] :
( ( P @ B7 )
=> ( ord_less_eq_nat @ A5 @ B7 ) )
=> ( Q @ A5 ) ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_706_Least__le,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ K ) ) ).
% Least_le
thf(fact_707_trace__comm,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_trace_a @ ( times_times_mat_a @ A @ B ) )
= ( complex_trace_a @ ( times_times_mat_a @ B @ A ) ) ) ) ) ).
% trace_comm
thf(fact_708_one__dim__mat__mult__is__scale,axiom,
! [B: mat_a,N: nat,A3: a] :
( ( member_mat_a @ B @ ( carrier_mat_a @ one_one_nat @ N ) )
=> ( ( times_times_mat_a
@ ( mat_a2 @ one_one_nat @ one_one_nat
@ ^ [K2: product_prod_nat_nat] : A3 )
@ B )
= ( smult_mat_a @ A3 @ B ) ) ) ).
% one_dim_mat_mult_is_scale
thf(fact_709_mat__of__row__dim_I1_J,axiom,
! [Y: vec_a] :
( ( dim_row_a @ ( mat_of_row_a @ Y ) )
= one_one_nat ) ).
% mat_of_row_dim(1)
thf(fact_710_mat__carrier,axiom,
! [Nr: nat,Nc: nat,F: product_prod_nat_nat > a] : ( member_mat_a @ ( mat_a2 @ Nr @ Nc @ F ) @ ( carrier_mat_a @ Nr @ Nc ) ) ).
% mat_carrier
thf(fact_711_dim__row__mat_I1_J,axiom,
! [Nr: nat,Nc: nat,F: product_prod_nat_nat > a] :
( ( dim_row_a @ ( mat_a2 @ Nr @ Nc @ F ) )
= Nr ) ).
% dim_row_mat(1)
thf(fact_712_card__1__singletonE,axiom,
! [A: set_mat_a] :
( ( ( finite_card_mat_a @ A )
= one_one_nat )
=> ~ ! [X2: mat_a] :
( A
!= ( insert_mat_a @ X2 @ bot_bot_set_mat_a ) ) ) ).
% card_1_singletonE
thf(fact_713_card__insert__le,axiom,
! [A: set_mat_a,X4: mat_a] : ( ord_less_eq_nat @ ( finite_card_mat_a @ A ) @ ( finite_card_mat_a @ ( insert_mat_a @ X4 @ A ) ) ) ).
% card_insert_le
thf(fact_714_ex__card,axiom,
! [N: nat,A: set_mat_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_mat_a @ A ) )
=> ? [S2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ S2 @ A )
& ( ( finite_card_mat_a @ S2 )
= N ) ) ) ).
% ex_card
thf(fact_715_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_716_finite__Collect__subsets,axiom,
! [A: set_mat_a] :
( ( finite_finite_mat_a @ A )
=> ( finite5775620362878804648_mat_a
@ ( collect_set_mat_a
@ ^ [B2: set_mat_a] : ( ord_le3318621148231462513_mat_a @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_717_finite__set__of__finite__funs,axiom,
! [A: set_nat,B: set_nat,D2: nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite2115694454571419734at_nat
@ ( collect_nat_nat
@ ^ [F2: nat > nat] :
! [X: nat] :
( ( ( member_nat @ X @ A )
=> ( member_nat @ ( F2 @ X ) @ B ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( F2 @ X )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_718_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_719_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_720_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_721_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_722_finite__Collect__not,axiom,
! [P: nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
~ ( P @ X ) ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Collect_not
thf(fact_723_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X: nat] : $true ) ) ).
% UNIV_def
thf(fact_724_UNIV__I,axiom,
! [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_725_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_726_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_727_top__le,axiom,
! [A3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ top_top_set_mat_a @ A3 )
=> ( A3 = top_top_set_mat_a ) ) ).
% top_le
thf(fact_728_top__unique,axiom,
! [A3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ top_top_set_mat_a @ A3 )
= ( A3 = top_top_set_mat_a ) ) ).
% top_unique
thf(fact_729_top_Oextremum,axiom,
! [A3: set_mat_a] : ( ord_le3318621148231462513_mat_a @ A3 @ top_top_set_mat_a ) ).
% top.extremum
thf(fact_730_finite__has__minimal2,axiom,
! [A: set_set_mat_a,A3: set_mat_a] :
( ( finite5775620362878804648_mat_a @ A )
=> ( ( member_set_mat_a @ A3 @ A )
=> ? [X2: set_mat_a] :
( ( member_set_mat_a @ X2 @ A )
& ( ord_le3318621148231462513_mat_a @ X2 @ A3 )
& ! [Xa: set_mat_a] :
( ( member_set_mat_a @ Xa @ A )
=> ( ( ord_le3318621148231462513_mat_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_731_finite__has__minimal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_732_finite__has__maximal2,axiom,
! [A: set_set_mat_a,A3: set_mat_a] :
( ( finite5775620362878804648_mat_a @ A )
=> ( ( member_set_mat_a @ A3 @ A )
=> ? [X2: set_mat_a] :
( ( member_set_mat_a @ X2 @ A )
& ( ord_le3318621148231462513_mat_a @ A3 @ X2 )
& ! [Xa: set_mat_a] :
( ( member_set_mat_a @ Xa @ A )
=> ( ( ord_le3318621148231462513_mat_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_733_finite__has__maximal2,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A3 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A3 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_734_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_735_rev__finite__subset,axiom,
! [B: set_mat_a,A: set_mat_a] :
( ( finite_finite_mat_a @ B )
=> ( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( finite_finite_mat_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_736_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_737_infinite__super,axiom,
! [S: set_mat_a,T2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ S @ T2 )
=> ( ~ ( finite_finite_mat_a @ S )
=> ~ ( finite_finite_mat_a @ T2 ) ) ) ).
% infinite_super
thf(fact_738_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_739_finite__subset,axiom,
! [A: set_mat_a,B: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( ( finite_finite_mat_a @ B )
=> ( finite_finite_mat_a @ A ) ) ) ).
% finite_subset
thf(fact_740_finite_Ointros_I1_J,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.intros(1)
thf(fact_741_finite_Ointros_I1_J,axiom,
finite_finite_mat_a @ bot_bot_set_mat_a ).
% finite.intros(1)
thf(fact_742_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_743_infinite__imp__nonempty,axiom,
! [S: set_mat_a] :
( ~ ( finite_finite_mat_a @ S )
=> ( S != bot_bot_set_mat_a ) ) ).
% infinite_imp_nonempty
thf(fact_744_finite_Ointros_I2_J,axiom,
! [A: set_mat_a,A3: mat_a] :
( ( finite_finite_mat_a @ A )
=> ( finite_finite_mat_a @ ( insert_mat_a @ A3 @ A ) ) ) ).
% finite.intros(2)
thf(fact_745_finite_Ointros_I2_J,axiom,
! [A: set_nat,A3: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A3 @ A ) ) ) ).
% finite.intros(2)
thf(fact_746_finite__insert,axiom,
! [A3: mat_a,A: set_mat_a] :
( ( finite_finite_mat_a @ ( insert_mat_a @ A3 @ A ) )
= ( finite_finite_mat_a @ A ) ) ).
% finite_insert
thf(fact_747_finite__insert,axiom,
! [A3: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A3 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_748_subset__UNIV,axiom,
! [A: set_mat_a] : ( ord_le3318621148231462513_mat_a @ A @ top_top_set_mat_a ) ).
% subset_UNIV
thf(fact_749_empty__not__UNIV,axiom,
bot_bot_set_mat_a != top_top_set_mat_a ).
% empty_not_UNIV
thf(fact_750_insert__UNIV,axiom,
! [X4: mat_a] :
( ( insert_mat_a @ X4 @ top_top_set_mat_a )
= top_top_set_mat_a ) ).
% insert_UNIV
thf(fact_751_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_752_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_nat
@ ^ [S3: nat] : P )
= top_top_set_nat ) )
& ( ~ P
=> ( ( collect_nat
@ ^ [S3: nat] : P )
= bot_bot_set_nat ) ) ) ).
% Collect_const
thf(fact_753_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_mat_a
@ ^ [S3: mat_a] : P )
= top_top_set_mat_a ) )
& ( ~ P
=> ( ( collect_mat_a
@ ^ [S3: mat_a] : P )
= bot_bot_set_mat_a ) ) ) ).
% Collect_const
thf(fact_754_finite__has__minimal,axiom,
! [A: set_set_mat_a] :
( ( finite5775620362878804648_mat_a @ A )
=> ( ( A != bot_bo8661580253428394715_mat_a )
=> ? [X2: set_mat_a] :
( ( member_set_mat_a @ X2 @ A )
& ! [Xa: set_mat_a] :
( ( member_set_mat_a @ Xa @ A )
=> ( ( ord_le3318621148231462513_mat_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_755_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_756_finite__has__maximal,axiom,
! [A: set_set_mat_a] :
( ( finite5775620362878804648_mat_a @ A )
=> ( ( A != bot_bo8661580253428394715_mat_a )
=> ? [X2: set_mat_a] :
( ( member_set_mat_a @ X2 @ A )
& ! [Xa: set_mat_a] :
( ( member_set_mat_a @ Xa @ A )
=> ( ( ord_le3318621148231462513_mat_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_757_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_758_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X2 @ F4 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_759_infinite__finite__induct,axiom,
! [P: set_mat_a > $o,A: set_mat_a] :
( ! [A6: set_mat_a] :
( ~ ( finite_finite_mat_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_mat_a )
=> ( ! [X2: mat_a,F4: set_mat_a] :
( ( finite_finite_mat_a @ F4 )
=> ( ~ ( member_mat_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_mat_a @ X2 @ F4 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_760_finite__ne__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X2 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_761_finite__ne__induct,axiom,
! [F3: set_mat_a,P: set_mat_a > $o] :
( ( finite_finite_mat_a @ F3 )
=> ( ( F3 != bot_bot_set_mat_a )
=> ( ! [X2: mat_a] : ( P @ ( insert_mat_a @ X2 @ bot_bot_set_mat_a ) )
=> ( ! [X2: mat_a,F4: set_mat_a] :
( ( finite_finite_mat_a @ F4 )
=> ( ( F4 != bot_bot_set_mat_a )
=> ( ~ ( member_mat_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_mat_a @ X2 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_762_finite__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X2 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_763_finite__induct,axiom,
! [F3: set_mat_a,P: set_mat_a > $o] :
( ( finite_finite_mat_a @ F3 )
=> ( ( P @ bot_bot_set_mat_a )
=> ( ! [X2: mat_a,F4: set_mat_a] :
( ( finite_finite_mat_a @ F4 )
=> ( ~ ( member_mat_a @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_mat_a @ X2 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_764_finite_Oinducts,axiom,
! [X4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ X4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: set_nat,A5: nat] :
( ( finite_finite_nat @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( insert_nat @ A5 @ A6 ) ) ) )
=> ( P @ X4 ) ) ) ) ).
% finite.inducts
thf(fact_765_finite_Oinducts,axiom,
! [X4: set_mat_a,P: set_mat_a > $o] :
( ( finite_finite_mat_a @ X4 )
=> ( ( P @ bot_bot_set_mat_a )
=> ( ! [A6: set_mat_a,A5: mat_a] :
( ( finite_finite_mat_a @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( insert_mat_a @ A5 @ A6 ) ) ) )
=> ( P @ X4 ) ) ) ) ).
% finite.inducts
thf(fact_766_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A2: set_nat,B4: nat] :
( ( A4
= ( insert_nat @ B4 @ A2 ) )
& ( finite_finite_nat @ A2 ) ) ) ) ) ).
% finite.simps
thf(fact_767_finite_Osimps,axiom,
( finite_finite_mat_a
= ( ^ [A4: set_mat_a] :
( ( A4 = bot_bot_set_mat_a )
| ? [A2: set_mat_a,B4: mat_a] :
( ( A4
= ( insert_mat_a @ B4 @ A2 ) )
& ( finite_finite_mat_a @ A2 ) ) ) ) ) ).
% finite.simps
thf(fact_768_finite_Ocases,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A5: nat] :
( A3
= ( insert_nat @ A5 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_769_finite_Ocases,axiom,
! [A3: set_mat_a] :
( ( finite_finite_mat_a @ A3 )
=> ( ( A3 != bot_bot_set_mat_a )
=> ~ ! [A6: set_mat_a] :
( ? [A5: mat_a] :
( A3
= ( insert_mat_a @ A5 @ A6 ) )
=> ~ ( finite_finite_mat_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_770_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_771_card__subset__eq,axiom,
! [B: set_mat_a,A: set_mat_a] :
( ( finite_finite_mat_a @ B )
=> ( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( ( ( finite_card_mat_a @ A )
= ( finite_card_mat_a @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_772_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ( finite_card_nat @ B5 )
= N )
& ( ord_less_eq_set_nat @ B5 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_773_infinite__arbitrarily__large,axiom,
! [A: set_mat_a,N: nat] :
( ~ ( finite_finite_mat_a @ A )
=> ? [B5: set_mat_a] :
( ( finite_finite_mat_a @ B5 )
& ( ( finite_card_mat_a @ B5 )
= N )
& ( ord_le3318621148231462513_mat_a @ B5 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_774_finite__subset__induct_H,axiom,
! [F3: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A5 @ A )
=> ( ( ord_less_eq_set_nat @ F4 @ A )
=> ( ~ ( member_nat @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A5 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_775_finite__subset__induct_H,axiom,
! [F3: set_mat_a,A: set_mat_a,P: set_mat_a > $o] :
( ( finite_finite_mat_a @ F3 )
=> ( ( ord_le3318621148231462513_mat_a @ F3 @ A )
=> ( ( P @ bot_bot_set_mat_a )
=> ( ! [A5: mat_a,F4: set_mat_a] :
( ( finite_finite_mat_a @ F4 )
=> ( ( member_mat_a @ A5 @ A )
=> ( ( ord_le3318621148231462513_mat_a @ F4 @ A )
=> ( ~ ( member_mat_a @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_mat_a @ A5 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_776_finite__subset__induct,axiom,
! [F3: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A5 @ A )
=> ( ~ ( member_nat @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A5 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_777_finite__subset__induct,axiom,
! [F3: set_mat_a,A: set_mat_a,P: set_mat_a > $o] :
( ( finite_finite_mat_a @ F3 )
=> ( ( ord_le3318621148231462513_mat_a @ F3 @ A )
=> ( ( P @ bot_bot_set_mat_a )
=> ( ! [A5: mat_a,F4: set_mat_a] :
( ( finite_finite_mat_a @ F4 )
=> ( ( member_mat_a @ A5 @ A )
=> ( ~ ( member_mat_a @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_mat_a @ A5 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_778_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_779_card__mono,axiom,
! [B: set_mat_a,A: set_mat_a] :
( ( finite_finite_mat_a @ B )
=> ( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_mat_a @ A ) @ ( finite_card_mat_a @ B ) ) ) ) ).
% card_mono
thf(fact_780_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_781_card__seteq,axiom,
! [B: set_mat_a,A: set_mat_a] :
( ( finite_finite_mat_a @ B )
=> ( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_mat_a @ B ) @ ( finite_card_mat_a @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_782_exists__subset__between,axiom,
! [A: set_nat,N: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B5: set_nat] :
( ( ord_less_eq_set_nat @ A @ B5 )
& ( ord_less_eq_set_nat @ B5 @ C2 )
& ( ( finite_card_nat @ B5 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_783_exists__subset__between,axiom,
! [A: set_mat_a,N: nat,C2: set_mat_a] :
( ( ord_less_eq_nat @ ( finite_card_mat_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_mat_a @ C2 ) )
=> ( ( ord_le3318621148231462513_mat_a @ A @ C2 )
=> ( ( finite_finite_mat_a @ C2 )
=> ? [B5: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ B5 )
& ( ord_le3318621148231462513_mat_a @ B5 @ C2 )
& ( ( finite_card_mat_a @ B5 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_784_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_785_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_mat_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_mat_a @ S ) )
=> ~ ! [T3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ T3 @ S )
=> ( ( ( finite_card_mat_a @ T3 )
= N )
=> ~ ( finite_finite_mat_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_786_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_nat,C2: nat] :
( ! [G3: set_nat] :
( ( ord_less_eq_set_nat @ G3 @ F3 )
=> ( ( finite_finite_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F3 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F3 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_787_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_mat_a,C2: nat] :
( ! [G3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ G3 @ F3 )
=> ( ( finite_finite_mat_a @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_mat_a @ G3 ) @ C2 ) ) )
=> ( ( finite_finite_mat_a @ F3 )
& ( ord_less_eq_nat @ ( finite_card_mat_a @ F3 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_788_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_789_finite__ranking__induct,axiom,
! [S: set_mat_a,P: set_mat_a > $o,F: mat_a > nat] :
( ( finite_finite_mat_a @ S )
=> ( ( P @ bot_bot_set_mat_a )
=> ( ! [X2: mat_a,S2: set_mat_a] :
( ( finite_finite_mat_a @ S2 )
=> ( ! [Y5: mat_a] :
( ( member_mat_a @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_mat_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_790_prod_Ofinite__Collect__op,axiom,
! [I2: set_nat,X4: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I2 )
& ( ( X4 @ I )
!= one_one_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I2 )
& ( ( Y @ I )
!= one_one_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I2 )
& ( ( times_times_nat @ ( X4 @ I ) @ ( Y @ I ) )
!= one_one_nat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_791_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B7: nat] :
( ( member_nat @ B7 @ B )
& ( R2 @ A5 @ B7 ) ) )
=> ( ! [A1: nat,A22: nat,B6: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B6 @ B )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_792_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_793_bounded__Max__nat,axiom,
! [P: nat > $o,X4: nat,M2: nat] :
( ( P @ X4 )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_794_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M4: nat] :
! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ord_less_eq_nat @ X @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_795_finite__less__ub,axiom,
! [F: nat > nat,U4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U4 ) ) ) ) ).
% finite_less_ub
thf(fact_796_class__semiring_Om__closed,axiom,
! [X4: nat,Y: nat] :
( ( member_nat @ X4 @ top_top_set_nat )
=> ( ( member_nat @ Y @ top_top_set_nat )
=> ( member_nat @ ( times_times_nat @ X4 @ Y ) @ top_top_set_nat ) ) ) ).
% class_semiring.m_closed
thf(fact_797_class__semiring_Om__assoc,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( member_nat @ X4 @ top_top_set_nat )
=> ( ( member_nat @ Y @ top_top_set_nat )
=> ( ( member_nat @ Z @ top_top_set_nat )
=> ( ( times_times_nat @ ( times_times_nat @ X4 @ Y ) @ Z )
= ( times_times_nat @ X4 @ ( times_times_nat @ Y @ Z ) ) ) ) ) ) ).
% class_semiring.m_assoc
thf(fact_798_class__semiring_Oone__unique,axiom,
! [U4: nat] :
( ( member_nat @ U4 @ top_top_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ top_top_set_nat )
=> ( ( times_times_nat @ U4 @ X2 )
= X2 ) )
=> ( U4 = one_one_nat ) ) ) ).
% class_semiring.one_unique
thf(fact_799_class__semiring_Oinv__unique,axiom,
! [Y: nat,X4: nat,Y6: nat] :
( ( ( times_times_nat @ Y @ X4 )
= one_one_nat )
=> ( ( ( times_times_nat @ X4 @ Y6 )
= one_one_nat )
=> ( ( member_nat @ X4 @ top_top_set_nat )
=> ( ( member_nat @ Y @ top_top_set_nat )
=> ( ( member_nat @ Y6 @ top_top_set_nat )
=> ( Y = Y6 ) ) ) ) ) ) ).
% class_semiring.inv_unique
thf(fact_800_class__semiring_Or__one,axiom,
! [X4: nat] :
( ( member_nat @ X4 @ top_top_set_nat )
=> ( ( times_times_nat @ X4 @ one_one_nat )
= X4 ) ) ).
% class_semiring.r_one
thf(fact_801_class__semiring_Ol__one,axiom,
! [X4: nat] :
( ( member_nat @ X4 @ top_top_set_nat )
=> ( ( times_times_nat @ one_one_nat @ X4 )
= X4 ) ) ).
% class_semiring.l_one
thf(fact_802_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M4: nat] :
? [N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
& ( member_nat @ N3 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_803_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_804_arg__min__least,axiom,
! [S: set_mat_a,Y: mat_a,F: mat_a > nat] :
( ( finite_finite_mat_a @ S )
=> ( ( S != bot_bot_set_mat_a )
=> ( ( member_mat_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic3922145225401787590_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_805_finite__transitivity__chain,axiom,
! [A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X2: nat] :
~ ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z5: nat] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z5 )
=> ( R @ X2 @ Z5 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ A )
& ( R @ X2 @ Y5 ) ) )
=> ( A = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_806_finite__transitivity__chain,axiom,
! [A: set_mat_a,R: mat_a > mat_a > $o] :
( ( finite_finite_mat_a @ A )
=> ( ! [X2: mat_a] :
~ ( R @ X2 @ X2 )
=> ( ! [X2: mat_a,Y2: mat_a,Z5: mat_a] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z5 )
=> ( R @ X2 @ Z5 ) ) )
=> ( ! [X2: mat_a] :
( ( member_mat_a @ X2 @ A )
=> ? [Y5: mat_a] :
( ( member_mat_a @ Y5 @ A )
& ( R @ X2 @ Y5 ) ) )
=> ( A = bot_bot_set_mat_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_807_finite__indexed__bound,axiom,
! [A: set_nat,P: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [X_12: nat] : ( P @ X2 @ X_12 ) )
=> ? [M5: nat] :
! [X6: nat] :
( ( member_nat @ X6 @ A )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ M5 )
& ( P @ X6 @ K3 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_808_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_809_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_810_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_811_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_812_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_813_Un__insert__right,axiom,
! [A: set_mat_a,A3: mat_a,B: set_mat_a] :
( ( sup_sup_set_mat_a @ A @ ( insert_mat_a @ A3 @ B ) )
= ( insert_mat_a @ A3 @ ( sup_sup_set_mat_a @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_814_Un__insert__left,axiom,
! [A3: mat_a,B: set_mat_a,C2: set_mat_a] :
( ( sup_sup_set_mat_a @ ( insert_mat_a @ A3 @ B ) @ C2 )
= ( insert_mat_a @ A3 @ ( sup_sup_set_mat_a @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_815_Un__empty__right,axiom,
! [A: set_mat_a] :
( ( sup_sup_set_mat_a @ A @ bot_bot_set_mat_a )
= A ) ).
% Un_empty_right
thf(fact_816_Un__empty__left,axiom,
! [B: set_mat_a] :
( ( sup_sup_set_mat_a @ bot_bot_set_mat_a @ B )
= B ) ).
% Un_empty_left
thf(fact_817_Un__empty,axiom,
! [A: set_mat_a,B: set_mat_a] :
( ( ( sup_sup_set_mat_a @ A @ B )
= bot_bot_set_mat_a )
= ( ( A = bot_bot_set_mat_a )
& ( B = bot_bot_set_mat_a ) ) ) ).
% Un_empty
thf(fact_818_Un__subset__iff,axiom,
! [A: set_mat_a,B: set_mat_a,C2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ ( sup_sup_set_mat_a @ A @ B ) @ C2 )
= ( ( ord_le3318621148231462513_mat_a @ A @ C2 )
& ( ord_le3318621148231462513_mat_a @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_819_subset__Un__eq,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A2: set_mat_a,B2: set_mat_a] :
( ( sup_sup_set_mat_a @ A2 @ B2 )
= B2 ) ) ) ).
% subset_Un_eq
thf(fact_820_subset__UnE,axiom,
! [C2: set_mat_a,A: set_mat_a,B: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ C2 @ ( sup_sup_set_mat_a @ A @ B ) )
=> ~ ! [A7: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A7 @ A )
=> ! [B8: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ B8 @ B )
=> ( C2
!= ( sup_sup_set_mat_a @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_821_Un__absorb2,axiom,
! [B: set_mat_a,A: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ B @ A )
=> ( ( sup_sup_set_mat_a @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_822_Un__absorb1,axiom,
! [A: set_mat_a,B: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( ( sup_sup_set_mat_a @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_823_Un__upper2,axiom,
! [B: set_mat_a,A: set_mat_a] : ( ord_le3318621148231462513_mat_a @ B @ ( sup_sup_set_mat_a @ A @ B ) ) ).
% Un_upper2
thf(fact_824_Un__upper1,axiom,
! [A: set_mat_a,B: set_mat_a] : ( ord_le3318621148231462513_mat_a @ A @ ( sup_sup_set_mat_a @ A @ B ) ) ).
% Un_upper1
thf(fact_825_Un__least,axiom,
! [A: set_mat_a,C2: set_mat_a,B: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ C2 )
=> ( ( ord_le3318621148231462513_mat_a @ B @ C2 )
=> ( ord_le3318621148231462513_mat_a @ ( sup_sup_set_mat_a @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_826_Un__mono,axiom,
! [A: set_mat_a,C2: set_mat_a,B: set_mat_a,D: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ C2 )
=> ( ( ord_le3318621148231462513_mat_a @ B @ D )
=> ( ord_le3318621148231462513_mat_a @ ( sup_sup_set_mat_a @ A @ B ) @ ( sup_sup_set_mat_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_827_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_828_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
| ( member_nat @ X @ B2 ) ) ) ) ) ).
% Un_def
thf(fact_829_insert__def,axiom,
( insert_mat_a
= ( ^ [A4: mat_a] :
( sup_sup_set_mat_a
@ ( collect_mat_a
@ ^ [X: mat_a] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_830_insert__def,axiom,
( insert_nat
= ( ^ [A4: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X: nat] : ( X = A4 ) ) ) ) ) ).
% insert_def
thf(fact_831_insert__is__Un,axiom,
( insert_mat_a
= ( ^ [A4: mat_a] : ( sup_sup_set_mat_a @ ( insert_mat_a @ A4 @ bot_bot_set_mat_a ) ) ) ) ).
% insert_is_Un
thf(fact_832_Un__singleton__iff,axiom,
! [A: set_mat_a,B: set_mat_a,X4: mat_a] :
( ( ( sup_sup_set_mat_a @ A @ B )
= ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) )
= ( ( ( A = bot_bot_set_mat_a )
& ( B
= ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) ) )
| ( ( A
= ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) )
& ( B = bot_bot_set_mat_a ) )
| ( ( A
= ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) )
& ( B
= ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_833_singleton__Un__iff,axiom,
! [X4: mat_a,A: set_mat_a,B: set_mat_a] :
( ( ( insert_mat_a @ X4 @ bot_bot_set_mat_a )
= ( sup_sup_set_mat_a @ A @ B ) )
= ( ( ( A = bot_bot_set_mat_a )
& ( B
= ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) ) )
| ( ( A
= ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) )
& ( B = bot_bot_set_mat_a ) )
| ( ( A
= ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) )
& ( B
= ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_834_insert__union,axiom,
( insert_mat_a
= ( ^ [X: mat_a,X5: set_mat_a] : ( sup_sup_set_mat_a @ X5 @ ( insert_mat_a @ X @ bot_bot_set_mat_a ) ) ) ) ).
% insert_union
thf(fact_835_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_mat_a] :
( ( sup_sup_set_mat_a @ X4 @ bot_bot_set_mat_a )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_836_sup__bot_Oright__neutral,axiom,
! [A3: set_mat_a] :
( ( sup_sup_set_mat_a @ A3 @ bot_bot_set_mat_a )
= A3 ) ).
% sup_bot.right_neutral
thf(fact_837_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A2 )
@ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ) ).
% sup_set_def
thf(fact_838_sup__Un__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_839_sup_OcoboundedI2,axiom,
! [C: set_mat_a,B3: set_mat_a,A3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ C @ B3 )
=> ( ord_le3318621148231462513_mat_a @ C @ ( sup_sup_set_mat_a @ A3 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_840_sup_OcoboundedI2,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% sup.coboundedI2
thf(fact_841_sup_OcoboundedI1,axiom,
! [C: set_mat_a,A3: set_mat_a,B3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ C @ A3 )
=> ( ord_le3318621148231462513_mat_a @ C @ ( sup_sup_set_mat_a @ A3 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_842_sup_OcoboundedI1,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A3 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% sup.coboundedI1
thf(fact_843_sup_Obounded__iff,axiom,
! [B3: set_mat_a,C: set_mat_a,A3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ ( sup_sup_set_mat_a @ B3 @ C ) @ A3 )
= ( ( ord_le3318621148231462513_mat_a @ B3 @ A3 )
& ( ord_le3318621148231462513_mat_a @ C @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_844_sup_Obounded__iff,axiom,
! [B3: nat,C: nat,A3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A3 )
= ( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ C @ A3 ) ) ) ).
% sup.bounded_iff
thf(fact_845_sup_Oabsorb__iff2,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [A4: set_mat_a,B4: set_mat_a] :
( ( sup_sup_set_mat_a @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_846_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_847_sup_Oabsorb__iff1,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [B4: set_mat_a,A4: set_mat_a] :
( ( sup_sup_set_mat_a @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_848_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_849_sup_Ocobounded2,axiom,
! [B3: set_mat_a,A3: set_mat_a] : ( ord_le3318621148231462513_mat_a @ B3 @ ( sup_sup_set_mat_a @ A3 @ B3 ) ) ).
% sup.cobounded2
thf(fact_850_sup_Ocobounded2,axiom,
! [B3: nat,A3: nat] : ( ord_less_eq_nat @ B3 @ ( sup_sup_nat @ A3 @ B3 ) ) ).
% sup.cobounded2
thf(fact_851_sup_Ocobounded1,axiom,
! [A3: set_mat_a,B3: set_mat_a] : ( ord_le3318621148231462513_mat_a @ A3 @ ( sup_sup_set_mat_a @ A3 @ B3 ) ) ).
% sup.cobounded1
thf(fact_852_sup_Ocobounded1,axiom,
! [A3: nat,B3: nat] : ( ord_less_eq_nat @ A3 @ ( sup_sup_nat @ A3 @ B3 ) ) ).
% sup.cobounded1
thf(fact_853_sup_Oorder__iff,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [B4: set_mat_a,A4: set_mat_a] :
( A4
= ( sup_sup_set_mat_a @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_854_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_855_sup_OboundedI,axiom,
! [B3: set_mat_a,A3: set_mat_a,C: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ B3 @ A3 )
=> ( ( ord_le3318621148231462513_mat_a @ C @ A3 )
=> ( ord_le3318621148231462513_mat_a @ ( sup_sup_set_mat_a @ B3 @ C ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_856_sup_OboundedI,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C @ A3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A3 ) ) ) ).
% sup.boundedI
thf(fact_857_sup_OboundedE,axiom,
! [B3: set_mat_a,C: set_mat_a,A3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ ( sup_sup_set_mat_a @ B3 @ C ) @ A3 )
=> ~ ( ( ord_le3318621148231462513_mat_a @ B3 @ A3 )
=> ~ ( ord_le3318621148231462513_mat_a @ C @ A3 ) ) ) ).
% sup.boundedE
thf(fact_858_sup_OboundedE,axiom,
! [B3: nat,C: nat,A3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A3 )
=> ~ ( ( ord_less_eq_nat @ B3 @ A3 )
=> ~ ( ord_less_eq_nat @ C @ A3 ) ) ) ).
% sup.boundedE
thf(fact_859_sup__absorb2,axiom,
! [X4: set_mat_a,Y: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X4 @ Y )
=> ( ( sup_sup_set_mat_a @ X4 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_860_sup__absorb2,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( sup_sup_nat @ X4 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_861_sup__absorb1,axiom,
! [Y: set_mat_a,X4: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ Y @ X4 )
=> ( ( sup_sup_set_mat_a @ X4 @ Y )
= X4 ) ) ).
% sup_absorb1
thf(fact_862_sup__absorb1,axiom,
! [Y: nat,X4: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ( ( sup_sup_nat @ X4 @ Y )
= X4 ) ) ).
% sup_absorb1
thf(fact_863_sup_Oabsorb2,axiom,
! [A3: set_mat_a,B3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A3 @ B3 )
=> ( ( sup_sup_set_mat_a @ A3 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_864_sup_Oabsorb2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ).
% sup.absorb2
thf(fact_865_sup_Oabsorb1,axiom,
! [B3: set_mat_a,A3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ B3 @ A3 )
=> ( ( sup_sup_set_mat_a @ A3 @ B3 )
= A3 ) ) ).
% sup.absorb1
thf(fact_866_sup_Oabsorb1,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ).
% sup.absorb1
thf(fact_867_sup__unique,axiom,
! [F: set_mat_a > set_mat_a > set_mat_a,X4: set_mat_a,Y: set_mat_a] :
( ! [X2: set_mat_a,Y2: set_mat_a] : ( ord_le3318621148231462513_mat_a @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_mat_a,Y2: set_mat_a] : ( ord_le3318621148231462513_mat_a @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_mat_a,Y2: set_mat_a,Z5: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ Y2 @ X2 )
=> ( ( ord_le3318621148231462513_mat_a @ Z5 @ X2 )
=> ( ord_le3318621148231462513_mat_a @ ( F @ Y2 @ Z5 ) @ X2 ) ) )
=> ( ( sup_sup_set_mat_a @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_868_sup__unique,axiom,
! [F: nat > nat > nat,X4: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat,Z5: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ Z5 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z5 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X4 @ Y )
= ( F @ X4 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_869_sup_OorderI,axiom,
! [A3: set_mat_a,B3: set_mat_a] :
( ( A3
= ( sup_sup_set_mat_a @ A3 @ B3 ) )
=> ( ord_le3318621148231462513_mat_a @ B3 @ A3 ) ) ).
% sup.orderI
thf(fact_870_sup_OorderI,axiom,
! [A3: nat,B3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B3 ) )
=> ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% sup.orderI
thf(fact_871_sup_OorderE,axiom,
! [B3: set_mat_a,A3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ B3 @ A3 )
=> ( A3
= ( sup_sup_set_mat_a @ A3 @ B3 ) ) ) ).
% sup.orderE
thf(fact_872_sup_OorderE,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( A3
= ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% sup.orderE
thf(fact_873_le__sup__iff,axiom,
! [X4: set_mat_a,Y: set_mat_a,Z: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ ( sup_sup_set_mat_a @ X4 @ Y ) @ Z )
= ( ( ord_le3318621148231462513_mat_a @ X4 @ Z )
& ( ord_le3318621148231462513_mat_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_874_le__sup__iff,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X4 @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_875_le__iff__sup,axiom,
( ord_le3318621148231462513_mat_a
= ( ^ [X: set_mat_a,Y4: set_mat_a] :
( ( sup_sup_set_mat_a @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_876_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( sup_sup_nat @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_877_sup__least,axiom,
! [Y: set_mat_a,X4: set_mat_a,Z: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ Y @ X4 )
=> ( ( ord_le3318621148231462513_mat_a @ Z @ X4 )
=> ( ord_le3318621148231462513_mat_a @ ( sup_sup_set_mat_a @ Y @ Z ) @ X4 ) ) ) ).
% sup_least
thf(fact_878_sup__least,axiom,
! [Y: nat,X4: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ( ( ord_less_eq_nat @ Z @ X4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X4 ) ) ) ).
% sup_least
thf(fact_879_sup__mono,axiom,
! [A3: set_mat_a,C: set_mat_a,B3: set_mat_a,D2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A3 @ C )
=> ( ( ord_le3318621148231462513_mat_a @ B3 @ D2 )
=> ( ord_le3318621148231462513_mat_a @ ( sup_sup_set_mat_a @ A3 @ B3 ) @ ( sup_sup_set_mat_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_880_sup__mono,axiom,
! [A3: nat,C: nat,B3: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ C )
=> ( ( ord_less_eq_nat @ B3 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B3 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_881_sup_Omono,axiom,
! [C: set_mat_a,A3: set_mat_a,D2: set_mat_a,B3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ C @ A3 )
=> ( ( ord_le3318621148231462513_mat_a @ D2 @ B3 )
=> ( ord_le3318621148231462513_mat_a @ ( sup_sup_set_mat_a @ C @ D2 ) @ ( sup_sup_set_mat_a @ A3 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_882_sup_Omono,axiom,
! [C: nat,A3: nat,D2: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ A3 )
=> ( ( ord_less_eq_nat @ D2 @ B3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% sup.mono
thf(fact_883_le__supI2,axiom,
! [X4: set_mat_a,B3: set_mat_a,A3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X4 @ B3 )
=> ( ord_le3318621148231462513_mat_a @ X4 @ ( sup_sup_set_mat_a @ A3 @ B3 ) ) ) ).
% le_supI2
thf(fact_884_le__supI2,axiom,
! [X4: nat,B3: nat,A3: nat] :
( ( ord_less_eq_nat @ X4 @ B3 )
=> ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% le_supI2
thf(fact_885_le__supI1,axiom,
! [X4: set_mat_a,A3: set_mat_a,B3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ X4 @ A3 )
=> ( ord_le3318621148231462513_mat_a @ X4 @ ( sup_sup_set_mat_a @ A3 @ B3 ) ) ) ).
% le_supI1
thf(fact_886_le__supI1,axiom,
! [X4: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ X4 @ A3 )
=> ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A3 @ B3 ) ) ) ).
% le_supI1
thf(fact_887_sup__ge2,axiom,
! [Y: set_mat_a,X4: set_mat_a] : ( ord_le3318621148231462513_mat_a @ Y @ ( sup_sup_set_mat_a @ X4 @ Y ) ) ).
% sup_ge2
thf(fact_888_sup__ge2,axiom,
! [Y: nat,X4: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X4 @ Y ) ) ).
% sup_ge2
thf(fact_889_sup__ge1,axiom,
! [X4: set_mat_a,Y: set_mat_a] : ( ord_le3318621148231462513_mat_a @ X4 @ ( sup_sup_set_mat_a @ X4 @ Y ) ) ).
% sup_ge1
thf(fact_890_sup__ge1,axiom,
! [X4: nat,Y: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y ) ) ).
% sup_ge1
thf(fact_891_le__supI,axiom,
! [A3: set_mat_a,X4: set_mat_a,B3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A3 @ X4 )
=> ( ( ord_le3318621148231462513_mat_a @ B3 @ X4 )
=> ( ord_le3318621148231462513_mat_a @ ( sup_sup_set_mat_a @ A3 @ B3 ) @ X4 ) ) ) ).
% le_supI
thf(fact_892_le__supI,axiom,
! [A3: nat,X4: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ X4 )
=> ( ( ord_less_eq_nat @ B3 @ X4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B3 ) @ X4 ) ) ) ).
% le_supI
thf(fact_893_le__supE,axiom,
! [A3: set_mat_a,B3: set_mat_a,X4: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ ( sup_sup_set_mat_a @ A3 @ B3 ) @ X4 )
=> ~ ( ( ord_le3318621148231462513_mat_a @ A3 @ X4 )
=> ~ ( ord_le3318621148231462513_mat_a @ B3 @ X4 ) ) ) ).
% le_supE
thf(fact_894_le__supE,axiom,
! [A3: nat,B3: nat,X4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B3 ) @ X4 )
=> ~ ( ( ord_less_eq_nat @ A3 @ X4 )
=> ~ ( ord_less_eq_nat @ B3 @ X4 ) ) ) ).
% le_supE
thf(fact_895_inf__sup__ord_I3_J,axiom,
! [X4: set_mat_a,Y: set_mat_a] : ( ord_le3318621148231462513_mat_a @ X4 @ ( sup_sup_set_mat_a @ X4 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_896_inf__sup__ord_I3_J,axiom,
! [X4: nat,Y: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_897_inf__sup__ord_I4_J,axiom,
! [Y: set_mat_a,X4: set_mat_a] : ( ord_le3318621148231462513_mat_a @ Y @ ( sup_sup_set_mat_a @ X4 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_898_inf__sup__ord_I4_J,axiom,
! [Y: nat,X4: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X4 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_899_sup__bot__left,axiom,
! [X4: set_mat_a] :
( ( sup_sup_set_mat_a @ bot_bot_set_mat_a @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_900_sup__bot__right,axiom,
! [X4: set_mat_a] :
( ( sup_sup_set_mat_a @ X4 @ bot_bot_set_mat_a )
= X4 ) ).
% sup_bot_right
thf(fact_901_bot__eq__sup__iff,axiom,
! [X4: set_mat_a,Y: set_mat_a] :
( ( bot_bot_set_mat_a
= ( sup_sup_set_mat_a @ X4 @ Y ) )
= ( ( X4 = bot_bot_set_mat_a )
& ( Y = bot_bot_set_mat_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_902_sup__eq__bot__iff,axiom,
! [X4: set_mat_a,Y: set_mat_a] :
( ( ( sup_sup_set_mat_a @ X4 @ Y )
= bot_bot_set_mat_a )
= ( ( X4 = bot_bot_set_mat_a )
& ( Y = bot_bot_set_mat_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_903_sup__bot_Oeq__neutr__iff,axiom,
! [A3: set_mat_a,B3: set_mat_a] :
( ( ( sup_sup_set_mat_a @ A3 @ B3 )
= bot_bot_set_mat_a )
= ( ( A3 = bot_bot_set_mat_a )
& ( B3 = bot_bot_set_mat_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_904_sup__bot_Oleft__neutral,axiom,
! [A3: set_mat_a] :
( ( sup_sup_set_mat_a @ bot_bot_set_mat_a @ A3 )
= A3 ) ).
% sup_bot.left_neutral
thf(fact_905_sup__bot_Oneutr__eq__iff,axiom,
! [A3: set_mat_a,B3: set_mat_a] :
( ( bot_bot_set_mat_a
= ( sup_sup_set_mat_a @ A3 @ B3 ) )
= ( ( A3 = bot_bot_set_mat_a )
& ( B3 = bot_bot_set_mat_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_906_add__right__imp__eq,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= ( plus_plus_nat @ C @ A3 ) )
=> ( B3 = C ) ) ).
% add_right_imp_eq
thf(fact_907_add__right__cancel,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ( plus_plus_nat @ B3 @ A3 )
= ( plus_plus_nat @ C @ A3 ) )
= ( B3 = C ) ) ).
% add_right_cancel
thf(fact_908_add__left__imp__eq,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= ( plus_plus_nat @ A3 @ C ) )
=> ( B3 = C ) ) ).
% add_left_imp_eq
thf(fact_909_add__left__cancel,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ( plus_plus_nat @ A3 @ B3 )
= ( plus_plus_nat @ A3 @ C ) )
= ( B3 = C ) ) ).
% add_left_cancel
thf(fact_910_add_Oleft__commute,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( plus_plus_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add.left_commute
thf(fact_911_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_912_add_Oassoc,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add.assoc
thf(fact_913_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B3: nat,A3: nat] :
( ( B
= ( plus_plus_nat @ K @ B3 ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% group_cancel.add2
thf(fact_914_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A3: nat,B3: nat] :
( ( A
= ( plus_plus_nat @ K @ A3 ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% group_cancel.add1
thf(fact_915_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( I3 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I3 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_916_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_917_trace__add__linear,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_trace_a @ ( plus_plus_mat_a @ A @ B ) )
= ( plus_plus_a @ ( complex_trace_a @ A ) @ ( complex_trace_a @ B ) ) ) ) ) ).
% trace_add_linear
thf(fact_918_add__smult__distrib__right__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat,K: a,L: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( smult_mat_a @ ( plus_plus_a @ K @ L ) @ A )
= ( plus_plus_mat_a @ ( smult_mat_a @ K @ A ) @ ( smult_mat_a @ L @ A ) ) ) ) ).
% add_smult_distrib_right_mat
thf(fact_919_add__smult__distrib__right__mat,axiom,
! [A: mat_nat,Nr: nat,Nc: nat,K: nat,L: nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( smult_mat_nat @ ( plus_plus_nat @ K @ L ) @ A )
= ( plus_plus_mat_nat @ ( smult_mat_nat @ K @ A ) @ ( smult_mat_nat @ L @ A ) ) ) ) ).
% add_smult_distrib_right_mat
thf(fact_920_combine__common__factor,axiom,
! [A3: nat,E2: nat,B3: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A3 @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B3 @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_921_distrib__right,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ).
% distrib_right
thf(fact_922_distrib__left,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( times_times_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ B3 ) @ ( times_times_nat @ A3 @ C ) ) ) ).
% distrib_left
thf(fact_923_comm__semiring__class_Odistrib,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_924_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_925_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( I3 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_926_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_927_add__mono,axiom,
! [A3: nat,B3: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D2 ) ) ) ) ).
% add_mono
thf(fact_928_add__left__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% add_left_mono
thf(fact_929_less__eqE,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ~ ! [C4: nat] :
( B3
!= ( plus_plus_nat @ A3 @ C4 ) ) ) ).
% less_eqE
thf(fact_930_add__right__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% add_right_mono
thf(fact_931_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C5: nat] :
( B4
= ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_932_add__le__cancel__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_cancel_left
thf(fact_933_add__le__imp__le__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_imp_le_left
thf(fact_934_add__le__cancel__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_cancel_right
thf(fact_935_add__le__imp__le__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% add_le_imp_le_right
thf(fact_936_class__semiring_Osemiring__simprules_I13_J,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( member_nat @ X4 @ top_top_set_nat )
=> ( ( member_nat @ Y @ top_top_set_nat )
=> ( ( member_nat @ Z @ top_top_set_nat )
=> ( ( times_times_nat @ Z @ ( plus_plus_nat @ X4 @ Y ) )
= ( plus_plus_nat @ ( times_times_nat @ Z @ X4 ) @ ( times_times_nat @ Z @ Y ) ) ) ) ) ) ).
% class_semiring.semiring_simprules(13)
thf(fact_937_class__semiring_Osemiring__simprules_I10_J,axiom,
! [X4: nat,Y: nat,Z: nat] :
( ( member_nat @ X4 @ top_top_set_nat )
=> ( ( member_nat @ Y @ top_top_set_nat )
=> ( ( member_nat @ Z @ top_top_set_nat )
=> ( ( times_times_nat @ ( plus_plus_nat @ X4 @ Y ) @ Z )
= ( plus_plus_nat @ ( times_times_nat @ X4 @ Z ) @ ( times_times_nat @ Y @ Z ) ) ) ) ) ) ).
% class_semiring.semiring_simprules(10)
thf(fact_938_some__in__eq,axiom,
! [A: set_nat] :
( ( member_nat
@ ( fChoice_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
@ A )
= ( A != bot_bot_set_nat ) ) ).
% some_in_eq
thf(fact_939_some__in__eq,axiom,
! [A: set_mat_a] :
( ( member_mat_a
@ ( fChoice_mat_a
@ ^ [X: mat_a] : ( member_mat_a @ X @ A ) )
@ A )
= ( A != bot_bot_set_mat_a ) ) ).
% some_in_eq
thf(fact_940_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_941_index__add__mat_I2_J,axiom,
! [A: mat_a,B: mat_a] :
( ( dim_row_a @ ( plus_plus_mat_a @ A @ B ) )
= ( dim_row_a @ B ) ) ).
% index_add_mat(2)
thf(fact_942_comm__add__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ A @ B )
= ( plus_plus_mat_a @ B @ A ) ) ) ) ).
% comm_add_mat
thf(fact_943_assoc__add__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat,B: mat_a,C2: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( plus_plus_mat_a @ ( plus_plus_mat_a @ A @ B ) @ C2 )
= ( plus_plus_mat_a @ A @ ( plus_plus_mat_a @ B @ C2 ) ) ) ) ) ) ).
% assoc_add_mat
thf(fact_944_add__carrier__mat,axiom,
! [B: mat_a,Nr: nat,Nc: nat,A: mat_a] :
( ( member_mat_a @ B @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( plus_plus_mat_a @ A @ B ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% add_carrier_mat
thf(fact_945_swap__plus__mat,axiom,
! [A: mat_a,N: nat,B: mat_a,C2: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ N ) )
=> ( ( plus_plus_mat_a @ ( plus_plus_mat_a @ A @ B ) @ C2 )
= ( plus_plus_mat_a @ ( plus_plus_mat_a @ A @ C2 ) @ B ) ) ) ) ) ).
% swap_plus_mat
thf(fact_946_add__carrier__mat_H,axiom,
! [A: mat_a,Nr: nat,Nc: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( plus_plus_mat_a @ A @ B ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% add_carrier_mat'
thf(fact_947_sumset__empty_I1_J,axiom,
! [A: set_mat_a] :
( ( plus_plus_set_mat_a @ A @ bot_bot_set_mat_a )
= bot_bot_set_mat_a ) ).
% sumset_empty(1)
thf(fact_948_sumset__empty_I2_J,axiom,
! [A: set_mat_a] :
( ( plus_plus_set_mat_a @ bot_bot_set_mat_a @ A )
= bot_bot_set_mat_a ) ).
% sumset_empty(2)
thf(fact_949_set__plus__mono2,axiom,
! [C2: set_mat_a,D: set_mat_a,E: set_mat_a,F3: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ C2 @ D )
=> ( ( ord_le3318621148231462513_mat_a @ E @ F3 )
=> ( ord_le3318621148231462513_mat_a @ ( plus_plus_set_mat_a @ C2 @ E ) @ ( plus_plus_set_mat_a @ D @ F3 ) ) ) ) ).
% set_plus_mono2
thf(fact_950_mult__add__distrib__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,B: mat_a,Nc: nat,C2: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ A @ ( plus_plus_mat_a @ B @ C2 ) )
= ( plus_plus_mat_a @ ( times_times_mat_a @ A @ B ) @ ( times_times_mat_a @ A @ C2 ) ) ) ) ) ) ).
% mult_add_distrib_mat
thf(fact_951_add__mult__distrib__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,B: mat_a,C2: mat_a,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ ( plus_plus_mat_a @ A @ B ) @ C2 )
= ( plus_plus_mat_a @ ( times_times_mat_a @ A @ C2 ) @ ( times_times_mat_a @ B @ C2 ) ) ) ) ) ) ).
% add_mult_distrib_mat
thf(fact_952_add__smult__distrib__left__mat,axiom,
! [A: mat_a,Nr: nat,Nc: nat,B: mat_a,K: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( smult_mat_a @ K @ ( plus_plus_mat_a @ A @ B ) )
= ( plus_plus_mat_a @ ( smult_mat_a @ K @ A ) @ ( smult_mat_a @ K @ B ) ) ) ) ) ).
% add_smult_distrib_left_mat
thf(fact_953_adjoint__add,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ M ) )
=> ( ( schur_mat_adjoint_a @ ( plus_plus_mat_a @ A @ B ) )
= ( plus_plus_mat_a @ ( schur_mat_adjoint_a @ A ) @ ( schur_mat_adjoint_a @ B ) ) ) ) ) ).
% adjoint_add
thf(fact_954_hermitian__add,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_hermitian_a @ A )
=> ( ( complex_hermitian_a @ B )
=> ( complex_hermitian_a @ ( plus_plus_mat_a @ A @ B ) ) ) ) ) ) ).
% hermitian_add
thf(fact_955_GreatestI__ex__nat,axiom,
! [P: nat > $o,B3: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_956_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_957_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_958_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_959_le__trans,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_960_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_961_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_962_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_963_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_964_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_965_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_966_mult__le__mono,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_967_mult__le__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_968_mult__le__mono2,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_969_someI2__bex,axiom,
! [A: set_nat,P: nat > $o,Q: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ A )
& ( P @ X6 ) )
=> ( ! [X2: nat] :
( ( ( member_nat @ X2 @ A )
& ( P @ X2 ) )
=> ( Q @ X2 ) )
=> ( Q
@ ( fChoice_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) ) ) ) ) ).
% someI2_bex
thf(fact_970_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_971_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_972_trans__le__add2,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_973_trans__le__add1,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_974_add__le__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_975_add__le__mono,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_976_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_977_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_978_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_979_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_980_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_981_Linear__Algebra__Complements_Otrace__add,axiom,
! [A: mat_a,B: mat_a] :
( ( square_mat_a @ A )
=> ( ( square_mat_a @ B )
=> ( ( ( dim_row_a @ A )
= ( dim_row_a @ B ) )
=> ( ( complex_trace_a @ ( plus_plus_mat_a @ A @ B ) )
= ( plus_plus_a @ ( complex_trace_a @ A ) @ ( complex_trace_a @ B ) ) ) ) ) ) ).
% Linear_Algebra_Complements.trace_add
thf(fact_982_square__mat_Osimps,axiom,
( square_mat_a
= ( ^ [A2: mat_a] :
( ( dim_col_a @ A2 )
= ( dim_row_a @ A2 ) ) ) ) ).
% square_mat.simps
thf(fact_983_square__mat_Oelims_I1_J,axiom,
! [X4: mat_a,Y: $o] :
( ( ( square_mat_a @ X4 )
= Y )
=> ( Y
= ( ( dim_col_a @ X4 )
= ( dim_row_a @ X4 ) ) ) ) ).
% square_mat.elims(1)
thf(fact_984_square__mat_Oelims_I2_J,axiom,
! [X4: mat_a] :
( ( square_mat_a @ X4 )
=> ( ( dim_col_a @ X4 )
= ( dim_row_a @ X4 ) ) ) ).
% square_mat.elims(2)
thf(fact_985_square__mat_Oelims_I3_J,axiom,
! [X4: mat_a] :
( ~ ( square_mat_a @ X4 )
=> ( ( dim_col_a @ X4 )
!= ( dim_row_a @ X4 ) ) ) ).
% square_mat.elims(3)
thf(fact_986_crossproduct__noteq,axiom,
! [A3: nat,B3: nat,C: nat,D2: nat] :
( ( ( A3 != B3 )
& ( C != D2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D2 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A3 @ D2 ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_987_crossproduct__eq,axiom,
! [W: nat,Y: nat,X4: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X4 @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X4 @ Y ) ) )
= ( ( W = X4 )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_988_vec__space_Orow__space__is__preserved,axiom,
! [P: mat_a,M: nat,A: mat_a,N: nat] :
( ( invertible_mat_a @ P )
=> ( ( member_mat_a @ P @ ( carrier_mat_a @ M @ M ) )
=> ( ( member_mat_a @ A @ ( carrier_mat_a @ M @ N ) )
=> ( ( vS_vec_row_space_a @ N @ ( times_times_mat_a @ P @ A ) )
= ( vS_vec_row_space_a @ N @ A ) ) ) ) ) ).
% vec_space.row_space_is_preserved
thf(fact_989_mult__hom_Ohom__add,axiom,
! [C: nat,X4: nat,Y: nat] :
( ( times_times_nat @ C @ ( plus_plus_nat @ X4 @ Y ) )
= ( plus_plus_nat @ ( times_times_nat @ C @ X4 ) @ ( times_times_nat @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_990_sum__mat__add,axiom,
! [A: mat_a,Nr: nat,Nc: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( comm_m5291664705200495434_mat_a @ ( plus_plus_mat_a @ A @ B ) )
= ( plus_plus_a @ ( comm_m5291664705200495434_mat_a @ A ) @ ( comm_m5291664705200495434_mat_a @ B ) ) ) ) ) ).
% sum_mat_add
thf(fact_991_sum__mat__add,axiom,
! [A: mat_nat,Nr: nat,Nc: nat,B: mat_nat] :
( ( member_mat_nat @ A @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( member_mat_nat @ B @ ( carrier_mat_nat @ Nr @ Nc ) )
=> ( ( comm_m4056229327131402372at_nat @ ( plus_plus_mat_nat @ A @ B ) )
= ( plus_plus_nat @ ( comm_m4056229327131402372at_nat @ A ) @ ( comm_m4056229327131402372at_nat @ B ) ) ) ) ) ).
% sum_mat_add
thf(fact_992_plus__mat__def,axiom,
( plus_plus_mat_a
= ( ^ [A2: mat_a,B2: mat_a] :
( mat_a2 @ ( dim_row_a @ B2 ) @ ( dim_col_a @ B2 )
@ ^ [Ij: product_prod_nat_nat] : ( plus_plus_a @ ( index_mat_a @ A2 @ Ij ) @ ( index_mat_a @ B2 @ Ij ) ) ) ) ) ).
% plus_mat_def
thf(fact_993_plus__mat__def,axiom,
( plus_plus_mat_nat
= ( ^ [A2: mat_nat,B2: mat_nat] :
( mat_nat2 @ ( dim_row_nat @ B2 ) @ ( dim_col_nat @ B2 )
@ ^ [Ij: product_prod_nat_nat] : ( plus_plus_nat @ ( index_mat_nat @ A2 @ Ij ) @ ( index_mat_nat @ B2 @ Ij ) ) ) ) ) ).
% plus_mat_def
thf(fact_994_update__mat__def,axiom,
( update_mat_a
= ( ^ [A2: mat_a,Ij: product_prod_nat_nat,A4: a] :
( mat_a2 @ ( dim_row_a @ A2 ) @ ( dim_col_a @ A2 )
@ ^ [Ij2: product_prod_nat_nat] : ( if_a @ ( Ij2 = Ij ) @ A4 @ ( index_mat_a @ A2 @ Ij2 ) ) ) ) ) ).
% update_mat_def
thf(fact_995_minus__mat__def,axiom,
( minus_minus_mat_a
= ( ^ [A2: mat_a,B2: mat_a] :
( mat_a2 @ ( dim_row_a @ B2 ) @ ( dim_col_a @ B2 )
@ ^ [Ij: product_prod_nat_nat] : ( minus_minus_a @ ( index_mat_a @ A2 @ Ij ) @ ( index_mat_a @ B2 @ Ij ) ) ) ) ) ).
% minus_mat_def
thf(fact_996_minus__mat__def,axiom,
( minus_minus_mat_nat
= ( ^ [A2: mat_nat,B2: mat_nat] :
( mat_nat2 @ ( dim_row_nat @ B2 ) @ ( dim_col_nat @ B2 )
@ ^ [Ij: product_prod_nat_nat] : ( minus_minus_nat @ ( index_mat_nat @ A2 @ Ij ) @ ( index_mat_nat @ B2 @ Ij ) ) ) ) ) ).
% minus_mat_def
thf(fact_997_diff__shunt__var,axiom,
! [X4: set_mat_a,Y: set_mat_a] :
( ( ( minus_4757590266979429866_mat_a @ X4 @ Y )
= bot_bot_set_mat_a )
= ( ord_le3318621148231462513_mat_a @ X4 @ Y ) ) ).
% diff_shunt_var
thf(fact_998_Diff__UNIV,axiom,
! [A: set_mat_a] :
( ( minus_4757590266979429866_mat_a @ A @ top_top_set_mat_a )
= bot_bot_set_mat_a ) ).
% Diff_UNIV
thf(fact_999_Diff__eq__empty__iff,axiom,
! [A: set_mat_a,B: set_mat_a] :
( ( ( minus_4757590266979429866_mat_a @ A @ B )
= bot_bot_set_mat_a )
= ( ord_le3318621148231462513_mat_a @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1000_subset__Diff__insert,axiom,
! [A: set_nat,B: set_nat,X4: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat @ X4 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) )
& ~ ( member_nat @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1001_subset__Diff__insert,axiom,
! [A: set_mat_a,B: set_mat_a,X4: mat_a,C2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ ( minus_4757590266979429866_mat_a @ B @ ( insert_mat_a @ X4 @ C2 ) ) )
= ( ( ord_le3318621148231462513_mat_a @ A @ ( minus_4757590266979429866_mat_a @ B @ C2 ) )
& ~ ( member_mat_a @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1002_Diff__not__in,axiom,
! [A3: nat,A: set_nat] :
~ ( member_nat @ A3 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ).
% Diff_not_in
thf(fact_1003_Diff__not__in,axiom,
! [A3: mat_a,A: set_mat_a] :
~ ( member_mat_a @ A3 @ ( minus_4757590266979429866_mat_a @ A @ ( insert_mat_a @ A3 @ bot_bot_set_mat_a ) ) ) ).
% Diff_not_in
thf(fact_1004_insert__Diff__single,axiom,
! [A3: mat_a,A: set_mat_a] :
( ( insert_mat_a @ A3 @ ( minus_4757590266979429866_mat_a @ A @ ( insert_mat_a @ A3 @ bot_bot_set_mat_a ) ) )
= ( insert_mat_a @ A3 @ A ) ) ).
% insert_Diff_single
thf(fact_1005_Diff__insert__absorb,axiom,
! [X4: nat,A: set_nat] :
( ~ ( member_nat @ X4 @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1006_Diff__insert__absorb,axiom,
! [X4: mat_a,A: set_mat_a] :
( ~ ( member_mat_a @ X4 @ A )
=> ( ( minus_4757590266979429866_mat_a @ ( insert_mat_a @ X4 @ A ) @ ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_1007_Diff__insert2,axiom,
! [A: set_mat_a,A3: mat_a,B: set_mat_a] :
( ( minus_4757590266979429866_mat_a @ A @ ( insert_mat_a @ A3 @ B ) )
= ( minus_4757590266979429866_mat_a @ ( minus_4757590266979429866_mat_a @ A @ ( insert_mat_a @ A3 @ bot_bot_set_mat_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_1008_insert__Diff,axiom,
! [A3: nat,A: set_nat] :
( ( member_nat @ A3 @ A )
=> ( ( insert_nat @ A3 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1009_insert__Diff,axiom,
! [A3: mat_a,A: set_mat_a] :
( ( member_mat_a @ A3 @ A )
=> ( ( insert_mat_a @ A3 @ ( minus_4757590266979429866_mat_a @ A @ ( insert_mat_a @ A3 @ bot_bot_set_mat_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_1010_Diff__insert,axiom,
! [A: set_mat_a,A3: mat_a,B: set_mat_a] :
( ( minus_4757590266979429866_mat_a @ A @ ( insert_mat_a @ A3 @ B ) )
= ( minus_4757590266979429866_mat_a @ ( minus_4757590266979429866_mat_a @ A @ B ) @ ( insert_mat_a @ A3 @ bot_bot_set_mat_a ) ) ) ).
% Diff_insert
thf(fact_1011_finite__Diff__insert,axiom,
! [A: set_mat_a,A3: mat_a,B: set_mat_a] :
( ( finite_finite_mat_a @ ( minus_4757590266979429866_mat_a @ A @ ( insert_mat_a @ A3 @ B ) ) )
= ( finite_finite_mat_a @ ( minus_4757590266979429866_mat_a @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1012_finite__Diff__insert,axiom,
! [A: set_nat,A3: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A3 @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_1013_Diff__subset__conv,axiom,
! [A: set_mat_a,B: set_mat_a,C2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ ( minus_4757590266979429866_mat_a @ A @ B ) @ C2 )
= ( ord_le3318621148231462513_mat_a @ A @ ( sup_sup_set_mat_a @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1014_Diff__partition,axiom,
! [A: set_mat_a,B: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( ( sup_sup_set_mat_a @ A @ ( minus_4757590266979429866_mat_a @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1015_trace__minus__linear,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_trace_a @ ( minus_minus_mat_a @ A @ B ) )
= ( minus_minus_a @ ( complex_trace_a @ A ) @ ( complex_trace_a @ B ) ) ) ) ) ).
% trace_minus_linear
thf(fact_1016_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ C ) @ B3 )
= ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B3 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1017_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_1018_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_1019_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_1020_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_1021_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_1022_insert__Diff__if,axiom,
! [X4: mat_a,B: set_mat_a,A: set_mat_a] :
( ( ( member_mat_a @ X4 @ B )
=> ( ( minus_4757590266979429866_mat_a @ ( insert_mat_a @ X4 @ A ) @ B )
= ( minus_4757590266979429866_mat_a @ A @ B ) ) )
& ( ~ ( member_mat_a @ X4 @ B )
=> ( ( minus_4757590266979429866_mat_a @ ( insert_mat_a @ X4 @ A ) @ B )
= ( insert_mat_a @ X4 @ ( minus_4757590266979429866_mat_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1023_insert__Diff__if,axiom,
! [X4: nat,B: set_nat,A: set_nat] :
( ( ( member_nat @ X4 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) )
& ( ~ ( member_nat @ X4 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A ) @ B )
= ( insert_nat @ X4 @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1024_insert__Diff1,axiom,
! [X4: mat_a,B: set_mat_a,A: set_mat_a] :
( ( member_mat_a @ X4 @ B )
=> ( ( minus_4757590266979429866_mat_a @ ( insert_mat_a @ X4 @ A ) @ B )
= ( minus_4757590266979429866_mat_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1025_insert__Diff1,axiom,
! [X4: nat,B: set_nat,A: set_nat] :
( ( member_nat @ X4 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_1026_Diff__insert0,axiom,
! [X4: mat_a,A: set_mat_a,B: set_mat_a] :
( ~ ( member_mat_a @ X4 @ A )
=> ( ( minus_4757590266979429866_mat_a @ A @ ( insert_mat_a @ X4 @ B ) )
= ( minus_4757590266979429866_mat_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1027_Diff__insert0,axiom,
! [X4: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X4 @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_1028_Diff__cancel,axiom,
! [A: set_mat_a] :
( ( minus_4757590266979429866_mat_a @ A @ A )
= bot_bot_set_mat_a ) ).
% Diff_cancel
thf(fact_1029_empty__Diff,axiom,
! [A: set_mat_a] :
( ( minus_4757590266979429866_mat_a @ bot_bot_set_mat_a @ A )
= bot_bot_set_mat_a ) ).
% empty_Diff
thf(fact_1030_Diff__empty,axiom,
! [A: set_mat_a] :
( ( minus_4757590266979429866_mat_a @ A @ bot_bot_set_mat_a )
= A ) ).
% Diff_empty
thf(fact_1031_double__diff,axiom,
! [A: set_mat_a,B: set_mat_a,C2: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ B )
=> ( ( ord_le3318621148231462513_mat_a @ B @ C2 )
=> ( ( minus_4757590266979429866_mat_a @ B @ ( minus_4757590266979429866_mat_a @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1032_Diff__subset,axiom,
! [A: set_mat_a,B: set_mat_a] : ( ord_le3318621148231462513_mat_a @ ( minus_4757590266979429866_mat_a @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1033_Diff__mono,axiom,
! [A: set_mat_a,C2: set_mat_a,D: set_mat_a,B: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ C2 )
=> ( ( ord_le3318621148231462513_mat_a @ D @ B )
=> ( ord_le3318621148231462513_mat_a @ ( minus_4757590266979429866_mat_a @ A @ B ) @ ( minus_4757590266979429866_mat_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_1034_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ~ ( member_nat @ X @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1035_left__diff__distrib_H,axiom,
! [B3: nat,C: nat,A3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B3 @ C ) @ A3 )
= ( minus_minus_nat @ ( times_times_nat @ B3 @ A3 ) @ ( times_times_nat @ C @ A3 ) ) ) ).
% left_diff_distrib'
thf(fact_1036_right__diff__distrib_H,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( times_times_nat @ A3 @ ( minus_minus_nat @ B3 @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A3 @ B3 ) @ ( times_times_nat @ A3 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1037_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A3 ) @ A3 )
= B3 ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1038_le__add__diff,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A3 ) ) ) ).
% le_add_diff
thf(fact_1039_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B3 @ A3 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1040_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1041_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A3 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1042_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A3 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1043_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A3 )
= ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A3 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1044_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B3 @ A3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A3 ) @ B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1045_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( plus_plus_nat @ A3 @ ( minus_minus_nat @ B3 @ A3 ) )
= B3 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1046_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( minus_minus_nat @ B3 @ A3 )
= C )
= ( B3
= ( plus_plus_nat @ C @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1047_add__le__imp__le__diff,axiom,
! [I3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
=> ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1048_le__add__diff__inverse,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A3 @ B3 ) )
= A3 ) ) ).
% le_add_diff_inverse
thf(fact_1049_le__add__diff__inverse2,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% le_add_diff_inverse2
thf(fact_1050_add__le__add__imp__diff__le,axiom,
! [I3: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1051_diff__diff__add,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A3 @ B3 ) @ C )
= ( minus_minus_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% diff_diff_add
thf(fact_1052_add__implies__diff,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( ( plus_plus_nat @ C @ B3 )
= A3 )
=> ( C
= ( minus_minus_nat @ A3 @ B3 ) ) ) ).
% add_implies_diff
thf(fact_1053_add__diff__cancel__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
= ( minus_minus_nat @ A3 @ B3 ) ) ).
% add_diff_cancel_left
thf(fact_1054_add__diff__cancel__left_H,axiom,
! [A3: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ A3 )
= B3 ) ).
% add_diff_cancel_left'
thf(fact_1055_add__diff__cancel__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
= ( minus_minus_nat @ A3 @ B3 ) ) ).
% add_diff_cancel_right
thf(fact_1056_add__diff__cancel__right_H,axiom,
! [A3: nat,B3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
= A3 ) ).
% add_diff_cancel_right'
thf(fact_1057_dim__update__mat_I1_J,axiom,
! [A: mat_a,Ij3: product_prod_nat_nat,A3: a] :
( ( dim_row_a @ ( update_mat_a @ A @ Ij3 @ A3 ) )
= ( dim_row_a @ A ) ) ).
% dim_update_mat(1)
thf(fact_1058_subset__insert__iff,axiom,
! [A: set_nat,X4: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X4 @ B ) )
= ( ( ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1059_subset__insert__iff,axiom,
! [A: set_mat_a,X4: mat_a,B: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ A @ ( insert_mat_a @ X4 @ B ) )
= ( ( ( member_mat_a @ X4 @ A )
=> ( ord_le3318621148231462513_mat_a @ ( minus_4757590266979429866_mat_a @ A @ ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) ) @ B ) )
& ( ~ ( member_mat_a @ X4 @ A )
=> ( ord_le3318621148231462513_mat_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1060_Diff__single__insert,axiom,
! [A: set_mat_a,X4: mat_a,B: set_mat_a] :
( ( ord_le3318621148231462513_mat_a @ ( minus_4757590266979429866_mat_a @ A @ ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) ) @ B )
=> ( ord_le3318621148231462513_mat_a @ A @ ( insert_mat_a @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1061_finite__empty__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ A )
=> ( ! [A5: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1062_finite__empty__induct,axiom,
! [A: set_mat_a,P: set_mat_a > $o] :
( ( finite_finite_mat_a @ A )
=> ( ( P @ A )
=> ( ! [A5: mat_a,A6: set_mat_a] :
( ( finite_finite_mat_a @ A6 )
=> ( ( member_mat_a @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_4757590266979429866_mat_a @ A6 @ ( insert_mat_a @ A5 @ bot_bot_set_mat_a ) ) ) ) ) )
=> ( P @ bot_bot_set_mat_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1063_infinite__coinduct,axiom,
! [X3: set_nat > $o,A: set_nat] :
( ( X3 @ A )
=> ( ! [A6: set_nat] :
( ( X3 @ A6 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A6 )
& ( ( X3 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_1064_infinite__coinduct,axiom,
! [X3: set_mat_a > $o,A: set_mat_a] :
( ( X3 @ A )
=> ( ! [A6: set_mat_a] :
( ( X3 @ A6 )
=> ? [X6: mat_a] :
( ( member_mat_a @ X6 @ A6 )
& ( ( X3 @ ( minus_4757590266979429866_mat_a @ A6 @ ( insert_mat_a @ X6 @ bot_bot_set_mat_a ) ) )
| ~ ( finite_finite_mat_a @ ( minus_4757590266979429866_mat_a @ A6 @ ( insert_mat_a @ X6 @ bot_bot_set_mat_a ) ) ) ) ) )
=> ~ ( finite_finite_mat_a @ A ) ) ) ).
% infinite_coinduct
thf(fact_1065_infinite__remove,axiom,
! [S: set_nat,A3: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1066_infinite__remove,axiom,
! [S: set_mat_a,A3: mat_a] :
( ~ ( finite_finite_mat_a @ S )
=> ~ ( finite_finite_mat_a @ ( minus_4757590266979429866_mat_a @ S @ ( insert_mat_a @ A3 @ bot_bot_set_mat_a ) ) ) ) ).
% infinite_remove
thf(fact_1067_card__le__sym__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1068_remove__induct,axiom,
! [P: set_nat > $o,B: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P @ B ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1069_remove__induct,axiom,
! [P: set_mat_a > $o,B: set_mat_a] :
( ( P @ bot_bot_set_mat_a )
=> ( ( ~ ( finite_finite_mat_a @ B )
=> ( P @ B ) )
=> ( ! [A6: set_mat_a] :
( ( finite_finite_mat_a @ A6 )
=> ( ( A6 != bot_bot_set_mat_a )
=> ( ( ord_le3318621148231462513_mat_a @ A6 @ B )
=> ( ! [X6: mat_a] :
( ( member_mat_a @ X6 @ A6 )
=> ( P @ ( minus_4757590266979429866_mat_a @ A6 @ ( insert_mat_a @ X6 @ bot_bot_set_mat_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1070_finite__remove__induct,axiom,
! [B: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1071_finite__remove__induct,axiom,
! [B: set_mat_a,P: set_mat_a > $o] :
( ( finite_finite_mat_a @ B )
=> ( ( P @ bot_bot_set_mat_a )
=> ( ! [A6: set_mat_a] :
( ( finite_finite_mat_a @ A6 )
=> ( ( A6 != bot_bot_set_mat_a )
=> ( ( ord_le3318621148231462513_mat_a @ A6 @ B )
=> ( ! [X6: mat_a] :
( ( member_mat_a @ X6 @ A6 )
=> ( P @ ( minus_4757590266979429866_mat_a @ A6 @ ( insert_mat_a @ X6 @ bot_bot_set_mat_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1072_card__Diff1__le,axiom,
! [A: set_mat_a,X4: mat_a] : ( ord_less_eq_nat @ ( finite_card_mat_a @ ( minus_4757590266979429866_mat_a @ A @ ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) ) ) @ ( finite_card_mat_a @ A ) ) ).
% card_Diff1_le
thf(fact_1073_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_1074_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1075_le__diff__iff_H,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ C )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A3 ) @ ( minus_minus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_1076_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1077_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1078_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1079_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1080_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1081_index__minus__mat_I2_J,axiom,
! [A: mat_a,B: mat_a] :
( ( dim_row_a @ ( minus_minus_mat_a @ A @ B ) )
= ( dim_row_a @ B ) ) ).
% index_minus_mat(2)
thf(fact_1082_minus__carrier__mat,axiom,
! [B: mat_a,Nr: nat,Nc: nat,A: mat_a] :
( ( member_mat_a @ B @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( minus_minus_mat_a @ A @ B ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ).
% minus_carrier_mat
thf(fact_1083_minus__carrier__mat_H,axiom,
! [A: mat_a,Nr: nat,Nc: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( member_mat_a @ ( minus_minus_mat_a @ A @ B ) @ ( carrier_mat_a @ Nr @ Nc ) ) ) ) ).
% minus_carrier_mat'
thf(fact_1084_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A2 )
@ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_1085_hermitian__minus,axiom,
! [A: mat_a,N: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( complex_hermitian_a @ A )
=> ( ( complex_hermitian_a @ B )
=> ( complex_hermitian_a @ ( minus_minus_mat_a @ A @ B ) ) ) ) ) ) ).
% hermitian_minus
thf(fact_1086_adjoint__minus,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ M ) )
=> ( ( schur_mat_adjoint_a @ ( minus_minus_mat_a @ A @ B ) )
= ( minus_minus_mat_a @ ( schur_mat_adjoint_a @ A ) @ ( schur_mat_adjoint_a @ B ) ) ) ) ) ).
% adjoint_minus
thf(fact_1087_smult__distrib__left__minus__mat,axiom,
! [A: mat_a,N: nat,B: mat_a,C: a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ N ) )
=> ( ( smult_mat_a @ C @ ( minus_minus_mat_a @ B @ A ) )
= ( minus_minus_mat_a @ ( smult_mat_a @ C @ B ) @ ( smult_mat_a @ C @ A ) ) ) ) ) ).
% smult_distrib_left_minus_mat
thf(fact_1088_minus__add__minus__mat,axiom,
! [U4: mat_a,Nr: nat,Nc: nat,V3: mat_a,W: mat_a] :
( ( member_mat_a @ U4 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ V3 @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( member_mat_a @ W @ ( carrier_mat_a @ Nr @ Nc ) )
=> ( ( minus_minus_mat_a @ U4 @ ( plus_plus_mat_a @ V3 @ W ) )
= ( minus_minus_mat_a @ ( minus_minus_mat_a @ U4 @ V3 ) @ W ) ) ) ) ) ).
% minus_add_minus_mat
thf(fact_1089_mat__minus__minus,axiom,
! [A: mat_a,N: nat,M: nat,B: mat_a,C2: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ N @ M ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ M ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ M ) )
=> ( ( minus_minus_mat_a @ A @ ( minus_minus_mat_a @ B @ C2 ) )
= ( plus_plus_mat_a @ ( minus_minus_mat_a @ A @ B ) @ C2 ) ) ) ) ) ).
% mat_minus_minus
thf(fact_1090_mult__minus__distrib__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,B: mat_a,Nc: nat,C2: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ A @ ( minus_minus_mat_a @ B @ C2 ) )
= ( minus_minus_mat_a @ ( times_times_mat_a @ A @ B ) @ ( times_times_mat_a @ A @ C2 ) ) ) ) ) ) ).
% mult_minus_distrib_mat
thf(fact_1091_minus__mult__distrib__mat,axiom,
! [A: mat_a,Nr: nat,N: nat,B: mat_a,C2: mat_a,Nc: nat] :
( ( member_mat_a @ A @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ B @ ( carrier_mat_a @ Nr @ N ) )
=> ( ( member_mat_a @ C2 @ ( carrier_mat_a @ N @ Nc ) )
=> ( ( times_times_mat_a @ ( minus_minus_mat_a @ A @ B ) @ C2 )
= ( minus_minus_mat_a @ ( times_times_mat_a @ A @ C2 ) @ ( times_times_mat_a @ B @ C2 ) ) ) ) ) ) ).
% minus_mult_distrib_mat
thf(fact_1092_le__diff__conv,axiom,
! [J: nat,K: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I3 @ K ) ) ) ).
% le_diff_conv
thf(fact_1093_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1094_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1095_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1096_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1097_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1098_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1099_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ( minus_minus_nat @ J @ I3 )
= K )
= ( J
= ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1100_nat__diff__add__eq2,axiom,
! [I3: nat,J: nat,U4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U4 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U4 ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1101_nat__diff__add__eq1,axiom,
! [J: nat,I3: nat,U4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U4 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U4 ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1102_nat__le__add__iff2,axiom,
! [I3: nat,J: nat,U4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U4 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U4 ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1103_nat__le__add__iff1,axiom,
! [J: nat,I3: nat,U4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I3 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U4 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U4 ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1104_nat__eq__add__iff2,axiom,
! [I3: nat,J: nat,U4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U4 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U4 ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1105_nat__eq__add__iff1,axiom,
! [J: nat,I3: nat,U4: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I3 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U4 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U4 ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U4 ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1106_card__Diff__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1107_card__Diff__subset,axiom,
! [B: set_mat_a,A: set_mat_a] :
( ( finite_finite_mat_a @ B )
=> ( ( ord_le3318621148231462513_mat_a @ B @ A )
=> ( ( finite_card_mat_a @ ( minus_4757590266979429866_mat_a @ A @ B ) )
= ( minus_minus_nat @ ( finite_card_mat_a @ A ) @ ( finite_card_mat_a @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1108_diff__card__le__card__Diff,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1109_card__Diff__insert,axiom,
! [A3: mat_a,A: set_mat_a,B: set_mat_a] :
( ( member_mat_a @ A3 @ A )
=> ( ~ ( member_mat_a @ A3 @ B )
=> ( ( finite_card_mat_a @ ( minus_4757590266979429866_mat_a @ A @ ( insert_mat_a @ A3 @ B ) ) )
= ( minus_minus_nat @ ( finite_card_mat_a @ ( minus_4757590266979429866_mat_a @ A @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1110_card__Diff__insert,axiom,
! [A3: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A3 @ A )
=> ( ~ ( member_nat @ A3 @ B )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A3 @ B ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1111_card__Diff__singleton__if,axiom,
! [X4: nat,A: set_nat] :
( ( ( member_nat @ X4 @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X4 @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1112_card__Diff__singleton__if,axiom,
! [X4: mat_a,A: set_mat_a] :
( ( ( member_mat_a @ X4 @ A )
=> ( ( finite_card_mat_a @ ( minus_4757590266979429866_mat_a @ A @ ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) ) )
= ( minus_minus_nat @ ( finite_card_mat_a @ A ) @ one_one_nat ) ) )
& ( ~ ( member_mat_a @ X4 @ A )
=> ( ( finite_card_mat_a @ ( minus_4757590266979429866_mat_a @ A @ ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) ) )
= ( finite_card_mat_a @ A ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1113_card__Diff__singleton,axiom,
! [X4: nat,A: set_nat] :
( ( member_nat @ X4 @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1114_card__Diff__singleton,axiom,
! [X4: mat_a,A: set_mat_a] :
( ( member_mat_a @ X4 @ A )
=> ( ( finite_card_mat_a @ ( minus_4757590266979429866_mat_a @ A @ ( insert_mat_a @ X4 @ bot_bot_set_mat_a ) ) )
= ( minus_minus_nat @ ( finite_card_mat_a @ A ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1115_remove__def,axiom,
( remove_mat_a
= ( ^ [X: mat_a,A2: set_mat_a] : ( minus_4757590266979429866_mat_a @ A2 @ ( insert_mat_a @ X @ bot_bot_set_mat_a ) ) ) ) ).
% remove_def
thf(fact_1116_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1117_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1118_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ A3 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1119_diff__zero,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% diff_zero
thf(fact_1120_zero__diff,axiom,
! [A3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1121_diff__add__zero,axiom,
! [A3: nat,B3: nat] :
( ( minus_minus_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1122_set__zero__plus2,axiom,
! [A: set_nat,B: set_nat] :
( ( member_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_set_nat @ B @ ( plus_plus_set_nat @ A @ B ) ) ) ).
% set_zero_plus2
thf(fact_1123_unitary__zero,axiom,
! [A: mat_a] :
( ( member_mat_a @ A @ ( carrier_mat_a @ zero_zero_nat @ zero_zero_nat ) )
=> ( complex_unitary_a @ A ) ) ).
% unitary_zero
thf(fact_1124_sum_Ofinite__Collect__op,axiom,
! [I2: set_nat,X4: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I2 )
& ( ( X4 @ I )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I2 )
& ( ( Y @ I )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I2 )
& ( ( plus_plus_nat @ ( X4 @ I ) @ ( Y @ I ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_1125_add__scale__eq__noteq,axiom,
! [R2: nat,A3: nat,B3: nat,C: nat,D2: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A3 = B3 )
& ( C != D2 ) )
=> ( ( plus_plus_nat @ A3 @ ( times_times_nat @ R2 @ C ) )
!= ( plus_plus_nat @ B3 @ ( times_times_nat @ R2 @ D2 ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1126_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1127_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1128_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1129_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1130_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_1131_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1132_set__zero,axiom,
( zero_zero_set_nat
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% set_zero
thf(fact_1133_member__remove,axiom,
! [X4: nat,Y: nat,A: set_nat] :
( ( member_nat @ X4 @ ( remove_nat @ Y @ A ) )
= ( ( member_nat @ X4 @ A )
& ( X4 != Y ) ) ) ).
% member_remove
thf(fact_1134_zero__reorient,axiom,
! [X4: nat] :
( ( zero_zero_nat = X4 )
= ( X4 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_1135_rel__simps_I46_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% rel_simps(46)
thf(fact_1136_zero__order_I2_J,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% zero_order(2)
thf(fact_1137_zero__order_I1_J,axiom,
! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).
% zero_order(1)
thf(fact_1138_card_Oempty,axiom,
( ( finite_card_mat_a @ bot_bot_set_mat_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_1139_class__semiring_Or__null,axiom,
! [X4: nat] :
( ( member_nat @ X4 @ top_top_set_nat )
=> ( ( times_times_nat @ X4 @ zero_zero_nat )
= zero_zero_nat ) ) ).
% class_semiring.r_null
thf(fact_1140_class__semiring_Ol__null,axiom,
! [X4: nat] :
( ( member_nat @ X4 @ top_top_set_nat )
=> ( ( times_times_nat @ zero_zero_nat @ X4 )
= zero_zero_nat ) ) ).
% class_semiring.l_null
thf(fact_1141_rel__simps_I45_J,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% rel_simps(45)
thf(fact_1142_rel__simps_I44_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% rel_simps(44)
thf(fact_1143_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1144_add__sign__intros_I8_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% add_sign_intros(8)
thf(fact_1145_add__sign__intros_I4_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).
% add_sign_intros(4)
thf(fact_1146_add__decreasing,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B3 ) ) ) ).
% add_decreasing
thf(fact_1147_add__increasing,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing
thf(fact_1148_add__decreasing2,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B3 ) ) ) ).
% add_decreasing2
thf(fact_1149_add__increasing2,axiom,
! [C: nat,B3: nat,A3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).
% add_increasing2
thf(fact_1150_add__nonneg__eq__0__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1151_add__nonpos__eq__0__iff,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1152_add__le__same__cancel1,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1153_add__le__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
= ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1154_le__add__same__cancel1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% le_add_same_cancel1
thf(fact_1155_le__add__same__cancel2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% le_add_same_cancel2
% Helper facts (11)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X4: a,Y: a] :
( ( if_a @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X4: a,Y: a] :
( ( if_a @ $true @ X4 @ Y )
= X4 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y: nat] :
( ( if_nat @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y: nat] :
( ( if_nat @ $true @ X4 @ Y )
= X4 ) ).
thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
! [P: nat > $o] :
( ( P @ ( fChoice_nat @ P ) )
= ( ? [X5: nat] : ( P @ X5 ) ) ) ).
thf(help_If_2_1_If_001t__Matrix__Omat_Itf__a_J_T,axiom,
! [X4: mat_a,Y: mat_a] :
( ( if_mat_a @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Matrix__Omat_Itf__a_J_T,axiom,
! [X4: mat_a,Y: mat_a] :
( ( if_mat_a @ $true @ X4 @ Y )
= X4 ) ).
thf(help_fChoice_1_1_fChoice_001t__Matrix__Omat_Itf__a_J_T,axiom,
! [P: mat_a > $o] :
( ( P @ ( fChoice_mat_a @ P ) )
= ( ? [X5: mat_a] : ( P @ X5 ) ) ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J_T,axiom,
! [X4: set_mat_a,Y: set_mat_a] :
( ( if_set_mat_a @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Matrix__Omat_Itf__a_J_J_T,axiom,
! [X4: set_mat_a,Y: set_mat_a] :
( ( if_set_mat_a @ $true @ X4 @ Y )
= X4 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( times_times_mat_a @ u @ ( schur_mat_adjoint_a @ u ) )
= ( one_mat_a @ ( dim_row_a @ b ) ) ) ).
%------------------------------------------------------------------------------